Research ArticleSimilarity Measures between Temporal ComplexIntuitionistic Fuzzy Sets and Application in PatternRecognition and Medical Diagnosis

MohammedM Khalaf 1 Sayer Obaid Alharbi2 andWathek Chammam 23

1Higher Institute of Engineering and Technology King Marriott PO Box 3135 Egypt2Department of Mathematics College of Science Al-Zulfi Al-Majmaah University PO Box 66 Al-Majmaah 11952 Saudi Arabia3Department of Mathematics Faculty of Sciences of Gabes Gabes University Gabes Tunisia

Correspondence should be addressed to Mohammed M Khalaf khalfmohammed2003yahoocom

Received 27 March 2019 Accepted 2 June 2019 Published 1 July 2019

Academic Editor Francisco R Villatoro

Copyright copy 2019 Mohammed M Khalaf et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This work addresses the issue of similarity measures between two temporal complex Atanassovrsquos intuitionistic fuzzy sets manymeasures of similarity between complex Atanassovrsquos intuitionistic fuzzy sets What was proposed before did not consider theabstention group influence which may lead to counterintuitive results in some cases A new structure of temporal complexAtanassovrsquos intuitionistic fuzzy sets is obtained This set is formally generalized from a conventional Atanassovrsquos intuitionisticcomplex fuzzy sets Here we analyze the limitations of the existing similarity measuresThen a new similarity measure of temporalcomplex Atanassovrsquos intuitionistic fuzzy sets is proposed and several numeric examples are given to demonstrate the validity of theproposed measure Finally the proposed similarity measure is applied to pattern recognition and medical diagnosis

1 Introduction

Fuzzy set theory was conferred by Zadeh [1] to solvedifficulties in dealing with uncertainties Since then thetheories of fuzzy sets and fuzzy logic have been examined bymany researchers to solve many real life problems involvingambiguous and uncertain environment By adding a newcomponent the idea of the concept of Atanassovrsquos intuition-istic fuzzy set (AIFS) was introduced [2] Applications ofthese sets have been broadly studied in other aspects suchas image processing [3] multicriteria decision making [4]pattern recognition [5] etc Buckley [6] and Nguyen et al[7] combined complex numbers with fuzzy sets On the otherhand the innovative complex fuzzy set is introduced Thecomplex fuzzy set is characterized by amembership function120583A(119909) whose range is not limited to [0 1] but extendedto the unit circle in the complex plane Hence 120583A(119909) is acomplex-Valued function that assigns a grade of membershipof the form 119903A(119909)119890119894120596A(119909) 119894 = radicminus1 to any element in theuniverse of discourseThe value of 120583119878(119909) is defined by the two

variables 119903A(119909) and 120596A(119909) both real-valued with 120583A(119909) isin[0 1] Complex fuzzy set theorymodifies the original conceptof fuzzy membership by asserting that at least in someinstances it is necessary to add a second dimension to theexpression of membership However this added dimensiondoes not alter the basic concept of fuzziness Membershipin a complex fuzzy set remains ldquoas fuzzyrdquo as membershipin a traditional fuzzy set The fuzziness of membershipie the representation of membership as a value in therange [0 1] is retained in complex fuzzy sets through theamplitude of the grade of membership 119903A(119909) The novelty ofcomplex fuzzy sets is manifested in the additional dimensionof membership the phase of the grade of membership120596A(119909) The properties of membership phase are discussedat length in this section Ramot et al [8 9] extended therange of membership to ldquounit circle in the complex planerdquounlike others who limited the range to [0 1] Omar [10]studied similarity measures between temporal intuitionisticfuzzy sets As the complex fuzzy membership grade is two-dimensional (amplitude and phase) a complex fuzzy set can

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 3246439 16 pageshttpsdoiorg10115520193246439

2 Discrete Dynamics in Nature and Society

15

1

05

0

minus05

minus1

minus15

minus1

0

10

2 4 6 8 10

D Im

Re

Figure 1 Complex fuzzy set defined in [11]

be visually represented by a three-dimensional graph wherethe universe of discourse is the third axis Figure 1 shows thecomplex fuzzy set

We divide the paper into four main sections In thefirst section preliminaries and basic definitions we providesome details about the complex fuzzy sets In the secondsection detail is given about the complex version of temporalcomplex intuitionistic fuzzy set which is an extension ofcomplex intuitionistic fuzzy set by adding the times andstudied the correlation coefficient between two temporalcomplex intuitionistic fuzzy set In the third section detailsis given about similarity measures between other extensionsof temporal complex intuitionistic fuzzy set and extend themethod proposed by Chaira [12] for intuitionistic fuzzy setbased on the Sugeno [13] and Omar [10] intuitionistic fuzzygenerator In the fourth section we give application in patternrecognition medical diagnosis and topology

2 Preliminaries and Basic Definitions

Definition 1 (see [8]) A complex fuzzy set (CFS) A definedon a universe 119883 is an object of the form A defined on auniverse of discourse119883 which is an object of the form

A = 119909 120583A (119909) 119890119894120596A(119909) 119909 isin 119883 (1)

where 119894 = radicminus1 120583A(119909) isin [0 1] and 0 le 120596A(119909) le 2120587Definition 2 (see [2]) A complex intuitionistic fuzzy set(CIFS) A defined on a universe of discourse 119883 is an objectof the form

A = 119909 120583A (119909) 119890119894120572A(119909) ]A (119909) 119890119894120573A(119909) 119909 isin 119883 (2)

where 119894 = radicminus1 120583A(119909) ]A(119909) isin [0 1] 120572A(119909) 120573A(119909) isin[0 2120587] and 0 le 120583119860(119909)+V119860(119909) le 1Definition 3 (see [14]) LetA andB be twoCIFSs in119883 where

A = 119909 120583A (119909) 119890119894120572A(119909) ]A (119909) 119890119894120573A(119909) 119909 isin 119883B = 119909 120583B (119909) 119890119894120572B(119909) ]B (119909) 119890119894120573B(119909) 119909 isin 119883 (3)

ThenA cupB is given as

A cupB

= 119909 120583AcupB (119909) 119890119894120572AcupB(119909) ]AcapB (119909) 119890119894120573AcapB(119909) 119909 isin 119883 (4)

where

120583AcupB (119909) 119890119894120572AcupB(119909)= 119909 [120583A (119909) or ]B (119909)] 119890119894120572A(119909)or120572B(119909) ]AcapB (119909) 119890119894120572AcapB(119909)= 119909 []A (119909) and ]B (119909)] 119890119894120573A(119909)and120573B(119909)

(5)

Definition 4 (see [15]) Let A and B be two CIF-sets in 119883where

A = 119909 120583A (119909) 119890119894120572A(119909) ]A (119909) 119890119894120573A(119909) 119909 isin 119883B = 119909 120583B (119909) 119890119894120572B(119909) ]B (119909) 119890119894120573B(119909) 119909 isin 119883 (6)

Then for all 119909 isin 119883(1)A sub B if and only if 120583A(119909) lt 120583B(119909) ]A(119909) gt ]B(119909)

For amplitude terms and 120572A(119909) lt 120572B(119909) 120573A(119909) gt 120573B(119909) forphase terms

(2)A = B if and only if 120583A(119909) = 120583B(119909) ]A(119909) = ]B(119909)For amplitude terms and 120572A(119909) = 120572B(119909) 120573A(119909) = 120573B(119909) forphase terms

Definition 5 (see [16]) LetA andB be twoCIFSs in119883 whereA = 119909 120583A (119909) 119890119894120572A(119909) ]A (119909) 119890119894120573A(119909) 119909 isin 119883B = 119909 120583B (119909) 119890119894120572B(119909) ]B (119909) 119890119894120573B(119909) 119909 isin 119883 (7)

Then for all 119909 isin 119883(1)A capB = 119909 120583AcapB(119909)119890119894120572AcapB(119909) ]AcupB(119909)119890119894120573AcupB(119909) 119909 isin119883

Discrete Dynamics in Nature and Society 3

where

120583AcapB (119909) 119890119894120572AcapB(119909)= 119909 [120583A (119909) and ]B (119909)] 119890119894120572A(119909)and120572B(119909) ]AcupB (119909) 119890119894120572AcupB(119909)= 119909 []A (119909) or ]B (119909)] 119890119894120573A(119909)or120573B(119909)

(8)

(2) The complex fuzzy complement of A denoted by Ais specified by a function

A = 119909 (1 minus 120583A (119909)) 119890119894(2120587minus120572A(119909)) (1 minus ]B (119909))sdot 119890119894(2120587minus120573A(119909)) 119909 isin 119883 (9)

(3) 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883 and 0 = 119909 (0 119890119894(2120587)) 119909 isin119883Example 6 Consider 119883 = 119886 119887 119888 119889 Let AB be a CIF-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(10)

Then

A capB = (0211989011989409120587 0411989011989405120587119886 011198901198942120587 0311989011989409120587119887 0111989011989401120587 0211989011989415120587119888 0511989011989402120587 0111989011989407120587119889 )

A = (0811989011989407120587 0611989011989415120587119886 0 1011989011989415120587119887 0311989011989417120587 0811989011989405120587119888 0211989011989409120587 0911989011989413120587119889 )

A cupB = (0211989011989413120587 0311989011989404120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989405120587119888 0811989011989411120587 0111989011989405120587119889 )

(11)

AndA subB andA =B

3 Temporal Complex Intuitionistic Fuzzy Set

Definition 7 Let119883 be a universe119879 be a nonempty set of timemoments and A sube 119883 A temporal complex intuitionisticfuzzy set (TCIFS) A defined on a universe of discourse 119883 isan object of the form

A (119879) = ((119909 119905) 120583A (119909 119905) 119890119894120572A(119909119905) ]A (119909 119905) 119890119894120573A(119909119905)) |(119909 119905) isin 119883 times 119879 (12)

where 120583A A times 119879 997888rarr [0 1] and ]A A times 119879 997888rarr[0 1] such that 0 le 120583A(119909 119905)119890119894120572A(119909119905)+]A(119909 119905)119890119894120573A(119909119905) le 1120583119860(119909 119905) and V119860(119909 119905) being the degrees of membership andnonmembership respectively of the element 119909 isin 119883 at themoment 119905 isin 119879 And 120572A(119909 119905) 120573A(119909 119905) isin [0 2120587] at themoment 119905 isin 119879 where 119894 = radicminus1

The hesitation degree of a TCIFS A is defined by120587A 1 minus 120583A(119909 119905)minusVA(119909 119905) such that 0 le 120587A(119909 119905) le 1 for each(119909 119905) isin A times 119879 For brevity we will write A instead of A(119879)when this does not cause confusions

Example 8 Suppose that119883 is a universal with respect to thetime set 1198791 = 119905 | 119905 = 6119896 119896 = 0 1 100 1198792 = 119905 | 119905 =2119896 119896 = 0 1 100 and 120572A(119909 119905) = 120573A(119909 119905) = 1198901198942120587 ThenTCIFSsAB are defined by

A (1198791) (119909) = (12 14) 119905 = 4119896 119896 = 1 100(19 35) 119905 = 4119896 + 2 119896 = 1 100

B (1198792) (119909) =

(23 16) 119905 = 3119896 119896 = 1 100(38 15) 119905 = 3119896 + 1 119896 = 1 100( 711 29) 119905 = 3119896 + 2 119896 = 1 100

(13)

Example 9 Suppose 119883 = 1199091 1199092 1199093 with respect to thetime set 119879 = 1199051 1199052 1199053 Then the details of a TCIFS A areexplained in Tables 1 2 and 3

Definition 10 LetA(1198791) andB(1198792) be two TCIFSs Then

A (1198791) capB (1198792) = (119909min (120583A (119909 119905) 119890119894120572A(119909119905) 120583B (119909 119905) 119890119894120572B(119909119905)) max (]A (119909 119905) 119890119894120573A(119909119905) ]B (119909 119905) 119890119894120573B(119909119905)) (119909 119905) isin 119883times (1198791 cup 1198792

A (1198791) cupB (1198792) = (119909max (120583A (119909 119905) 119890119894120572A(119909119905) 120583B (119909 119905) 119890119894120572B(119909119905)) min (]A (119909 119905) 119890119894120573A(119909119905) ]B (119909 119905) 119890119894120573B(119909119905)) (119909 119905) isin 119883times (1198791 cup 1198792

4 Discrete Dynamics in Nature and Society

Table 1 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 2 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 3 TCIFS A1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)A (1198791) = (119909(]A (119909 119905) 119890119894(2120587minus120573A(119909)) 120583A (119909 119905) 119890119894(2120587minus120572A(119909)) ) (119909 119905)isin 119883 times 1198791

(14)

where

120583A (119909 119905) = 120583A (119909 119905) 119905 isin 11987910 119905 isin 1198792 minus 1198791

]A (119909 119905) = ]A (119909 119905) 119905 isin 11987911 119905 isin 1198792 minus 1198791

120583B (119909 119905) = 120583B (119909 119905) 119905 isin 11987910 119905 isin 1198792 minus 1198791

]B (119909 119905) = ]B (119909 119905) 119905 isin 11987911 119905 isin 1198792 minus 1198791

(15)

Definition 11 Wedefine the following two operators119891 119886119899119889 119892over a TCIFSA

119891 (A (119879)) = (119909 (119909 119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) 119898119894119899119905 isin 119879sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

ℎ (A (119879)) = (119909 (119909 119898119886119909119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905) 119898119894119899119905 isin 119883sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

119892 (A (119879)) = (119909 (119909 119898119894119899119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) 119898119886119909119905 isin 119879sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

