7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
1/96
CLASSICAL AND FUZZY SETS
Dr S.Natarajan
Professor,
Department of Information Science and Engineering
PESIT, Bangalore
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
2/96
2
Classical Sets and Fu! Sets
Classical Sets " #perations on Classical Sets, Properties ofClassical $Crisp% Sets, &apping of Classical Sets to Functions,
Fu! Sets " Fu! Set operations, Properties of Fu! Sets,
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
3/96
'
( )oolean logic
( fu! logic
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
4/96
*
Crisp set vs. Fuzzy set
A traditional crisp set A fuzzy set
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
5/96
5
DEFINITIONS- CLASSICAL SETS
Sets Classical sets either an element belongs to the set or
it does not. For example, for the set of integers, eitheran integer is even or it is not (it is odd). However,either o! are in the "S# or o! are not. $hat abo!t%ing into "S#, what happens as o! are crossing&
#nother example is for blac' and white photographs,one cannot sa either a pixel is white or it is blac'.However, when o! digitie a bw *g!re, o! t!rn allthe bw and gra scales into +5 discrete tones.
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
6/96
+
Crisp set vs. Fuzzy set
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
7/96-
# classical(crisp) set Ain the !niverse of disco!rse
Ucan be de*ned in three was/ b enumerating(listing) elements (often called listlistor extensionalextensionalde*nition)
/ b specifing the common propertiesof elements(intensionalintensionalor r!ler!le de*nition)
the notation A = {x !"x#$means that set Aiscomposed of elements x s!ch that everyxhas theproperty!"x#
/ b introd!cing a %ero-one mem&ers'ip (unction
(characteristiccharacteristicorindicatorindicatorde*nition)
3bydivisiblenotis!ndU"i#$"
3bydivisibleis!ndU"i#"%&'('&
3)bydivisibleis*U+ Anu(bers'"inte,ero#&setU
AA
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
8/960
CLASSICAL SETS ")#
Classical sets are also called crisp(sets).
1ists # 2 3apples, oranges, cherries,mangoes4
# 2 3a,a+,a64
# 2 3+, 7, , 0, 84
Form!las # 2 3x 9 x is an even nat!ral
n!mber4 # 2 3x 9 x 2 +n, n is a nat!ral
n!mber4
:embership or characteristic f!nction
=
Ax
Axx
A if0
if1)(
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
9/96
9
Characteristic function
Let A be any subset of X, the characteristic
function of A, denoted by , is defined by
Characteristic function of the set of real numbers
from to 10
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
10/96
-
CLASSICAL SETS &-'/ 0 collection of o)jects all a/ing te same caracteristics 00
1NIE3SE #F DISC#13SE 000 indi/idual elements in te
1ni/erse 4 5ill )e denoted as 6E6amples7
( Te cloc8 speeds of computer CP1s 9ert
( Te operating currents of an electronic motor
( Te operating temperature of a eat pump $ in degrees Celsius%
( Te 3icter magnitude of an eart:ua8e( Te integers to -
First * items are e6amples of real 5orld engineering elements
For te purpose of modeling , tese engineering pro)lems are
simplified and onl! integer /alues of te elements are considered&agnitudes of 3icter Scale greater tan ;
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
11/96
CLASSICAL SETS &3'
( 1seful metric " cardinalit! or te cardinal num)er " total num)er of
elements in a uni/erse 4 is called its cardinal num)er denoted )! n6,
6 is te inde6
Discrete uni/erses tat are composed of a counta)l! finite collection
of elements 5ill a/e a finite cardinal num)er A continuous uni/erses
comprising of an infinite collection of elements 5ill a/e an infinitecardinalit!
Sets " collections of elements 5itin an uni/erse
Su)sets " collection of elements 5itin te sets
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
12/96
1!
"asic concepts
#et$ a collection of items
%o &epresent sets
' List method Aa, b, c*' &ule method C x + (x) *
' -amily of sets Ai+ i. *
' /niersal set X and empty set
%he set C is composed of elementsx
ery x has property
i$ inde2 .$ inde2 set
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
13/96
13
#et .nclusion
A" $xA implies thatx"
A " $ A" and " A
A" $ A" and A"
A is a subset of "
A and " are e4ual set
A is a proper subset of "
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
14/96
15
o6er set
All the possible subsets of a 7ien set X is call the
po6er set of X, denoted byP(X) A+ AX*
|P(X) + !n6hen +X+ n
Xa, b, c*
(X) , a, b, c, a, b*, b, c*, a, c*, X*
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
15/96
1per!tions on Cl!ssi2!l Sets
1nion7B @ 6 6 or 6 B
Intersection7B @ 6 6 and 6 B
Complement7
@ 6 6 , 6 4
4 " 1ni/ersal Set
Set Difference7
B @ 6 6 and 6 B
Set difference is also denoted )! 0 B
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
16/96
18
#et perations
Complement
/nion
.ntersection
difference
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
17/96
UNI1N 1F T1 SETS
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
18/96
G
INTE4SECTI1N 1F T1 SETS
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
19/96
;
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
20/96
2-
C156LE5ENT 1F A SET
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
21/96
2
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
22/96
22
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
23/96
6roperties o# Cl!ssi2!l Sets
B @ B B @ B
$B C% @ $ B% C$B C% @ $ B% C
$B C% @ $ B% $C%$B C% @ $ B% $C%
@
@
4 @ 44 @ @
@
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
24/96
6roperties o# Cl!ssi2!l Sets
If B C, ten C;e :organ:s 1aw
(#
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
25/96
!