119897 (A (119879)) = (119909 (119909 119898119894119899119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905) 119898119886119909119905 isin 119883sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

(16)

Theorem 12 119891(A(119879)) and 119892(A(119879)) are TCIFSsProof Suppose that119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) = 120583A (119909 1199051) 119890119894120572A(1199091199051)

for some 1199051 isin 119879 (17)

and119898119894119899119905 isin 119879]A (119909 119905) 119890119894120572A(119909119905) = ]A (119909 1199052) 119890119894120572A(1199091199052)for some 1199052 isin 119879 (18)

Therefore

]A (119909 1199052) 119890119894120573A(1199091199052) le V119860 (119909 1199051) 119890119894120573A(1199091199051) And119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) + 119898119894119899119905 isin 119879V119860 (119909 119905) 119890119894120572A(119909119905)= 120583A (119909 1199051) 119890119894120572A(1199091199051) + V119860 (119909 1199052) 119890119894120572A(1199091199052)le 120583A (119909 1199051) 119890119894120572A(1199091199051) + V119860 (119909 1199051) 119890119894120573A(1199091199051) le 1(19)

Then 119891(A(119879)) is TCIFSs Also by the same fashion 119892(A(119879))are TCIFSs

Theorem 13 For every TCIFSA(119879)(1) 119891(119891(A(119879))) = 119891(A(119879))(2) 119892(119892(A(119879))) = 119892(A(119879))(3) 119891(119892(A(119879))) = 119892(A(119879))(4) 119892(119891(A(119879))) = 119891(A(119879))

Proof The proof is obvious

Theorem 14 For every TCIFSA(119879)(1) ℎ(119891(A(119879))) = 119891(ℎ(A(119879)))(2) 119897(119892(A(119879))) = 119892(119897(A(119879)))

Discrete Dynamics in Nature and Society 5

Proof (1)

ℎ (119891 (A (119879))) = (119909 (119909 119898119886119909119905 isin 119883 119898119886119909119905 isin 119879120583A (119909 119905)sdot 119890119894120572A(119909119905) 119898119894119899119905 isin 119883 119898119894119899119905 isin 119879]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905)isin 119883 times 119879 = (119909 (119909 119898119886119909119905 isin 119879 119898119886119909119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905)119898119894119899119905 isin 119879 119898119894119899119905 isin 119883]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879= 119891 (ℎ (A (119879)))

(20)

(2) By the same fashion one has the following

Definition 15 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (21)

where119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))

(22)

is the correlation of two TCIFSsA andB and

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

119879 (B) = 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905))+ ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(23)

are the information temporal complex intuitionistic energiesofA andB respectively

Example 16 Suppose that 119883 = 1199091 1199092 1199093 with respect tothe time set 119879 = 1199051 1199052 1199053 The details of a TCIFS A(119879)are explained in Table 4 Table 5 explained TCIFS B(119879) andTable 6 explained the correlation coefficient 119896(AB) betweenTCIFSA(119879) and TCIFSB(119879)Proposition 17 LetA(1198791) andB(1198792) be two TCIFS Then

(1) 119879(A) = 119879(A)(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A)(3) 119862(AB) = 119862(BA)

Table 4 TCIFSA(119879)1199051 1199052 11990531199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894)1199094 (06 01) (01119894 09119894) (06119894 04119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894)

Table 5 TCIFSB(119879)1199051 1199052 11990531199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894)1199092 (04 03) (07119894 02119894) (09119894 02119894)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (06 01) (01119894 09119894) (06119894 04119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894)

Table 6 Then 119896(AB)1199051 1199052 11990531199091 119896 (AB) = 0509 119896 (AB) = 1923 119896 (AB) = 29731199092 119896(AB) =3041 119896 (AB) = 6592 119896(AB) =66481199093 119896(AB) =5000 119896(AB) =2544 119896 (AB) = 13341199094 119896(AB) =4301 119896(AB) =6403 119896(AB) =68021199095 119896 (AB) = 1360 119896(AB) =5508 119896(AB) =42041199096 119896 (AB) = 1360 119896(AB) =5581 119896 (AB) = 4204

Proof Let 119905 isin 1198791 From Definition 10 120583A(119909 119905) = 120583A(119909 119905)]A(119909 119905) = ]A(119909 119905) 120583B(119909 119905) = 120583B(119909 119905) ]B(119909 119905) = ]B(119909 119905)and then

(1)

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 119899sum119894=1

119898sum119895=1

(]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))+ 120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))) = 119879 (A)

(24)

(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587119862 (AA)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905))

6 Discrete Dynamics in Nature and Society

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 1198902119894120572A(119909119905)+ ]A2 (119909119894 119905119894) 1198902119894120573A(119909119905))

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))) = 119879 (A)

(25)

(3)

119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))= 119899sum119894=1

119898sum119895=1

(]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)+ 120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)) = 119862 (BA)

(26)

Theorem 18 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) if 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119896(AB) = 1(2) 119896(AB) = 119896(BA)(3) 0 lt 119896(AB) lt 1

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (27)

IfA =B then radic119879(A)119879(B) = radic119879(A)2 = 119879(A)Then fromProposition 17 (2) one has the following

If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A) then119896 (AB) = 119879 (A)119879 (A) = 1 (28)

(2) From Proposition 17 (3) 119862(AB) = 119862(BA)Then

119896 (AB) = 119862 (BA)radic119879 (B) 119879 (A) = 119896 (BA) (29)

(3) We will prove that 119896(AB) lt 1 such that it is evident0 lt 119896(AB) so suppose that119899sum119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205721119899sum119894=1

119898sum119895=1

120583B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205722119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205723

119899sum119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205724

(30)

Then119879 (A) 119879 (B)= ( 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894)sdot 119890119894(2120587minus120573A(119909)) + ]A

2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))))times ( 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905)) + ]B

2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))) )= (( 119899sum

119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))) + ( 119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894)

sdot 119890119894(2120587minus120573A(119909))))times (( 119899sum

119894=1

119898sum119895=1

120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905))) + ( 119899sum

119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))))

(31)

Thenradic119879 (A) 119879 (B) = [(1205721 + 1205723) times (1205722 + 1205724)]12= (1205721 + 1205723)12 times (1205722 + 1205724)12119862 (AB)= (( 119899sum

119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905))) + ( 119899sum119894=1

119898sum119895=1

]A (119909119894 119905119894)sdot 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)))

(32)

Then

119862 (AB) = (1205721 times 1205722)12 + (1205723 times 1205724)12 (33)

Then

Discrete Dynamics in Nature and Society 7

1198962 (AB)le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724)

(34)

But

1198962 (AB) minus 1 le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724) minus 1= 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724 minus (1205721 + 1205723) times (1205722 + 1205724)(1205721 + 1205723) times (1205722 + 1205724) = [(1205721 times 1205724)12 minus (1205722 times 1205723)12]2(1205721 + 1205723) times (1205722 + 1205724)le 0

(35)

Hence 1198962(AB) minus 1 le 0 and then 119896 (AB) le 1Definition 19 Let 119878 119879119868119862119865119878119904 (119883 119879) times 119879119868119862119865119878119904 (119883 119879) 997888rarr[0 1] be a function and letA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSsin the universal 119883 = 1199091 1199092 1199093 119909119899 with respect to thetime set 119879 = 1199051 1199052 1199053 119905119898 Then 119878(AB) is said to be thesimilarity degree between 119879119868119862119865119878119904 A and B satisfying thefollowing statements

(1) 0 le 119878(AB) le 1(2) 119878(AB) = 1 ifA =B

(3) 119878(AB) = 119878(BA)(4) If A sube B sube 119862 Then 119878(A 119862) le 119878(AB) 119878(A 119862) le119878(B 119862)

Now we can have the following degrees of the similaritybetweenAB and satisfy the conditions from (1) to (4) Let119878A (119894 119895) = 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)119878B (119894 119895) = 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)120595A (119894 119895)= 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) + 1 minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)2

120595B (119894 119895)= ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) + 1 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)2

(36)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 Then

1198781 (AB) = 1 minus 12119898119899 119899sum119894=1

119898sum119895=1

1003816100381610038161003816119878A (119894 119895) minus 119878B (119894 119895)1003816100381610038161003816 1198782 (AB) = 1 minus 12119898119899 119899sum

119894=1

119898sum119895=1

10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816 + 10038161003816100381610038161003816]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816 1198780 (AB)= 1 minus 1radic2119898119899 119899sum119894=1

119898sum119895=1

radic(120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895))2 + (]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895))21198783 (AB) = 1 minus 1

119910radic2119898119899radic 119899sum119894=1119898sum119895=1

1003816100381610038161003816120595A (119894 119895) minus 120595B (119894 119895)1003816100381610038161003816119910 1 le 119910 lt +infin1198784 (AB)= sum119899119894=1sum119898119895=1min(|120583A(119909119894 119905119895)119890

119894120572A(119909119894 119905119895)||120583B(119909119894 119905119895)119890119894120572B(119909119894 119905119895)|)+min(|1minus]A(119909119894 119905119894)119890

119894120573A(119909119894 119905119895)||1minus]A(119909119894 119905119894)119890119894120573A(119909119894 119905119895)|)sum119899119894=1sum119898119895=1max (10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895)10038161003816100381610038161003816 10038161003816100381610038161003816120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816) +max (100381610038161003816100381610038161 minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)10038161003816100381610038161003816 100381610038161003816100381610038161 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816)

(37)

From a comparison between similarity measures 1198780(AB)1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) we give the follow-ing example

Example 20 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056The details of a TCIFS A(119879)

8 Discrete Dynamics in Nature and Society

Table 7 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 8 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

are explained in Table 7 Table 8 explained TCIFS B(119879)and Table 9 explained a comparison between similaritymeasures 1198780(AB) 1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) TCIFS A(119879) and TCIFS B(119879)4 Similarity Measures between OtherExtensions of Temporal ComplexIntuitionistic Fuzzy Set

The following definition extend the method proposed byChaira [12] for intuitionistic fuzzy set based on the Sugeno[13] and Omar [10] intuitionistic fuzzy generator

Definition 21 If 120583A(119909 119905)119890119894120572A(119909119905) is the degrees of membershipfunction of the element 119909 isin 119883 at the moment 119905 isin 119879 thennonmembership function ]A(119909)119890119894120573A(119909) = 119866(120583A(119909 119905)119890119894120572A(119909119905))where

119866(120583A (119909 119905) 119890119894120572A(119909119905)) = 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) 120572 gt 0 (38)

And 119866(1) = 0 119866(0) = 1 and by help of the Sugeno [6]intuitionistic fuzzy generator TCIFSA is given by

A120572 (119879) = ((119909 119905) 120583A (119909 119905)sdot 119890119894120572A(119909119905) 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905)) | (119909 119905) isin 119883 times 119879

(39)

The hesitation degree of a TCIFSA is

120587A120572 (119909 119905) = 1 minus 120583A (119909 119905) minus 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) (40)

Example 22 Suppose that A(119879) is TCIFS defined on 119883 =1199091 1199092 1199093 with respect to the time set 119879 = 1199051 1199052 1199053 Thedetails of a TCIFS A(119879) are explained in Tables 10 11 and12 Table 13 explained TCIFS A1 when 120572 = 1 and Table 14explained the hesitation degree of a TCIFSA

If 120572 = 1 then one has the following (see Tables 13 and 14)Definition 23 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the timeset 119879 = 1199051 1199052 1199053 119905119898 Then a cosine similarity measurebetweenA(119879) 119886119899119889 B(119879) is proposed as follows

119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(41)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 3

where

120583AcapB (119909) 119890119894120572AcapB(119909)= 119909 [120583A (119909) and ]B (119909)] 119890119894120572A(119909)and120572B(119909) ]AcupB (119909) 119890119894120572AcupB(119909)= 119909 []A (119909) or ]B (119909)] 119890119894120573A(119909)or120573B(119909)

(8)

(2) The complex fuzzy complement of A denoted by Ais specified by a function

A = 119909 (1 minus 120583A (119909)) 119890119894(2120587minus120572A(119909)) (1 minus ]B (119909))sdot 119890119894(2120587minus120573A(119909)) 119909 isin 119883 (9)

(3) 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883 and 0 = 119909 (0 119890119894(2120587)) 119909 isin119883Example 6 Consider 119883 = 119886 119887 119888 119889 Let AB be a CIF-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(10)

Then

A capB = (0211989011989409120587 0411989011989405120587119886 011198901198942120587 0311989011989409120587119887 0111989011989401120587 0211989011989415120587119888 0511989011989402120587 0111989011989407120587119889 )

A = (0811989011989407120587 0611989011989415120587119886 0 1011989011989415120587119887 0311989011989417120587 0811989011989405120587119888 0211989011989409120587 0911989011989413120587119889 )

A cupB = (0211989011989413120587 0311989011989404120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989405120587119888 0811989011989411120587 0111989011989405120587119889 )

(11)

AndA subB andA =B

3 Temporal Complex Intuitionistic Fuzzy Set

Definition 7 Let119883 be a universe119879 be a nonempty set of timemoments and A sube 119883 A temporal complex intuitionisticfuzzy set (TCIFS) A defined on a universe of discourse 119883 isan object of the form

A (119879) = ((119909 119905) 120583A (119909 119905) 119890119894120572A(119909119905) ]A (119909 119905) 119890119894120573A(119909119905)) |(119909 119905) isin 119883 times 119879 (12)

where 120583A A times 119879 997888rarr [0 1] and ]A A times 119879 997888rarr[0 1] such that 0 le 120583A(119909 119905)119890119894120572A(119909119905)+]A(119909 119905)119890119894120573A(119909119905) le 1120583119860(119909 119905) and V119860(119909 119905) being the degrees of membership andnonmembership respectively of the element 119909 isin 119883 at themoment 119905 isin 119879 And 120572A(119909 119905) 120573A(119909 119905) isin [0 2120587] at themoment 119905 isin 119879 where 119894 = radicminus1

The hesitation degree of a TCIFS A is defined by120587A 1 minus 120583A(119909 119905)minusVA(119909 119905) such that 0 le 120587A(119909 119905) le 1 for each(119909 119905) isin A times 119879 For brevity we will write A instead of A(119879)when this does not cause confusions

Example 8 Suppose that119883 is a universal with respect to thetime set 1198791 = 119905 | 119905 = 6119896 119896 = 0 1 100 1198792 = 119905 | 119905 =2119896 119896 = 0 1 100 and 120572A(119909 119905) = 120573A(119909 119905) = 1198901198942120587 ThenTCIFSsAB are defined by

A (1198791) (119909) = (12 14) 119905 = 4119896 119896 = 1 100(19 35) 119905 = 4119896 + 2 119896 = 1 100

B (1198792) (119909) =

(23 16) 119905 = 3119896 119896 = 1 100(38 15) 119905 = 3119896 + 1 119896 = 1 100( 711 29) 119905 = 3119896 + 2 119896 = 1 100

(13)

Example 9 Suppose 119883 = 1199091 1199092 1199093 with respect to thetime set 119879 = 1199051 1199052 1199053 Then the details of a TCIFS A areexplained in Tables 1 2 and 3

Definition 10 LetA(1198791) andB(1198792) be two TCIFSs Then

A (1198791) capB (1198792) = (119909min (120583A (119909 119905) 119890119894120572A(119909119905) 120583B (119909 119905) 119890119894120572B(119909119905)) max (]A (119909 119905) 119890119894120573A(119909119905) ]B (119909 119905) 119890119894120573B(119909119905)) (119909 119905) isin 119883times (1198791 cup 1198792

A (1198791) cupB (1198792) = (119909max (120583A (119909 119905) 119890119894120572A(119909119905) 120583B (119909 119905) 119890119894120572B(119909119905)) min (]A (119909 119905) 119890119894120573A(119909119905) ]B (119909 119905) 119890119894120573B(119909119905)) (119909 119905) isin 119883times (1198791 cup 1198792

4 Discrete Dynamics in Nature and Society

Table 1 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 2 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 3 TCIFS A1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)A (1198791) = (119909(]A (119909 119905) 119890119894(2120587minus120573A(119909)) 120583A (119909 119905) 119890119894(2120587minus120572A(119909)) ) (119909 119905)isin 119883 times 1198791

(14)

where

120583A (119909 119905) = 120583A (119909 119905) 119905 isin 11987910 119905 isin 1198792 minus 1198791

]A (119909 119905) = ]A (119909 119905) 119905 isin 11987911 119905 isin 1198792 minus 1198791

120583B (119909 119905) = 120583B (119909 119905) 119905 isin 11987910 119905 isin 1198792 minus 1198791

]B (119909 119905) = ]B (119909 119905) 119905 isin 11987911 119905 isin 1198792 minus 1198791

(15)

Definition 11 Wedefine the following two operators119891 119886119899119889 119892over a TCIFSA

119891 (A (119879)) = (119909 (119909 119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) 119898119894119899119905 isin 119879sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

ℎ (A (119879)) = (119909 (119909 119898119886119909119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905) 119898119894119899119905 isin 119883sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

119892 (A (119879)) = (119909 (119909 119898119894119899119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) 119898119886119909119905 isin 119879sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

119897 (A (119879)) = (119909 (119909 119898119894119899119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905) 119898119886119909119905 isin 119883sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

(16)

Theorem 12 119891(A(119879)) and 119892(A(119879)) are TCIFSsProof Suppose that119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) = 120583A (119909 1199051) 119890119894120572A(1199091199051)

for some 1199051 isin 119879 (17)

and119898119894119899119905 isin 119879]A (119909 119905) 119890119894120572A(119909119905) = ]A (119909 1199052) 119890119894120572A(1199091199052)for some 1199052 isin 119879 (18)

Therefore

]A (119909 1199052) 119890119894120573A(1199091199052) le V119860 (119909 1199051) 119890119894120573A(1199091199051) And119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) + 119898119894119899119905 isin 119879V119860 (119909 119905) 119890119894120572A(119909119905)= 120583A (119909 1199051) 119890119894120572A(1199091199051) + V119860 (119909 1199052) 119890119894120572A(1199091199052)le 120583A (119909 1199051) 119890119894120572A(1199091199051) + V119860 (119909 1199051) 119890119894120573A(1199091199051) le 1(19)

Then 119891(A(119879)) is TCIFSs Also by the same fashion 119892(A(119879))are TCIFSs

Theorem 13 For every TCIFSA(119879)(1) 119891(119891(A(119879))) = 119891(A(119879))(2) 119892(119892(A(119879))) = 119892(A(119879))(3) 119891(119892(A(119879))) = 119892(A(119879))(4) 119892(119891(A(119879))) = 119891(A(119879))

Proof The proof is obvious

Theorem 14 For every TCIFSA(119879)(1) ℎ(119891(A(119879))) = 119891(ℎ(A(119879)))(2) 119897(119892(A(119879))) = 119892(119897(A(119879)))

Discrete Dynamics in Nature and Society 5

Proof (1)

ℎ (119891 (A (119879))) = (119909 (119909 119898119886119909119905 isin 119883 119898119886119909119905 isin 119879120583A (119909 119905)sdot 119890119894120572A(119909119905) 119898119894119899119905 isin 119883 119898119894119899119905 isin 119879]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905)isin 119883 times 119879 = (119909 (119909 119898119886119909119905 isin 119879 119898119886119909119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905)119898119894119899119905 isin 119879 119898119894119899119905 isin 119883]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879= 119891 (ℎ (A (119879)))

(20)

(2) By the same fashion one has the following

Definition 15 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (21)

where119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))

(22)

is the correlation of two TCIFSsA andB and

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

119879 (B) = 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905))+ ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(23)

are the information temporal complex intuitionistic energiesofA andB respectively

Example 16 Suppose that 119883 = 1199091 1199092 1199093 with respect tothe time set 119879 = 1199051 1199052 1199053 The details of a TCIFS A(119879)are explained in Table 4 Table 5 explained TCIFS B(119879) andTable 6 explained the correlation coefficient 119896(AB) betweenTCIFSA(119879) and TCIFSB(119879)Proposition 17 LetA(1198791) andB(1198792) be two TCIFS Then

(1) 119879(A) = 119879(A)(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A)(3) 119862(AB) = 119862(BA)

Table 4 TCIFSA(119879)1199051 1199052 11990531199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894)1199094 (06 01) (01119894 09119894) (06119894 04119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894)

Table 5 TCIFSB(119879)1199051 1199052 11990531199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894)1199092 (04 03) (07119894 02119894) (09119894 02119894)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (06 01) (01119894 09119894) (06119894 04119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894)

Table 6 Then 119896(AB)1199051 1199052 11990531199091 119896 (AB) = 0509 119896 (AB) = 1923 119896 (AB) = 29731199092 119896(AB) =3041 119896 (AB) = 6592 119896(AB) =66481199093 119896(AB) =5000 119896(AB) =2544 119896 (AB) = 13341199094 119896(AB) =4301 119896(AB) =6403 119896(AB) =68021199095 119896 (AB) = 1360 119896(AB) =5508 119896(AB) =42041199096 119896 (AB) = 1360 119896(AB) =5581 119896 (AB) = 4204

Proof Let 119905 isin 1198791 From Definition 10 120583A(119909 119905) = 120583A(119909 119905)]A(119909 119905) = ]A(119909 119905) 120583B(119909 119905) = 120583B(119909 119905) ]B(119909 119905) = ]B(119909 119905)and then

(1)

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 119899sum119894=1

119898sum119895=1

(]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))+ 120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))) = 119879 (A)

(24)

(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587119862 (AA)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905))

6 Discrete Dynamics in Nature and Society

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 1198902119894120572A(119909119905)+ ]A2 (119909119894 119905119894) 1198902119894120573A(119909119905))

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))) = 119879 (A)

(25)

(3)

119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))= 119899sum119894=1

119898sum119895=1

(]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)+ 120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)) = 119862 (BA)

(26)

Theorem 18 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) if 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119896(AB) = 1(2) 119896(AB) = 119896(BA)(3) 0 lt 119896(AB) lt 1

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (27)

IfA =B then radic119879(A)119879(B) = radic119879(A)2 = 119879(A)Then fromProposition 17 (2) one has the following

If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A) then119896 (AB) = 119879 (A)119879 (A) = 1 (28)

(2) From Proposition 17 (3) 119862(AB) = 119862(BA)Then

119896 (AB) = 119862 (BA)radic119879 (B) 119879 (A) = 119896 (BA) (29)

(3) We will prove that 119896(AB) lt 1 such that it is evident0 lt 119896(AB) so suppose that119899sum119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205721119899sum119894=1

119898sum119895=1

120583B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205722119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205723

119899sum119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205724

(30)

Then119879 (A) 119879 (B)= ( 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894)sdot 119890119894(2120587minus120573A(119909)) + ]A

2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))))times ( 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905)) + ]B

2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))) )= (( 119899sum

119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))) + ( 119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894)

sdot 119890119894(2120587minus120573A(119909))))times (( 119899sum

119894=1

119898sum119895=1

120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905))) + ( 119899sum

119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))))

(31)

Thenradic119879 (A) 119879 (B) = [(1205721 + 1205723) times (1205722 + 1205724)]12= (1205721 + 1205723)12 times (1205722 + 1205724)12119862 (AB)= (( 119899sum

119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905))) + ( 119899sum119894=1

119898sum119895=1

]A (119909119894 119905119894)sdot 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)))

(32)

Then

119862 (AB) = (1205721 times 1205722)12 + (1205723 times 1205724)12 (33)

Then

Discrete Dynamics in Nature and Society 7

1198962 (AB)le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724)

(34)

But

1198962 (AB) minus 1 le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724) minus 1= 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724 minus (1205721 + 1205723) times (1205722 + 1205724)(1205721 + 1205723) times (1205722 + 1205724) = [(1205721 times 1205724)12 minus (1205722 times 1205723)12]2(1205721 + 1205723) times (1205722 + 1205724)le 0

(35)

Hence 1198962(AB) minus 1 le 0 and then 119896 (AB) le 1Definition 19 Let 119878 119879119868119862119865119878119904 (119883 119879) times 119879119868119862119865119878119904 (119883 119879) 997888rarr[0 1] be a function and letA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSsin the universal 119883 = 1199091 1199092 1199093 119909119899 with respect to thetime set 119879 = 1199051 1199052 1199053 119905119898 Then 119878(AB) is said to be thesimilarity degree between 119879119868119862119865119878119904 A and B satisfying thefollowing statements

(1) 0 le 119878(AB) le 1(2) 119878(AB) = 1 ifA =B

(3) 119878(AB) = 119878(BA)(4) If A sube B sube 119862 Then 119878(A 119862) le 119878(AB) 119878(A 119862) le119878(B 119862)

Now we can have the following degrees of the similaritybetweenAB and satisfy the conditions from (1) to (4) Let119878A (119894 119895) = 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)119878B (119894 119895) = 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)120595A (119894 119895)= 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) + 1 minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)2

120595B (119894 119895)= ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) + 1 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)2

(36)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 Then

1198781 (AB) = 1 minus 12119898119899 119899sum119894=1

119898sum119895=1

1003816100381610038161003816119878A (119894 119895) minus 119878B (119894 119895)1003816100381610038161003816 1198782 (AB) = 1 minus 12119898119899 119899sum

119894=1

119898sum119895=1

10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816 + 10038161003816100381610038161003816]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816 1198780 (AB)= 1 minus 1radic2119898119899 119899sum119894=1

119898sum119895=1

radic(120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895))2 + (]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895))21198783 (AB) = 1 minus 1

119910radic2119898119899radic 119899sum119894=1119898sum119895=1

1003816100381610038161003816120595A (119894 119895) minus 120595B (119894 119895)1003816100381610038161003816119910 1 le 119910 lt +infin1198784 (AB)= sum119899119894=1sum119898119895=1min(|120583A(119909119894 119905119895)119890

119894120572A(119909119894 119905119895)||120583B(119909119894 119905119895)119890119894120572B(119909119894 119905119895)|)+min(|1minus]A(119909119894 119905119894)119890

119894120573A(119909119894 119905119895)||1minus]A(119909119894 119905119894)119890119894120573A(119909119894 119905119895)|)sum119899119894=1sum119898119895=1max (10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895)10038161003816100381610038161003816 10038161003816100381610038161003816120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816) +max (100381610038161003816100381610038161 minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)10038161003816100381610038161003816 100381610038161003816100381610038161 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816)

(37)

From a comparison between similarity measures 1198780(AB)1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) we give the follow-ing example

Example 20 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056The details of a TCIFS A(119879)

8 Discrete Dynamics in Nature and Society

Table 7 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 8 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

are explained in Table 7 Table 8 explained TCIFS B(119879)and Table 9 explained a comparison between similaritymeasures 1198780(AB) 1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) TCIFS A(119879) and TCIFS B(119879)4 Similarity Measures between OtherExtensions of Temporal ComplexIntuitionistic Fuzzy Set