&eal numbers
%otal orderin7$ a b
&eal a2is$ the set of real number (x;a2is)
.nteral$
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
26/96
2+
De 5or,!n7s l!8s
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
27/96
2H
Distri)uti/e la5s
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
28/96
2G
Asso2i!tive l!8s
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
29/96
Can be extended to n sets
?eneralied ;e :organ 1aw# #:
@@
"sing ( ) to 'eep original processing order
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
30/96
?eneralied ;!alit 1aw
@
@
"sing ( ) to 'eep original processing order
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
31/96
1aw of the excl!ded middle
# #:2 @
1aw of the Contradiction
# #:2
T'ese la*s are not true (orFu%%+ Sets,
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
32/96
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
33/96
''
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
34/96
'*
5!ppin, o# Cl!ssi2!l Sets to Fun2tions( 3elates set0teoretic forms to function teoretic representation of
information( &ore generall!, it can )e used to map elements or su)sets on one
uni/erse of discourse to elements or sets in anoter uni/erse
( Suppose 4 and 1 are t5o uni/erses of discourse $information%( If an element 6 is contained in 4 and corresponds to an element !
contained in , it is generall!termed as mapping from 4 to ,of fA 4 0?
. As a mapping, the characteristic (indicator) function
expresses membership in set A for the element x in the universe
This membership idea is a mapping from an element x in the universe X to
one of the two elements in universe Y i!e!, to the elements " or #$or an% set A defined on the universe X, there exists a function&theoretic set,
called a 'alue et, denoted b% '(A), under the mapping of the characteristic
function, x! % convention the null set is assigned the membershipvalue " and the whole set X is assigned the membership value #
= Ax
Axx
A if0
if1
)(
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
35/96
'J
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
36/96
'+
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
37/96
'H
Fuzzy T9eory
Fuzzy Fuzzy Un2ert!inty : 5!t9e(!ti2s De2ision;5!
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
38/96
'G
( fu! sets " the truth of a statement
becomes a matter of degree
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
39/96
';
( fu! mem)ersip functions
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
40/96
*-
( fu! mem)ersip functions$anoter e6ample%
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
41/96
*
De#initions 0 #uzzy sets
Fuzzy sets"
admits gradation suc as all tones )et5een)lac8 and 5ite. fu! set as a grapicaldescription tat e6presses o5 te transition fromone to anoter ta8es place. Tis grapicaldescription is called a mem)ersip function.
De#inition 7 Ket 4 )e some set of o)jects, 5itelements noted as 6.
" Tus, 4 @ 6. For e6ample, if 4 5ere to e:ualte set of all common ouse pets, ten
" 4 @ dogs, cats, fis, )irds, 5ere 6 @ dogs,62 @ cats, 6' @ fis, and 6* @ )irds.
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
42/96
Fuzzy Sets
Caracteristic function 4, indicating te )elongingness
of 6 to te set
4$6% @ 6 - 6
or called (e(bers9ip
9ence,B 4B$6%
@ 4$6% 4B$6%@ ma6$4$6%,4B$6%%
Note Some )oo8s use L for , )ut still it is not ordinar!additionM
Some more e6planations follo5
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
43/96
Fuzzy Sets
B 4B$6%
@ 4$6% 4B$6%
@ min$4$6%,4B$6%%
4$6% @ " 4$6%
B 4$6%
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
44/96
Fuzzy Sets
Note $6% O-,
not -, li8e Crisp set
@
$6% Q 6 L
$62% Q 62 L
@ $6i% Q 6i
Note7 RL add
RQ di/ide
#nl! for representing element and its
mem)ersip.
lso some )oo8s use $6% for Crisp Sets too.
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
45/96
Fuzzy Set 1per!tions
B$6% @ $6% B$6%
@ ma6$$6%, B$6%%
B$6% @ $6% B$6%
@ min$$6%, B$6%%
$6% @ 0 $6%
De &organs Ka5 also olds7$B% @ B$B% @ B
But, in general
X
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
46/96
*+
Fu! SetFu! Setis a set 5it a smoot )oundaries
Fu! Set Teor!Fu! Set Teor!generalies classical set teor!to allo5partial membership
Fu! Setis a uni/ersal set 4 is determined )! a
(e(bers9ip #un2tion$6%$6%tat assigns to eac
element 64 a num)er $6% in te unit inter/alO-,
1ni/ersal set 4 $Universe o# Dis2ourse% contains all
possi)le elements of concern for a particular
application Fu! set as a one0to0one correspondenceone0to0one correspondence
5it its mem)ersip function
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
47/96
*H
Fu! setis defined as
A = + &" A&'' )" /" A&'>$"%?