The following definition extend the method proposed byChaira [12] for intuitionistic fuzzy set based on the Sugeno[13] and Omar [10] intuitionistic fuzzy generator

Definition 21 If 120583A(119909 119905)119890119894120572A(119909119905) is the degrees of membershipfunction of the element 119909 isin 119883 at the moment 119905 isin 119879 thennonmembership function ]A(119909)119890119894120573A(119909) = 119866(120583A(119909 119905)119890119894120572A(119909119905))where

119866(120583A (119909 119905) 119890119894120572A(119909119905)) = 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) 120572 gt 0 (38)

And 119866(1) = 0 119866(0) = 1 and by help of the Sugeno [6]intuitionistic fuzzy generator TCIFSA is given by

A120572 (119879) = ((119909 119905) 120583A (119909 119905)sdot 119890119894120572A(119909119905) 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905)) | (119909 119905) isin 119883 times 119879

(39)

The hesitation degree of a TCIFSA is

120587A120572 (119909 119905) = 1 minus 120583A (119909 119905) minus 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) (40)

Example 22 Suppose that A(119879) is TCIFS defined on 119883 =1199091 1199092 1199093 with respect to the time set 119879 = 1199051 1199052 1199053 Thedetails of a TCIFS A(119879) are explained in Tables 10 11 and12 Table 13 explained TCIFS A1 when 120572 = 1 and Table 14explained the hesitation degree of a TCIFSA

If 120572 = 1 then one has the following (see Tables 13 and 14)Definition 23 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the timeset 119879 = 1199051 1199052 1199053 119905119898 Then a cosine similarity measurebetweenA(119879) 119886119899119889 B(119879) is proposed as follows

119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(41)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Discrete Dynamics in Nature and Society

Table 1 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 2 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 3 TCIFS A1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)A (1198791) = (119909(]A (119909 119905) 119890119894(2120587minus120573A(119909)) 120583A (119909 119905) 119890119894(2120587minus120572A(119909)) ) (119909 119905)isin 119883 times 1198791

(14)

where

120583A (119909 119905) = 120583A (119909 119905) 119905 isin 11987910 119905 isin 1198792 minus 1198791

]A (119909 119905) = ]A (119909 119905) 119905 isin 11987911 119905 isin 1198792 minus 1198791

120583B (119909 119905) = 120583B (119909 119905) 119905 isin 11987910 119905 isin 1198792 minus 1198791

]B (119909 119905) = ]B (119909 119905) 119905 isin 11987911 119905 isin 1198792 minus 1198791

(15)

Definition 11 Wedefine the following two operators119891 119886119899119889 119892over a TCIFSA

119891 (A (119879)) = (119909 (119909 119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) 119898119894119899119905 isin 119879sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

ℎ (A (119879)) = (119909 (119909 119898119886119909119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905) 119898119894119899119905 isin 119883sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

119892 (A (119879)) = (119909 (119909 119898119894119899119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) 119898119886119909119905 isin 119879sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

119897 (A (119879)) = (119909 (119909 119898119894119899119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905) 119898119886119909119905 isin 119883sdot ]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879

(16)

Theorem 12 119891(A(119879)) and 119892(A(119879)) are TCIFSsProof Suppose that119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) = 120583A (119909 1199051) 119890119894120572A(1199091199051)

for some 1199051 isin 119879 (17)

and119898119894119899119905 isin 119879]A (119909 119905) 119890119894120572A(119909119905) = ]A (119909 1199052) 119890119894120572A(1199091199052)for some 1199052 isin 119879 (18)

Therefore

]A (119909 1199052) 119890119894120573A(1199091199052) le V119860 (119909 1199051) 119890119894120573A(1199091199051) And119898119886119909119905 isin 119879120583A (119909 119905) 119890119894120572A(119909119905) + 119898119894119899119905 isin 119879V119860 (119909 119905) 119890119894120572A(119909119905)= 120583A (119909 1199051) 119890119894120572A(1199091199051) + V119860 (119909 1199052) 119890119894120572A(1199091199052)le 120583A (119909 1199051) 119890119894120572A(1199091199051) + V119860 (119909 1199051) 119890119894120573A(1199091199051) le 1(19)

Then 119891(A(119879)) is TCIFSs Also by the same fashion 119892(A(119879))are TCIFSs

Theorem 13 For every TCIFSA(119879)(1) 119891(119891(A(119879))) = 119891(A(119879))(2) 119892(119892(A(119879))) = 119892(A(119879))(3) 119891(119892(A(119879))) = 119892(A(119879))(4) 119892(119891(A(119879))) = 119891(A(119879))

Proof The proof is obvious

Theorem 14 For every TCIFSA(119879)(1) ℎ(119891(A(119879))) = 119891(ℎ(A(119879)))(2) 119897(119892(A(119879))) = 119892(119897(A(119879)))

Discrete Dynamics in Nature and Society 5

Proof (1)

ℎ (119891 (A (119879))) = (119909 (119909 119898119886119909119905 isin 119883 119898119886119909119905 isin 119879120583A (119909 119905)sdot 119890119894120572A(119909119905) 119898119894119899119905 isin 119883 119898119894119899119905 isin 119879]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905)isin 119883 times 119879 = (119909 (119909 119898119886119909119905 isin 119879 119898119886119909119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905)119898119894119899119905 isin 119879 119898119894119899119905 isin 119883]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879= 119891 (ℎ (A (119879)))

(20)

(2) By the same fashion one has the following

Definition 15 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (21)

where119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))

(22)

is the correlation of two TCIFSsA andB and

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

119879 (B) = 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905))+ ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(23)

are the information temporal complex intuitionistic energiesofA andB respectively

Example 16 Suppose that 119883 = 1199091 1199092 1199093 with respect tothe time set 119879 = 1199051 1199052 1199053 The details of a TCIFS A(119879)are explained in Table 4 Table 5 explained TCIFS B(119879) andTable 6 explained the correlation coefficient 119896(AB) betweenTCIFSA(119879) and TCIFSB(119879)Proposition 17 LetA(1198791) andB(1198792) be two TCIFS Then

(1) 119879(A) = 119879(A)(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A)(3) 119862(AB) = 119862(BA)

Table 4 TCIFSA(119879)1199051 1199052 11990531199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894)1199094 (06 01) (01119894 09119894) (06119894 04119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894)

Table 5 TCIFSB(119879)1199051 1199052 11990531199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894)1199092 (04 03) (07119894 02119894) (09119894 02119894)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (06 01) (01119894 09119894) (06119894 04119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894)

Table 6 Then 119896(AB)1199051 1199052 11990531199091 119896 (AB) = 0509 119896 (AB) = 1923 119896 (AB) = 29731199092 119896(AB) =3041 119896 (AB) = 6592 119896(AB) =66481199093 119896(AB) =5000 119896(AB) =2544 119896 (AB) = 13341199094 119896(AB) =4301 119896(AB) =6403 119896(AB) =68021199095 119896 (AB) = 1360 119896(AB) =5508 119896(AB) =42041199096 119896 (AB) = 1360 119896(AB) =5581 119896 (AB) = 4204

Proof Let 119905 isin 1198791 From Definition 10 120583A(119909 119905) = 120583A(119909 119905)]A(119909 119905) = ]A(119909 119905) 120583B(119909 119905) = 120583B(119909 119905) ]B(119909 119905) = ]B(119909 119905)and then

(1)

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 119899sum119894=1

119898sum119895=1

(]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))+ 120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))) = 119879 (A)

(24)

(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587119862 (AA)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905))

6 Discrete Dynamics in Nature and Society

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 1198902119894120572A(119909119905)+ ]A2 (119909119894 119905119894) 1198902119894120573A(119909119905))

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))) = 119879 (A)

(25)

(3)

119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))= 119899sum119894=1

119898sum119895=1

(]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)+ 120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)) = 119862 (BA)

(26)

Theorem 18 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) if 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119896(AB) = 1(2) 119896(AB) = 119896(BA)(3) 0 lt 119896(AB) lt 1

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (27)

IfA =B then radic119879(A)119879(B) = radic119879(A)2 = 119879(A)Then fromProposition 17 (2) one has the following

If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A) then119896 (AB) = 119879 (A)119879 (A) = 1 (28)

(2) From Proposition 17 (3) 119862(AB) = 119862(BA)Then

119896 (AB) = 119862 (BA)radic119879 (B) 119879 (A) = 119896 (BA) (29)

(3) We will prove that 119896(AB) lt 1 such that it is evident0 lt 119896(AB) so suppose that119899sum119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205721119899sum119894=1

119898sum119895=1

120583B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205722119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205723

119899sum119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205724

(30)

Then119879 (A) 119879 (B)= ( 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894)sdot 119890119894(2120587minus120573A(119909)) + ]A

2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))))times ( 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905)) + ]B

2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))) )= (( 119899sum

119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))) + ( 119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894)

sdot 119890119894(2120587minus120573A(119909))))times (( 119899sum

119894=1

119898sum119895=1

120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905))) + ( 119899sum

119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))))

(31)

Thenradic119879 (A) 119879 (B) = [(1205721 + 1205723) times (1205722 + 1205724)]12= (1205721 + 1205723)12 times (1205722 + 1205724)12119862 (AB)= (( 119899sum

119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905))) + ( 119899sum119894=1

119898sum119895=1

]A (119909119894 119905119894)sdot 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)))

(32)

Then

119862 (AB) = (1205721 times 1205722)12 + (1205723 times 1205724)12 (33)

Then

Discrete Dynamics in Nature and Society 7

1198962 (AB)le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724)

(34)

But

1198962 (AB) minus 1 le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724) minus 1= 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724 minus (1205721 + 1205723) times (1205722 + 1205724)(1205721 + 1205723) times (1205722 + 1205724) = [(1205721 times 1205724)12 minus (1205722 times 1205723)12]2(1205721 + 1205723) times (1205722 + 1205724)le 0

(35)

Hence 1198962(AB) minus 1 le 0 and then 119896 (AB) le 1Definition 19 Let 119878 119879119868119862119865119878119904 (119883 119879) times 119879119868119862119865119878119904 (119883 119879) 997888rarr[0 1] be a function and letA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSsin the universal 119883 = 1199091 1199092 1199093 119909119899 with respect to thetime set 119879 = 1199051 1199052 1199053 119905119898 Then 119878(AB) is said to be thesimilarity degree between 119879119868119862119865119878119904 A and B satisfying thefollowing statements

(1) 0 le 119878(AB) le 1(2) 119878(AB) = 1 ifA =B

(3) 119878(AB) = 119878(BA)(4) If A sube B sube 119862 Then 119878(A 119862) le 119878(AB) 119878(A 119862) le119878(B 119862)

Now we can have the following degrees of the similaritybetweenAB and satisfy the conditions from (1) to (4) Let119878A (119894 119895) = 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)119878B (119894 119895) = 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)120595A (119894 119895)= 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) + 1 minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)2

120595B (119894 119895)= ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) + 1 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)2

(36)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 Then

1198781 (AB) = 1 minus 12119898119899 119899sum119894=1

119898sum119895=1

1003816100381610038161003816119878A (119894 119895) minus 119878B (119894 119895)1003816100381610038161003816 1198782 (AB) = 1 minus 12119898119899 119899sum

119894=1

119898sum119895=1

10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816 + 10038161003816100381610038161003816]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816 1198780 (AB)= 1 minus 1radic2119898119899 119899sum119894=1

119898sum119895=1

radic(120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895))2 + (]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895))21198783 (AB) = 1 minus 1

119910radic2119898119899radic 119899sum119894=1119898sum119895=1

1003816100381610038161003816120595A (119894 119895) minus 120595B (119894 119895)1003816100381610038161003816119910 1 le 119910 lt +infin1198784 (AB)= sum119899119894=1sum119898119895=1min(|120583A(119909119894 119905119895)119890

119894120572A(119909119894 119905119895)||120583B(119909119894 119905119895)119890119894120572B(119909119894 119905119895)|)+min(|1minus]A(119909119894 119905119894)119890

119894120573A(119909119894 119905119895)||1minus]A(119909119894 119905119894)119890119894120573A(119909119894 119905119895)|)sum119899119894=1sum119898119895=1max (10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895)10038161003816100381610038161003816 10038161003816100381610038161003816120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816) +max (100381610038161003816100381610038161 minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)10038161003816100381610038161003816 100381610038161003816100381610038161 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816)

(37)

From a comparison between similarity measures 1198780(AB)1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) we give the follow-ing example

Example 20 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056The details of a TCIFS A(119879)

8 Discrete Dynamics in Nature and Society

Table 7 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 8 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

are explained in Table 7 Table 8 explained TCIFS B(119879)and Table 9 explained a comparison between similaritymeasures 1198780(AB) 1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) TCIFS A(119879) and TCIFS B(119879)4 Similarity Measures between OtherExtensions of Temporal ComplexIntuitionistic Fuzzy Set

The following definition extend the method proposed byChaira [12] for intuitionistic fuzzy set based on the Sugeno[13] and Omar [10] intuitionistic fuzzy generator

Definition 21 If 120583A(119909 119905)119890119894120572A(119909119905) is the degrees of membershipfunction of the element 119909 isin 119883 at the moment 119905 isin 119879 thennonmembership function ]A(119909)119890119894120573A(119909) = 119866(120583A(119909 119905)119890119894120572A(119909119905))where

119866(120583A (119909 119905) 119890119894120572A(119909119905)) = 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) 120572 gt 0 (38)

And 119866(1) = 0 119866(0) = 1 and by help of the Sugeno [6]intuitionistic fuzzy generator TCIFSA is given by

A120572 (119879) = ((119909 119905) 120583A (119909 119905)sdot 119890119894120572A(119909119905) 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905)) | (119909 119905) isin 119883 times 119879

(39)

The hesitation degree of a TCIFSA is

120587A120572 (119909 119905) = 1 minus 120583A (119909 119905) minus 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) (40)

Example 22 Suppose that A(119879) is TCIFS defined on 119883 =1199091 1199092 1199093 with respect to the time set 119879 = 1199051 1199052 1199053 Thedetails of a TCIFS A(119879) are explained in Tables 10 11 and12 Table 13 explained TCIFS A1 when 120572 = 1 and Table 14explained the hesitation degree of a TCIFSA

If 120572 = 1 then one has the following (see Tables 13 and 14)Definition 23 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the timeset 119879 = 1199051 1199052 1199053 119905119898 Then a cosine similarity measurebetweenA(119879) 119886119899119889 B(119879) is proposed as follows

119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(41)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 5

Proof (1)

ℎ (119891 (A (119879))) = (119909 (119909 119898119886119909119905 isin 119883 119898119886119909119905 isin 119879120583A (119909 119905)sdot 119890119894120572A(119909119905) 119898119894119899119905 isin 119883 119898119894119899119905 isin 119879]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905)isin 119883 times 119879 = (119909 (119909 119898119886119909119905 isin 119879 119898119886119909119905 isin 119883120583A (119909 119905) 119890119894120572A(119909119905)119898119894119899119905 isin 119879 119898119894119899119905 isin 119883]A (119909 119905) 119890119894120573A(119909119905)) (119909 119905) isin 119883 times 119879= 119891 (ℎ (A (119879)))

(20)

(2) By the same fashion one has the following

Definition 15 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (21)

where119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))

(22)

is the correlation of two TCIFSsA andB and

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

119879 (B) = 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905))+ ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(23)

are the information temporal complex intuitionistic energiesofA andB respectively

Example 16 Suppose that 119883 = 1199091 1199092 1199093 with respect tothe time set 119879 = 1199051 1199052 1199053 The details of a TCIFS A(119879)are explained in Table 4 Table 5 explained TCIFS B(119879) andTable 6 explained the correlation coefficient 119896(AB) betweenTCIFSA(119879) and TCIFSB(119879)Proposition 17 LetA(1198791) andB(1198792) be two TCIFS Then

(1) 119879(A) = 119879(A)(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A)(3) 119862(AB) = 119862(BA)

Table 4 TCIFSA(119879)1199051 1199052 11990531199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894)1199094 (06 01) (01119894 09119894) (06119894 04119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894)

Table 5 TCIFSB(119879)1199051 1199052 11990531199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894)1199092 (04 03) (07119894 02119894) (09119894 02119894)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (06 01) (01119894 09119894) (06119894 04119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894)

Table 6 Then 119896(AB)1199051 1199052 11990531199091 119896 (AB) = 0509 119896 (AB) = 1923 119896 (AB) = 29731199092 119896(AB) =3041 119896 (AB) = 6592 119896(AB) =66481199093 119896(AB) =5000 119896(AB) =2544 119896 (AB) = 13341199094 119896(AB) =4301 119896(AB) =6403 119896(AB) =68021199095 119896 (AB) = 1360 119896(AB) =5508 119896(AB) =42041199096 119896 (AB) = 1360 119896(AB) =5581 119896 (AB) = 4204

Proof Let 119905 isin 1198791 From Definition 10 120583A(119909 119905) = 120583A(119909 119905)]A(119909 119905) = ]A(119909 119905) 120583B(119909 119905) = 120583B(119909 119905) ]B(119909 119905) = ]B(119909 119905)and then

(1)

119879 (A) = 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 119899sum119894=1

119898sum119895=1

(]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))+ 120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))) = 119879 (A)

(24)

(2) If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587119862 (AA)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905))

6 Discrete Dynamics in Nature and Society

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 1198902119894120572A(119909119905)+ ]A2 (119909119894 119905119894) 1198902119894120573A(119909119905))

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))) = 119879 (A)

(25)

(3)

119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))= 119899sum119894=1

119898sum119895=1

(]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)+ 120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)) = 119862 (BA)

(26)

Theorem 18 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) if 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119896(AB) = 1(2) 119896(AB) = 119896(BA)(3) 0 lt 119896(AB) lt 1

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (27)

IfA =B then radic119879(A)119879(B) = radic119879(A)2 = 119879(A)Then fromProposition 17 (2) one has the following

If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A) then119896 (AB) = 119879 (A)119879 (A) = 1 (28)

(2) From Proposition 17 (3) 119862(AB) = 119862(BA)Then

119896 (AB) = 119862 (BA)radic119879 (B) 119879 (A) = 119896 (BA) (29)

(3) We will prove that 119896(AB) lt 1 such that it is evident0 lt 119896(AB) so suppose that119899sum119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205721119899sum119894=1

119898sum119895=1

120583B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205722119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205723

119899sum119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205724

(30)

Then119879 (A) 119879 (B)= ( 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894)sdot 119890119894(2120587minus120573A(119909)) + ]A

2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))))times ( 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905)) + ]B

2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))) )= (( 119899sum

119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))) + ( 119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894)

sdot 119890119894(2120587minus120573A(119909))))times (( 119899sum

119894=1

119898sum119895=1

120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905))) + ( 119899sum

119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))))

(31)

Thenradic119879 (A) 119879 (B) = [(1205721 + 1205723) times (1205722 + 1205724)]12= (1205721 + 1205723)12 times (1205722 + 1205724)12119862 (AB)= (( 119899sum

119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905))) + ( 119899sum119894=1

119898sum119895=1

]A (119909119894 119905119894)sdot 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)))

(32)

Then

119862 (AB) = (1205721 times 1205722)12 + (1205723 times 1205724)12 (33)

Then

Discrete Dynamics in Nature and Society 7

1198962 (AB)le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724)

(34)

But

1198962 (AB) minus 1 le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724) minus 1= 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724 minus (1205721 + 1205723) times (1205722 + 1205724)(1205721 + 1205723) times (1205722 + 1205724) = [(1205721 times 1205724)12 minus (1205722 times 1205723)12]2(1205721 + 1205723) times (1205722 + 1205724)le 0

(35)

Hence 1198962(AB) minus 1 le 0 and then 119896 (AB) le 1Definition 19 Let 119878 119879119868119862119865119878119904 (119883 119879) times 119879119868119862119865119878119904 (119883 119879) 997888rarr[0 1] be a function and letA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSsin the universal 119883 = 1199091 1199092 1199093 119909119899 with respect to thetime set 119879 = 1199051 1199052 1199053 119905119898 Then 119878(AB) is said to be thesimilarity degree between 119879119868119862119865119878119904 A and B satisfying thefollowing statements

(1) 0 le 119878(AB) le 1(2) 119878(AB) = 1 ifA =B

(3) 119878(AB) = 119878(BA)(4) If A sube B sube 119862 Then 119878(A 119862) le 119878(AB) 119878(A 119862) le119878(B 119862)

Now we can have the following degrees of the similaritybetweenAB and satisfy the conditions from (1) to (4) Let119878A (119894 119895) = 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)119878B (119894 119895) = 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)120595A (119894 119895)= 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) + 1 minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)2

120595B (119894 119895)= ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) + 1 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)2

(36)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 Then

1198781 (AB) = 1 minus 12119898119899 119899sum119894=1

119898sum119895=1

1003816100381610038161003816119878A (119894 119895) minus 119878B (119894 119895)1003816100381610038161003816 1198782 (AB) = 1 minus 12119898119899 119899sum

119894=1

119898sum119895=1

10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816 + 10038161003816100381610038161003816]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816 1198780 (AB)= 1 minus 1radic2119898119899 119899sum119894=1

119898sum119895=1

radic(120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895))2 + (]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895))21198783 (AB) = 1 minus 1

119910radic2119898119899radic 119899sum119894=1119898sum119895=1

1003816100381610038161003816120595A (119894 119895) minus 120595B (119894 119895)1003816100381610038161003816119910 1 le 119910 lt +infin1198784 (AB)= sum119899119894=1sum119898119895=1min(|120583A(119909119894 119905119895)119890

119894120572A(119909119894 119905119895)||120583B(119909119894 119905119895)119890119894120572B(119909119894 119905119895)|)+min(|1minus]A(119909119894 119905119894)119890

119894120573A(119909119894 119905119895)||1minus]A(119909119894 119905119894)119890119894120573A(119909119894 119905119895)|)sum119899119894=1sum119898119895=1max (10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895)10038161003816100381610038161003816 10038161003816100381610038161003816120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816) +max (100381610038161003816100381610038161 minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)10038161003816100381610038161003816 100381610038161003816100381610038161 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816)

(37)

From a comparison between similarity measures 1198780(AB)1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) we give the follow-ing example

Example 20 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056The details of a TCIFS A(119879)

8 Discrete Dynamics in Nature and Society

Table 7 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 8 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

are explained in Table 7 Table 8 explained TCIFS B(119879)and Table 9 explained a comparison between similaritymeasures 1198780(AB) 1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) TCIFS A(119879) and TCIFS B(119879)4 Similarity Measures between OtherExtensions of Temporal ComplexIntuitionistic Fuzzy Set

The following definition extend the method proposed byChaira [12] for intuitionistic fuzzy set based on the Sugeno[13] and Omar [10] intuitionistic fuzzy generator

Definition 21 If 120583A(119909 119905)119890119894120572A(119909119905) is the degrees of membershipfunction of the element 119909 isin 119883 at the moment 119905 isin 119879 thennonmembership function ]A(119909)119890119894120573A(119909) = 119866(120583A(119909 119905)119890119894120572A(119909119905))where

119866(120583A (119909 119905) 119890119894120572A(119909119905)) = 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) 120572 gt 0 (38)

And 119866(1) = 0 119866(0) = 1 and by help of the Sugeno [6]intuitionistic fuzzy generator TCIFSA is given by

A120572 (119879) = ((119909 119905) 120583A (119909 119905)sdot 119890119894120572A(119909119905) 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905)) | (119909 119905) isin 119883 times 119879

(39)

The hesitation degree of a TCIFSA is

120587A120572 (119909 119905) = 1 minus 120583A (119909 119905) minus 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) (40)

Example 22 Suppose that A(119879) is TCIFS defined on 119883 =1199091 1199092 1199093 with respect to the time set 119879 = 1199051 1199052 1199053 Thedetails of a TCIFS A(119879) are explained in Tables 10 11 and12 Table 13 explained TCIFS A1 when 120572 = 1 and Table 14explained the hesitation degree of a TCIFSA

If 120572 = 1 then one has the following (see Tables 13 and 14)Definition 23 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the timeset 119879 = 1199051 1199052 1199053 119905119898 Then a cosine similarity measurebetweenA(119879) 119886119899119889 B(119879) is proposed as follows

119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(41)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Discrete Dynamics in Nature and Society

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 1198902119894120572A(119909119905)+ ]A2 (119909119894 119905119894) 1198902119894120573A(119909119905))

= 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120572A(119909))+ ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))) = 119879 (A)

(25)

(3)

119862 (AB)= 119899sum119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905)+ ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905))= 119899sum119894=1

119898sum119895=1

(]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)+ 120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905)) = 119862 (BA)

(26)

Theorem 18 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) if 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119896(AB) = 1(2) 119896(AB) = 119896(BA)(3) 0 lt 119896(AB) lt 1

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 The correlation coefficientofA andB is given by

119896 (AB) = 119862 (AB)radic119879 (A) 119879 (B) (27)

IfA =B then radic119879(A)119879(B) = radic119879(A)2 = 119879(A)Then fromProposition 17 (2) one has the following

If 120572A(119909 119905) = 2120587 and 120573A(119909 119905) = 2120587 then 119862(AA) =119879(A) then119896 (AB) = 119879 (A)119879 (A) = 1 (28)

(2) From Proposition 17 (3) 119862(AB) = 119862(BA)Then

119896 (AB) = 119862 (BA)radic119879 (B) 119879 (A) = 119896 (BA) (29)

(3) We will prove that 119896(AB) lt 1 such that it is evident0 lt 119896(AB) so suppose that119899sum119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205721119899sum119894=1

119898sum119895=1

120583B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205722119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) = 1205723

119899sum119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)) = 1205724

(30)

Then119879 (A) 119879 (B)= ( 119899sum119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894)sdot 119890119894(2120587minus120573A(119909)) + ]A

2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))))times ( 119899sum119894=1

119898sum119895=1

(120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905)) + ]B

2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))) )= (( 119899sum

119894=1

119898sum119895=1

(120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))) + ( 119899sum119894=1

119898sum119895=1

]A2 (119909119894 119905119894)

sdot 119890119894(2120587minus120573A(119909))))times (( 119899sum

119894=1

119898sum119895=1

120583B2 (119909119894 119905119894)sdot 119890119894(2120587minus120572B(119909119905))) + ( 119899sum

119894=1

119898sum119895=1

]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))))

(31)

Thenradic119879 (A) 119879 (B) = [(1205721 + 1205723) times (1205722 + 1205724)]12= (1205721 + 1205723)12 times (1205722 + 1205724)12119862 (AB)= (( 119899sum

119894=1

119898sum119895=1

(120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905))) + ( 119899sum119894=1

119898sum119895=1

]A (119909119894 119905119894)sdot 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)))