$6% @ Degree$6Degree$6%%is a ,r!de o# (e(bers9ipof
element 64 in set
/% /- /3 . $ %@- %
.. unit interv!l
N
.
.
U &universe o# dis2ourse'
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
48/96
*G
&em)ersip functions can )e represented &!'graphically,
&b'in a tabularor listform, &2 'analyticallyand &d'
geometrically$as a points in te unit 2ube%
eometrical representation for t5o0element uni/ersal set
U= &+%"-)'as a follo5ing /iualiation7 (e(bers9ip v!lues
%.$ &$"%' &%"%'
-
%
$.$ U &$"$'
% - % &%"$'
,r!p9i2!l &st!nd!rd' set o# (!i(u( #uzzinessB represent!tion #or(
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
49/96
*;
[see the previous figure] ertices $-,-%, $-,%,
$,-% and $,% represent !ll 2risp setstat can )e
defined for te uni/ersal set U, e.g. te point $,-%
corresponds to te crisp set +%) $element -as
no membership%
&em)ersip functions can )e symmetricalor
asymmetrical, and te most commonl! used forms
are triangulartriangular, trapeoidaltrapeoidal, aussianaussianand )ell)ell$te
first t5o dominate in applications due to si(pli2ity
and computational e##i2ien2y%
&em)ersip functions are typicallydefined on
one0dimensional uni/erses, and in most cases, te
mem)ersip function appears in te continuoscontinuos
formform
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
50/96
6roperties o# Fuzzy Sets
B @ B B @ B
$B C% @ $ B% C$B C% @ $ B% C
$B C% @ $ B% $C%$B C% @ $ B% $C%
@ @
4 @ 4 4 @ @ @
If B C, ten C
@
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
51/96
J
Te eigt of a fu! set eigt of a fu! set is te igest $ma6imum% /alue
of its mem)ersip function, i.e. 9ei,9t&A' =
If a fu! set as a 9ei,9t %, ten it is called a normal fu!normal fu!
setsetA in contrast, if eigt$% , te fu! set is said to )e
su)normalsu)normal
su)normal set is a fu! set tat contains onl! elements
5itpartial (
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
52/96
J2
set of all elements of te uni/ersal set U5it a propert!
A&' = %$ is a fu! set% is called te core of a fu! set core of a fu! set
$2ore&A'%
%
$
! b U = >!"b?
2ore&A'
supp&A'
9ei,9t&A' = % (normal fuzzy set)
5e(bers9ip#un2tion 9!s !
tr!pezoid!l #or(
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
53/96
J'
Fu! SetFu! Setis a set 5it a smoot )oundaries
Fu! Set Teor!Fu! Set Teor!generalies classical set teor!
to allo5partial membership
Fu! Setis a uni/ersal set 11is determined )! a
(e(bers9ip #un2tion$6%$6%tat assigns to eacelement 61 a num)er $6% in te unit inter/alO-,
1ni/ersal set 11$Universe o# Dis2ourse% contains all
possi)le elements of concern for a particular
application
Fu! set as a one0to0one correspondenceone0to0one correspondence
5it its mem)ersip function
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
54/96
J*
In#or(!tion orld
Crisp set as a uni:ue mem)ersip function
$6% @ 6
- 6
$6% -,
Fu! Set can a/e an infinite num)er of mem)ersip
functionsO-,
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
55/96
55
Fuzziness
E6amples7
num)er is close to J
Fuzziness
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
56/96
J+
Fuzziness
E6amples7
9eQse is tall
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
57/96
JH
E6ample7 =oung>
( E6ample7
" nn is 2G, -.G in set =oung>
" Bo) is 'J, -. in set =oung>
" Carlie is 2', .- in set =oung>
( 1nli8e statistics and pro)a)ilities, te degreeis not descri)ing
probabilitiestat te item is in te set, )ut instead descri)es to what
extent te item is te set.
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
58/96
JG
Fuzzy Subset
1 @ , 2, ', *,.,-
@ , 2, ', *, J
B @ 2, ', *
B in C3ISP SET T9E#3
$6% ?@ B$6%, 6
In terms of mem)ersip predicate
Crisp su)setood
S,$6% @ S2$6%, 6
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
59/96
J;
eo(etri2 Interpret!tion&%'
$-,%$,%
$-,-% $,-%
B
B2 B'
62
6
1 @ 6 , 62
Bis are suc tat
Bi$6% @ $6%, 6
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
60/96
+-
eo(etri2 Interpret!tion &-'
( Te points 5itin te !percu)e for 5ic is te upper
rigt corner are te su)sets of .
( Space defined )! te s:uare is te po5er set of .
( Formulation of ZADE, classical fu! set teor!
( For B to )e a su)set of , B$6% @ $6%, 6.
Tis means B P$% crispl!.
B
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
61/96
+
eo(etri2 Interpret!tion &3'
( Eac Bi is a su)set of to some degree.
BB2
B'
( 3esult of 1nion, Intersection, Complement is a SET
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
62/96
+2
5e(bers9ip #un2tion o# #uzzy lo,i2
ge2J *- JJ
Youn, 1ld
5iddle
-.J
D#&
Degree of
Membership
Fu! /alues
Fu! /alues a/e