(32)

Then

119862 (AB) = (1205721 times 1205722)12 + (1205723 times 1205724)12 (33)

Then

Discrete Dynamics in Nature and Society 7

1198962 (AB)le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724)

(34)

But

1198962 (AB) minus 1 le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724) minus 1= 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724 minus (1205721 + 1205723) times (1205722 + 1205724)(1205721 + 1205723) times (1205722 + 1205724) = [(1205721 times 1205724)12 minus (1205722 times 1205723)12]2(1205721 + 1205723) times (1205722 + 1205724)le 0

(35)

Hence 1198962(AB) minus 1 le 0 and then 119896 (AB) le 1Definition 19 Let 119878 119879119868119862119865119878119904 (119883 119879) times 119879119868119862119865119878119904 (119883 119879) 997888rarr[0 1] be a function and letA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSsin the universal 119883 = 1199091 1199092 1199093 119909119899 with respect to thetime set 119879 = 1199051 1199052 1199053 119905119898 Then 119878(AB) is said to be thesimilarity degree between 119879119868119862119865119878119904 A and B satisfying thefollowing statements

(1) 0 le 119878(AB) le 1(2) 119878(AB) = 1 ifA =B

(3) 119878(AB) = 119878(BA)(4) If A sube B sube 119862 Then 119878(A 119862) le 119878(AB) 119878(A 119862) le119878(B 119862)

Now we can have the following degrees of the similaritybetweenAB and satisfy the conditions from (1) to (4) Let119878A (119894 119895) = 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)119878B (119894 119895) = 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)120595A (119894 119895)= 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) + 1 minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)2

120595B (119894 119895)= ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) + 1 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)2

(36)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 Then

1198781 (AB) = 1 minus 12119898119899 119899sum119894=1

119898sum119895=1

1003816100381610038161003816119878A (119894 119895) minus 119878B (119894 119895)1003816100381610038161003816 1198782 (AB) = 1 minus 12119898119899 119899sum

119894=1

119898sum119895=1

10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816 + 10038161003816100381610038161003816]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816 1198780 (AB)= 1 minus 1radic2119898119899 119899sum119894=1

119898sum119895=1

radic(120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895))2 + (]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895))21198783 (AB) = 1 minus 1

119910radic2119898119899radic 119899sum119894=1119898sum119895=1

1003816100381610038161003816120595A (119894 119895) minus 120595B (119894 119895)1003816100381610038161003816119910 1 le 119910 lt +infin1198784 (AB)= sum119899119894=1sum119898119895=1min(|120583A(119909119894 119905119895)119890

119894120572A(119909119894 119905119895)||120583B(119909119894 119905119895)119890119894120572B(119909119894 119905119895)|)+min(|1minus]A(119909119894 119905119894)119890

119894120573A(119909119894 119905119895)||1minus]A(119909119894 119905119894)119890119894120573A(119909119894 119905119895)|)sum119899119894=1sum119898119895=1max (10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895)10038161003816100381610038161003816 10038161003816100381610038161003816120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816) +max (100381610038161003816100381610038161 minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)10038161003816100381610038161003816 100381610038161003816100381610038161 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816)

(37)

From a comparison between similarity measures 1198780(AB)1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) we give the follow-ing example

Example 20 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056The details of a TCIFS A(119879)

8 Discrete Dynamics in Nature and Society

Table 7 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 8 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

are explained in Table 7 Table 8 explained TCIFS B(119879)and Table 9 explained a comparison between similaritymeasures 1198780(AB) 1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) TCIFS A(119879) and TCIFS B(119879)4 Similarity Measures between OtherExtensions of Temporal ComplexIntuitionistic Fuzzy Set

The following definition extend the method proposed byChaira [12] for intuitionistic fuzzy set based on the Sugeno[13] and Omar [10] intuitionistic fuzzy generator

Definition 21 If 120583A(119909 119905)119890119894120572A(119909119905) is the degrees of membershipfunction of the element 119909 isin 119883 at the moment 119905 isin 119879 thennonmembership function ]A(119909)119890119894120573A(119909) = 119866(120583A(119909 119905)119890119894120572A(119909119905))where

119866(120583A (119909 119905) 119890119894120572A(119909119905)) = 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) 120572 gt 0 (38)

And 119866(1) = 0 119866(0) = 1 and by help of the Sugeno [6]intuitionistic fuzzy generator TCIFSA is given by

A120572 (119879) = ((119909 119905) 120583A (119909 119905)sdot 119890119894120572A(119909119905) 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905)) | (119909 119905) isin 119883 times 119879

(39)

The hesitation degree of a TCIFSA is

120587A120572 (119909 119905) = 1 minus 120583A (119909 119905) minus 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) (40)

Example 22 Suppose that A(119879) is TCIFS defined on 119883 =1199091 1199092 1199093 with respect to the time set 119879 = 1199051 1199052 1199053 Thedetails of a TCIFS A(119879) are explained in Tables 10 11 and12 Table 13 explained TCIFS A1 when 120572 = 1 and Table 14explained the hesitation degree of a TCIFSA

If 120572 = 1 then one has the following (see Tables 13 and 14)Definition 23 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the timeset 119879 = 1199051 1199052 1199053 119905119898 Then a cosine similarity measurebetweenA(119879) 119886119899119889 B(119879) is proposed as follows

119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(41)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 7

1198962 (AB)le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724)

(34)

But

1198962 (AB) minus 1 le 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724(1205721 + 1205723) times (1205722 + 1205724) minus 1= 1205721 times 1205722 + 2 (1205721 times 1205722 times 1205723 times 1205724)12 + 1205723 times 1205724 minus (1205721 + 1205723) times (1205722 + 1205724)(1205721 + 1205723) times (1205722 + 1205724) = [(1205721 times 1205724)12 minus (1205722 times 1205723)12]2(1205721 + 1205723) times (1205722 + 1205724)le 0

(35)

Hence 1198962(AB) minus 1 le 0 and then 119896 (AB) le 1Definition 19 Let 119878 119879119868119862119865119878119904 (119883 119879) times 119879119868119862119865119878119904 (119883 119879) 997888rarr[0 1] be a function and letA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSsin the universal 119883 = 1199091 1199092 1199093 119909119899 with respect to thetime set 119879 = 1199051 1199052 1199053 119905119898 Then 119878(AB) is said to be thesimilarity degree between 119879119868119862119865119878119904 A and B satisfying thefollowing statements

(1) 0 le 119878(AB) le 1(2) 119878(AB) = 1 ifA =B

(3) 119878(AB) = 119878(BA)(4) If A sube B sube 119862 Then 119878(A 119862) le 119878(AB) 119878(A 119862) le119878(B 119862)

Now we can have the following degrees of the similaritybetweenAB and satisfy the conditions from (1) to (4) Let119878A (119894 119895) = 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)119878B (119894 119895) = 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)120595A (119894 119895)= 120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) + 1 minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)2

120595B (119894 119895)= ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) + 1 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)2

(36)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 Then

1198781 (AB) = 1 minus 12119898119899 119899sum119894=1

119898sum119895=1

1003816100381610038161003816119878A (119894 119895) minus 119878B (119894 119895)1003816100381610038161003816 1198782 (AB) = 1 minus 12119898119899 119899sum

119894=1

119898sum119895=1

10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816 + 10038161003816100381610038161003816]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816 1198780 (AB)= 1 minus 1radic2119898119899 119899sum119894=1

119898sum119895=1

radic(120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895) minus 120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895))2 + (]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895) minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895))21198783 (AB) = 1 minus 1

119910radic2119898119899radic 119899sum119894=1119898sum119895=1

1003816100381610038161003816120595A (119894 119895) minus 120595B (119894 119895)1003816100381610038161003816119910 1 le 119910 lt +infin1198784 (AB)= sum119899119894=1sum119898119895=1min(|120583A(119909119894 119905119895)119890

119894120572A(119909119894 119905119895)||120583B(119909119894 119905119895)119890119894120572B(119909119894 119905119895)|)+min(|1minus]A(119909119894 119905119894)119890

119894120573A(119909119894 119905119895)||1minus]A(119909119894 119905119894)119890119894120573A(119909119894 119905119895)|)sum119899119894=1sum119898119895=1max (10038161003816100381610038161003816120583A (119909119894 119905119895) 119890119894120572A(119909119894 119905119895)10038161003816100381610038161003816 10038161003816100381610038161003816120583B (119909119894 119905119895) 119890119894120572B(119909119894 119905119895)10038161003816100381610038161003816) +max (100381610038161003816100381610038161 minus ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)10038161003816100381610038161003816 100381610038161003816100381610038161 minus ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)10038161003816100381610038161003816)

(37)

From a comparison between similarity measures 1198780(AB)1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) we give the follow-ing example

Example 20 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056The details of a TCIFS A(119879)

8 Discrete Dynamics in Nature and Society

Table 7 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 8 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

are explained in Table 7 Table 8 explained TCIFS B(119879)and Table 9 explained a comparison between similaritymeasures 1198780(AB) 1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) TCIFS A(119879) and TCIFS B(119879)4 Similarity Measures between OtherExtensions of Temporal ComplexIntuitionistic Fuzzy Set

The following definition extend the method proposed byChaira [12] for intuitionistic fuzzy set based on the Sugeno[13] and Omar [10] intuitionistic fuzzy generator

Definition 21 If 120583A(119909 119905)119890119894120572A(119909119905) is the degrees of membershipfunction of the element 119909 isin 119883 at the moment 119905 isin 119879 thennonmembership function ]A(119909)119890119894120573A(119909) = 119866(120583A(119909 119905)119890119894120572A(119909119905))where

119866(120583A (119909 119905) 119890119894120572A(119909119905)) = 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) 120572 gt 0 (38)

And 119866(1) = 0 119866(0) = 1 and by help of the Sugeno [6]intuitionistic fuzzy generator TCIFSA is given by

A120572 (119879) = ((119909 119905) 120583A (119909 119905)sdot 119890119894120572A(119909119905) 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905)) | (119909 119905) isin 119883 times 119879

(39)

The hesitation degree of a TCIFSA is

120587A120572 (119909 119905) = 1 minus 120583A (119909 119905) minus 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) (40)

Example 22 Suppose that A(119879) is TCIFS defined on 119883 =1199091 1199092 1199093 with respect to the time set 119879 = 1199051 1199052 1199053 Thedetails of a TCIFS A(119879) are explained in Tables 10 11 and12 Table 13 explained TCIFS A1 when 120572 = 1 and Table 14explained the hesitation degree of a TCIFSA

If 120572 = 1 then one has the following (see Tables 13 and 14)Definition 23 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the timeset 119879 = 1199051 1199052 1199053 119905119898 Then a cosine similarity measurebetweenA(119879) 119886119899119889 B(119879) is proposed as follows

119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(41)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Discrete Dynamics in Nature and Society

Table 7 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 8 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

are explained in Table 7 Table 8 explained TCIFS B(119879)and Table 9 explained a comparison between similaritymeasures 1198780(AB) 1198781(AB) 1198782(AB) 1198783(AB) 1198784(AB) TCIFS A(119879) and TCIFS B(119879)4 Similarity Measures between OtherExtensions of Temporal ComplexIntuitionistic Fuzzy Set

The following definition extend the method proposed byChaira [12] for intuitionistic fuzzy set based on the Sugeno[13] and Omar [10] intuitionistic fuzzy generator

Definition 21 If 120583A(119909 119905)119890119894120572A(119909119905) is the degrees of membershipfunction of the element 119909 isin 119883 at the moment 119905 isin 119879 thennonmembership function ]A(119909)119890119894120573A(119909) = 119866(120583A(119909 119905)119890119894120572A(119909119905))where

119866(120583A (119909 119905) 119890119894120572A(119909119905)) = 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) 120572 gt 0 (38)

And 119866(1) = 0 119866(0) = 1 and by help of the Sugeno [6]intuitionistic fuzzy generator TCIFSA is given by

A120572 (119879) = ((119909 119905) 120583A (119909 119905)sdot 119890119894120572A(119909119905) 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905)) | (119909 119905) isin 119883 times 119879

(39)

The hesitation degree of a TCIFSA is

120587A120572 (119909 119905) = 1 minus 120583A (119909 119905) minus 1 minus 120583A (119909 119905) 119890119894120572A(119909119905)1 + 120572120583A (119909 119905) 119890119894120572A(119909119905) (40)

Example 22 Suppose that A(119879) is TCIFS defined on 119883 =1199091 1199092 1199093 with respect to the time set 119879 = 1199051 1199052 1199053 Thedetails of a TCIFS A(119879) are explained in Tables 10 11 and12 Table 13 explained TCIFS A1 when 120572 = 1 and Table 14explained the hesitation degree of a TCIFSA

If 120572 = 1 then one has the following (see Tables 13 and 14)Definition 23 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the timeset 119879 = 1199051 1199052 1199053 119905119898 Then a cosine similarity measurebetweenA(119879) 119886119899119889 B(119879) is proposed as follows

119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(41)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 9

Table9Acomparis

onbetweensim

ilaritymeasures119878 0(A

B)119878 1(A

B)119878 2(A

B)119878 3(A

B)119878 4(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119878 0(AB )=

09795

119878 0(AB )=

9646119878 0(A

B )=1minus0

235119894119878 0(A

B )=1

119878 0(AB )=

09795

119878 0(AB )=

1119878 1(A

B )=9986

119878 1(AB )=

9958119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B )=9980

119878 1(AB )=

1119878 2(A

B )=100

119878 2(AB )=

1119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=9980

119878 2(AB )=

1119878 3(A

B )=9941

119878 3(AB )=

9823119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B )=9916

119878 3(AB )=

1119878 4(A

B )=8882

119878 4(AB )=

9035119878 4(A

B )=8589

119878 4(AB )=

1119878 4(A

B )=9128

119878 4(AB )=

1119909 2

119878 0(AB )=

1119878 0(A

B )=9575

119878 0(AB)=

1minus0263i

119878 0(AB )=

9882119878 0(A

B )=0930

2119878 0(A

B )=1minus0

263119894119878 1(A

B )=9944

119878 1(AB )=

9819119878 1(A

B )=9930

119878 1(AB)=

9980119878 1(A

B )=9901

119878 1(AB)=

9986119878 2(A

B )=1

119878 2(AB )=

1119878 2(A

B )=9986

119878 2(AB )=

9980119878 2(A

B )=1

119878 2(AB )=

1119878 3(A

B )=9764

119878 3(AB )=

9233119878 3(A

B )=9705

119878 3(AB)=

9916119878 3(A

B )=9583

119878 3(AB)=

9941119878 4(A

B )=7333

119878 4(AB )=

5474119878 4(A

B )=8193

119878 4(AB )=

1119878 4(A

B )=7448

119878 4(AB )=

5454119909 3

119878 0(AB )=

1119878 0(A

B )=1minus0

790119894119878 0(A

B )=1minus0

677119894119878 0(A

B )=9764

119878 0(AB )=

9882119878 0(A

B )=1minus5

892119894119878 1(A

B )=1

119878 1(AB )=

9875119878 1(A

B )=9958

119878 1(AB )=

9968119878 1(A

B )=9986

119878 1(AB)=

9930119878 2(A

B )=1

119878 2(AB )=

9930119878 2(A

B )=9958

119878 2(AB )=

9968119878 2(A

B )=1

119878 2(AB )=

9930119878 3(A

B )=1

119878 3(AB )=

9469119878 3(A

B )=9823

119878 3(AB )=

9868119878 3(A

B )=9941

119878 3(AB)=

9705119878 4(A

B )=1

119878 4(AB )=

5992119878 4(A

B )=5761

119878 4(AB )=

9090119878 4(A

B )=9552

119878 4(AB )=

7018119909 4

119878 0(AB )=

9302minus08

16119894119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9302119878 0(A

B )=8129

119878 0(AB )=

1minus0204119894

119878 1(AB )=

9891119878 1(A

B )=9805

119878 1(AB )=

9986119878 1(A

B )=9891

119878 1(AB )=

9805119878 1(A

B)=9986

119878 2(AB )=

9891119878 2(A

B )=1

119878 2(AB )=

9986119878 2(A

B )=9989

1119878 2(A

B )=1

119878 2(AB )=

9986119878 3(A

B )=9539

119878 3(AB )=

9175119878 3(A

B )=9941

119878 3(AB )=

9539119878 3(A

B )=9175

119878 3(AB)=

9941119878 4(A

B )=8797

7119878 4(A

B )=9137

119878 4(AB )=

8743119878 4(A

B )=8797

119878 4(AB )=

9137119878 4(A

B )=8743

119909 5119878 0(A

B )=9298

+0217119894

119878 0(AB )=

9845+06

97119894119878 0(A

B )=1minus0

333119894119878 0(A

B )=1minus0

236119894119878 0(A

B )=1

119878 0(AB )=

1minus0117119894

119878 1(AB )=

9929119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9972

119878 1(AB )=

1119878 1(A

B)=9986

119878 2(AB )=

9931119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9972

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9899119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9882

119878 3(AB )=

1119878 3(A

B)=9941

119878 4(AB )=

6853119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8576

119878 4(AB )=

1119878 4(A

B )=9170

119909 6119878 0(A

B )=1minus0

471119894119878 0(A

B )=8945

+0697119894

119878 0(AB )=

1minus0333119894

119878 0(AB )=

1minus0204119894

119878 0(AB )=

9923119878 0(A

B )=9973

119878 1(AB )=

9944119878 1(A

B )=9868

119878 1(AB )=

9944119878 1(A

B )=9958

119878 1(AB)=

9929119878 1(A

B)=9968

119878 2(AB )=

9944119878 2(A

B )=1

119878 2(AB )=

9972119878 2(A

B )=9986

119878 2(AB )=

1119878 2(A

B )=1

119878 3(AB )=

9764119878 3(A

B )=9444

119878 3(AB )=

9764119878 3(A

B )=9823

119878 3(AB)=

9699119878 3(A

B)=9868

119878 4(AB )=

7507119878 4(A

B )=8502

119878 4(AB )=

8015119878 4(A

B )=8372

119878 4(AB )=

6138119878 4(A

B )=5364

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Discrete Dynamics in Nature and Society

Theorem 24 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 if A = B and 120572A(119909 119905) = 120573A(119909 119905) =2120587(2) 119862119879(AB) = 119862119879(BA)

(3) minus1 le 119862119879(AB) le 1(4) if 119899 = 119898 = 1 then 119862119879(AB) = 119896(AB)

Proof (1) Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and thetime moments 119879 = 1199051 1199052 1199053 119905119898 The cosine similaritymeasure betweenA(119879) 119886119899119889 B(119879) is given by

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))(42)

IfA =B and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))

= 1(43)

(2)

119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= 1119899119898 ( 119898sum

119895=1

119899sum119894=1

120583B (119909119894 119905119894) 119890119894120572B(119909119905)120583A (119909119894 119905119894) 119890119894120572A(119909119905) + ]B (119909119894 119905119894) 119890119894120573B(119909119905)]A (119909119894 119905119894) 119890119894120573A(119909119905)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909))radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)))= 119862119879 (BA)

(44)

(3) By the same way in (3) in Theorem 21 one has thefollowing

(4) If 119899 = 119898 = 1 then119862119879 (AB)= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572A(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573A(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (AB)= ( 119899sum119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119905)120583B (119909119894 119905119894) 119890119894120572B(119909119905) + ]A (119909119894 119905119894) 119890119894120573A(119909119905)]B (119909119894 119905119894) 119890119894120573B(119909119905)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))= 119896 (AB)

(45)

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 11

Definition 25 Suppose thatA(119879) 119886119899119889 B(119879) is TCIFSs in theuniversal 119883 = 1199091 1199092 1199093 119909119899 with respect to the time set119879 = 1199051 1199052 1199053 119905119898Then the distancemeasure of the angleis proposed as follows

119889 (AB) = cosminus1 (119862119879 (AB)) (46)

Theorem 26 Suppose thatA(119879) andB(119879) are TCIFSs in theuniversal119883 with respect to the time set 119879 Then

(1) 119862119879(AB) = 1 and 120572A(119909 119905) = 120573A(119909 119905) = 2120587 then119889(AB) = 0(2) 119862119879(AB) = 119862119879(BA) then 119889(AB) = 119889(BA)(3) if minus1 le 119862119879(AB) le 1 then 119889(AB) ge 0(4) ifA subeB sube 119862 then 119889(A 119862) le 119889(AB) + 119889(B 119862)

Proof (1) (2) and (3) are simple proof(4) LetA(119879)B(119879) 119886119899119889 119862(119879) be TCIFSs in the universal119883 = 1199091 1199092 1199093 119909119899 with respect to the time set 119879 =1199051 1199052 1199053 119905119898 Then the distance measure of the angle is

proposed as follows

119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) B (119909119894 119905119895)))119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895)))119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))= cosminus1 (119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895)))(47)

where 119894 = 1 2 3 119899 119895 = 1 2 3 119898 and119862119879 (A (119909119894 119905119895) B (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (B (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583B (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))119862119879 (A (119909119894 119905119895) 119862 (119909119894 119905119895))= 1119898119899 ( 119899sum

119894=1

119898sum119895=1

120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583119862 (119909119894 119905119894) 119890119894120572119862(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]119862 (119909119894 119905119894) 119890119894120573119862(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic1205831198622 (119909119894 119905119894) 119890119894(2120587minus120572119862(119909119905)) + ]1198622 (119909119894 119905119894) 119890119894(2120587minus120573119862(119909)))

(48)

If A(119909119894 119905119895) sube B(119909119894 119905119895) sube 119862(119909119894 119905119895) for each 119894 = 1 2 3 119899119895 = 1 2 3 119898 then119889(119894119895) (A (119909119894 119905119895) B (119909119894 119905119895))+ 119889(119894119895) (B (119909119894 119905119895) 119862 (119909119894 119905119895))ge 119889(119894119895) (A (119909119894 119905119895) 119862 (119909119894 119905119895))

(49)

Definition 27 Let A and B be two TCIFSs defined on theuniverse of discourse 119883 = 1199091 1199092 1199093 119909119899 and the timemoments 119879 = 1199051 1199052 1199053 119905119898 Suppose that 119896(AB) iscorrelation coefficient of A and B Then a weight similaritymeasure between TCIFSs A(119879) 119886119899119889 B(119879) is proposed asfollows

120588119879 (A (119909119894 119905119895) B (119909119894 119905119895))= ( 119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895))sdot 120583A (119909119894 119905119894) 119890119894120572A(119909119894 119905119895)120583B (119909119894 119905119894) 119890119894120572B(119909119894 119905119895) + ]A (119909119894 119905119894) 119890119894120573A(119909119894 119905119895)]B (119909119894 119905119894) 119890119894120573B(119909119894 119905119895)radic120583A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909)) + ]A2 (119909119894 119905119894) 119890119894(2120587minus120573A(119909))radic120583B2 (119909119894 119905119894) 119890119894(2120587minus120572B(119909119905)) + ]B2 (119909119894 119905119894) 119890119894(2120587minus120573B(119909)))

(50)

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Discrete Dynamics in Nature and Society

Table 10 TIFSA1199051 1199052 11990531199091 (02 01) (01 06) (03 05)1199092 (06 01) (01 09) (06 04)1199093 (07 01) (01 07) (08 05)Table 11 120572A(119909 119905) = 120573A(119909 119905)1199051 1199052 11990531199091 119890119894(1205872) 119890119894120587 11989011989421205871199092 1198901198942120587 119890119894(1205872) 119890119894(1205872)1199093 119890119894(31205872) 119890119894(31205872) 119890119894(1205872)

Table 12 TCIFSA1199051 1199052 11990531199091 (02119894 01119894) (minus01 minus06) (03 05)1199092 (06 01) (01119894 09119894) (06119894 04119894)1199093 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)Table 13 TCIFSA11199051 1199052 11990531199091 (02119894 1 minus 021198941 + 02119894) (minus01 12) (03 053)

1199092 (06 025) (01119894 1 minus 011198941 + 01119894) (06119894 1 minus 061198941 + 06119894)1199093 (minus07119894 1 + 071198941 minus 07119894) (minus01119894 1 + 011198941 minus 01119894 (08119894 1 minus 081198941 + 08119894)Table 14 The hesitation degree of a TCIFSA

1199051 1199052 11990531199091 0133 minus12 0161199092 015 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 06119894 minus 1 minus 061198941 + 061198941199093 1 minus 07119894 minus 1 + 071198941 minus 07119894 1 minus 01119894 minus 1 minus 011198941 + 01119894 1 minus 08119894 minus 1 minus 081198941 + 08119894And we have the following properties

(1) 120588119879(AB) = 1 thenA =B

(2) 120588119879(AB) = 120588119879(BA)(3) minus1 le 120588119879(AB) le 1

Remark 28 120588119879(A(119909119894 119905119895)B(119909119894 119905119895)) = 119862119879(A(119909119894 119905119895)B(119909119894119905119895)) if119899sum119894=1

119898sum119895=1

119896 (A (119909119894 119905119895) B (119909119894 119905119895)) = 1119898119899 (51)

From a comparison between similarity measures 119862119879(AB)120588119879(AB) we give the following example (the same data inExample 16)

Example 29 Suppose that A(119879) 119886119899119889 B(119879) is TCIFSsdefined on 119883 = 1199091 1199092 1199093 1199094 1199095 1199096 with respect to thetime set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056 The details of a TCIFS A(119879)

are explained in Table 15 Table 16 explained TCIFS B(119879)and Table 17 explained a comparison between similaritymeasures between 119862119879(AB) 120588119879(AB)41 Application in Pattern Recognition andMedical DiagnosisLet 119871 = 1199091 1199092 1199093 1199094 1199095 1199096 be the set of symptoms of thediseases with respect to the time set 119879 = 1199051 1199052 1199053 1199054 1199055 1199056and 1198711 be the set of diagnoses By using the similaritymeasures 119862119879(119871 1198711) we try to discover that the patient maysuffer from one from diseases 119871 which have symptoms 1199091at the time 1199051 and we let 119871 be standard case symptoms ofone of diseases (Table 18) and 1198711 be any case (Table 19)Table 20 explained the similaritymeasures119862119879(119871 1198711) betweena standard case 119871 and any case 1198711

And we define the symptoms of case by Table 19Then Table 20 explained the similarity measures119862119879(119871 1198711) between a standard case 119871 and any case 1198711When the similarity measures minus1 le 119862119879(119871 1198711) le 1 are

small then probability that the patient is suffering from thedisease 119909 at the time 119905 is big and the conversely is true

42 Complex Intuitionistic Fuzzy Topology

Definition 30 An intuitionistic complex fuzzy topology on119883 is a family 120591 of 119862119868119865-sets in 119883 which satisfies the followingproperties

(1) 1 0 isin 120591(2) ifAB isin 120591 thenA capB isin 120591(3) ifB119894 isin 120591 for each 119894 isin Γ then⋃119894isinΓB119894 isin 120591

Then (119883 120591) is called complex intuitionistic fuzzy topologicalspace The elements of 120591 are called 119862119868119865119874-sets and thecomplement of the 119862119868119874-sets is called 119862119868119865119862-setsExample 31 Consider 119883 = 119886 119887 119888 119889 Let AB be a 119862119868119874-subset of119883 as given by

A = (0211989011989413120587 0411989011989405120587119886 1011989011989415120587 0011989011989405120587119887 0711989011989403120587 0211989011989415120587119888 0811989011989411120587 0111989011989407120587119889 )

B = (0211989011989409120587 0311989011989404120587119886 0111989011989402120587 0311989011989409120587119887 0111989011989401120587 0211989011989405120587119888 0511989011989402120587 0111989011989405120587119889 )

(52)

Then 120591 = 0 1ABA cap BA cup B is an complexintuitionistic fuzzy topology on119883

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 13

Table 15 TCIFSA(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 16 TCIFSB(119879)1199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)

Definition 32 If (119883 120591) is called complex intuitionistic fuzzytopological space 119860 sube 119883 then the interior of 119860 is defined asthe union of all 119862119868119865119874-subsets of 119860 and it is denoted by 119860∘That is 119860∘ is the largest 119862119868119865119874-subset of 119860 The closure of 119860is defined as the intersection of all 119862119868119865119862 sets containing 119860and it is denoted by 119860minus That is 119860minus is the smallest 119862119868119865119862-setcontaining 119860Example 33 Consider 119883 = 119886 119887 Let AB be a 119862119868119865-subsetof119883 as given by

A = (0211989011989403120587 0411989011989405120587119886 0211989011989401120587 0111989011989407120587119887 ) B = (0211989011989409120587 0311989011989404120587119886 0511989011989402120587 0111989011989405120587119887 ) 119862 = (0111989011989407120587 0211989011989401120587119886 0411989011989401120587 0111989011989401120587119887 )

(53)

Then 120591 = 0 1AB is an intuitionistic complex fuzzytopology on 119883 119862∘ = 0 = 119909 (0 119890119894(2120587)) 119909 isin 119883 and119862minus = 1 = 119909 (119890119894(2120587) 0) 119909 isin 119883Definition 34 An intuitionistic complex fuzzy topologicalspace (119883 120591) is said to be extremely disconnected if theclosure of each 119862119868119865119874-set is 119862119868119865119874-setDefinition 35 Let (119883 120591) be an intuitionistic complex fuzzytopological space A subsetA of119883 is said to be119862119868119865 semiopenset (by short119862119868119865119878119874-set) (resp CIF preopen (by short119862119868119875119874-set) 119862119868119865 120572-open (by short 119862119868119865120572119874-set) 119862119868119865 120573-open (byshort 119862119868120573119874-119904119890119905) and 119862119868b-open (by short 119862119868119887119874-set) If A subeA∘minus (resp A sube Aminus∘ A sube A∘minus∘ A sube Aminus∘minus and sube A∘minus cupAminus∘) the family of all 119862119868119878119874-set (resp 119862119868119875119874-set 119862119868120572119874-set

119862119868120573119874-119904119890119905 and 119862119868119887119874-set) in119883 is denoted by 119862119868119878119874(119883) (resp119862119868119875119874(119883) 119862119868120572119874(119883) and 119862119868119887119874(119883))The implications between these concepts in the following

diagram and the converse are not true in general

119862119868119865119874dArr dArr119862119868119878119865119874 119862119868119875119865119874dArr dArr119862119868119865120572119874dArr119862119868119887119874dArr119862119868119865 120573-119874

(54)

5 Conclusion

In this paper we introduced and studied a temporal com-plex intuitionistic fuzzy sets as generalization of complexAtanassovrsquos intuitionistic fuzzy sets by taking the time in themoving of the point a correlation between two temporalcomplex intuitionistic fuzzy sets is discussed A similaritybetween temporal complex intuitionistic fuzzy sets is mainpoints in the paper as a generalization of the similarityintroduced by Omar [10] and Sugeno [13] We calculate theresults by the programMaple 7 Finallywe give an applicationsto know if the patient is suffering from the diseases or notand introduce the main building in a topology by usingthe same the set In future research similarity measures

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Discrete Dynamics in Nature and Society

Table17Th

enthes

imilaritymeasures119862 119879(

AB)a

nd120588 119879(A

B)119905 1

119905 2119905 3

119905 4119905 5

119905 6119909 1

119862 119879(AB )=

minus02723

119862 119879(AB )=

minus0010

119862 119879(AB )=

minus0023

119862 119879(AB )=

minus0006

119862 119879(AB)=

minus006119862 119879(A

B )=minus006

120588 119879(AB )=

minus1278

120588 119879(AB )=

minus1900

120588 119879(AB )=

minus2800

120588 119879(AB )=

minus1000

120588 119879(AB )=

0707120588 119879(A

B )=minus100

0119909 2

119862 119879(AB )=

0246119862 119879(A

B )=minus004

5119862 119879(A

B )=minus114

4119862 119879(A

B )=000

119862 119879(AB )=

003119862 119879(A

B )=004

120588 119879(AB )=

3818120588 119879(A

B )=minus072

9120588 119879(A

B )=minus277

2120588 119879(A

B )=minus100

120588 119879(AB )minus

0500120588 119879(A

B )=1356

119909 3119862 119879(A

B )=minus027

7119862 119879(A

B )=minus000

7119862 119879(A

B )=minus000

4119862 119879(A

B )=000

119862 119879(AB )=

minus014119862 119879(A

B )=minus001

120588 119879(AB )=

minus3041

120588 119879(AB )=

minus2844

120588 119879(AB )=

minus6478

120588 119879(AB )=

minus100120588 119879(A

B )minus7300

120588 119879(AB )=

minus0964

119909 4119862 119879(A

B )=minus027

7119862 119879(A

B )=0113

119862 119879(AB )=

minus0128

119862 119879(AB )=

0051119862 119879(A

B )=011

119862 119879(AB )=

minus012120588 119879(A

B )=minus304

0120588 119879(A

B )=2548

120588 119879(AB )=

minus1333

120588 119879(AB )=

minus9997

120588 119879(AB )=

7297120588 119879(A

B )=minus297

1119909 5

119862 119879(AB )=

0265119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )minus

006119862 119879(A

B )=minus002

120588 119879(AB )=

4110120588 119879(A

B )=minus639

3120588 119879(A

B )=minus614

8120588 119879(A

B )=minus419

8120588 119879(A

B )=minus100

120588 119879(AB )=

minus6453

119909 6119862 119879(A

B )=minus026

5119862 119879(A

B )=0048

119862 119879(AB )=

minus0044

119862 119879(AB )=

minus0008

119862 119879(AB )=

003119862 119879(A

B )=minus003

120588 119879(AB )=

minus1300

120588 119879(AB )=

minus5500

120588 119879(AB )=

minus3800

120588 119879(AB )=

minus6519

120588 119879(AB )=

minus588120588 119879(A

B )=0266

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 15

Table 18 Standard case symptoms of one of diseases 1198711199051 1199052 1199053 1199054 1199055 11990561199091 (02119894 03119894) (minus01119894 minus03119894) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199092 (06 01) (01119894 09119894) (06119894 04119894) (01119894 01119894) (minus01 minus06) (03 05)1199093 (minus07119894 minus01119894) (minus08119894 minus01119894) (08119894 05119894) (02 01) (01119894 09119894) (06119894 04119894)1199094 (06 01) (01119894 09119894) (06119894 04119894) (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894)1199095 (02119894 01119894) (minus01 minus06) (03119894 05119894) (02119894 01119894) (02119894 01119894) (02119894 01119894)1199096 (06119894 01119894) (01119894 09119894) (06119894 04119894) (01119894 02119894) (minus01 minus06) (03 05)

Table 19 The symptoms of case L11199051 1199052 1199053 1199054 1199055 11990561199091 (01119894 01119894) (minus01119894 minus06119894) (01119894 05119894) (02119894 01119894) (02 01) (02119894 01119894)1199092 (04 03) (07119894 02119894) (09119894 02119894) (01 01119894) (01119894 06119894) (01 02)1199093 (minus07119894 minus01119894) (minus01119894 minus03119894) (01119894 01119894) (01119894 01) (01119894 08119894) (01119894 04119894)1199094 (minus07119894 minus01119894) (minus01119894 minus07119894) (08119894 05119894) (06 01) (01119894 09119894) (06119894 04119894)1199095 (06 01) (01119894 09119894) (06119894 04119894) (04119894 01119894) (02119894 01119894) (01119894 01119894)1199096 (02119894 01119894) (minus01 minus06) (03119894 05119894) (03119894 01119894) (02119894 01119894) (02119894 01119894)Table 20 Similarity measures 119862119879(119871 1198711)1199051 1199052 1199053 1199054 1199055 11990561199091 119862119879 (119871 1198711) = minus02723 119862119879 (119871 1198711) = minus0010 119862119879 (119871 1198711) = minus0023 119862119879 (119871 1198711) = minus0006 119862119879(119871 1198711) = minus006 119862119879 (119871 1198711) = minus0061199092 119862119879 (119871 1198711) = 0246 119862119879 (119871 1198711) = minus0045 119862119879 (119871 1198711) = minus1144 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = 0041199093 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = minus0007 119862119879 (119871 1198711) = minus0004 119862119879 (119871 1198711) = 000 119862119879 (119871 1198711) = minus014 119862119879 (119871 1198711) = minus0011199094 119862119879 (119871 1198711) = minus0277 119862119879 (119871 1198711) = 0113 119862119879 (119871 1198711) = minus0128 119862119879 (119871 1198711) = 0051 119862119879 (119871 1198711) = 011 119862119879 (119871 1198711) = minus0121199095 119862119879 (119871 1198711) = 0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) minus 006 119862119879 (119871 1198711) = minus0021199096 119862119879 (119871 1198711) = minus0265 119862119879 (119871 1198711) = 0048 119862119879 (119871 1198711) = minus0044 119862119879 (119871 1198711) = minus0008 119862119879 (119871 1198711) = 003 119862119879 (119871 1198711) = minus003

between temporal complex multifuzzy soft and applicationsin engineering medical physics and automobiles will bestudied

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally

Acknowledgments

Sayer Obaid Alharbi thanks Deanship of Scientific Research(DSR) for providing excellent research facilitiesThe publica-tion costs of this article were partially covered by the EstonianAcademy of Sciences

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[3] N Palaniappan andR Srinivasan ldquoApplications of intuitionisticfuzzy sets of root type in image processingrdquo in Proceedings ofthe NAFIPS 2009 - 2009 Annual Meeting of the North AmericanFuzzy Information Processing Society pp 1ndash5 Cincinnati OHUSA June 2009

[4] E Szmidt and J Kacprzyk ldquoAn application of intuitionisticfuzzy set similaritymeasures to amulti-criteria decisionmakingproblemrdquo in Proceedings of the Artificial Intelligence and SoftComputingmdashICAISC vol 4029 pp 314ndash323 Zakopane Poland2006

[5] I K Vlachos and G D Sergiadis ldquoIntuitionistic fuzzyinformationmdashapplications to pattern recognitionrdquo PatternRecognition Letters vol 28 no 2 pp 197ndash206 2007

[6] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets and Systemsvol 33 no 3 pp 333ndash345 1989

[7] H Nguyen A Kandel and V Kreinovich ldquoComplex fuzzy setstowards new foundationsrdquo in Proceedings of the Ninth IEEEInternational Conference on Fuzzy Systems FUZZ-IEEE 2000Soft Computing in the Information Age pp 1045ndash1048 SanAntonio TX USA 2000

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

16 Discrete Dynamics in Nature and Society

[8] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[9] D RamotM Friedman G Langholz andA Kandel ldquoComplexfuzzy logicrdquo IEEE Transactions on Fuzzy Systems vol 11 no 4pp 450ndash461 2003

[10] Omar and G Maejo ldquoSimilarity measures between temporalintuitionistic fuzzy setsrdquoThe International Journal of Science ampTechnology vol 11 no 3 pp 275ndash284 2017

[11] S Dick ldquoToward complex fuzzy logicrdquo IEEE Transactions onFuzzy Systems vol 13 no 3 pp 405ndash414 2005

[12] T Chaira ldquoIntuitionistic fuzzy segmentation of medicalimagesrdquo IEEE Transactions on Biomedical Engineering vol 57no 6 pp 1430ndash1436 2010

[13] M Sugeno ldquoFuzzy measures and fuzzy integrals a surveyrdquo inFuzzy Automata and Decision Processes M M Gupta G NSaridis and B R Gaines Eds pp 89ndash102 Noth Holland NewYork NY USA 1977

[14] A S Alkouri and A R Salleh ldquoComplex intuitionistic fuzzysetsrdquo in Proceedings of the 2nd International Conference onFundamental and Applied Sciences 2012 ICFAS 2012 pp 464ndash470 Malaysia June 2012

[15] A M Alkouri and A R Salleh ldquoComplex atanassovrsquos intuition-istic fuzzy relationrdquo Abstract and Applied Analysis vol 2013Article ID 287382 18 pages 2013

[16] G Zhang T S Dillon K Y Cai J Ma and J Lu ldquoOperationproperties and 120575-equalities of complex fuzzy setsrdquo InternationalJournal of Approximate Reasoning vol 50 no 8 pp 1227ndash12492009

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom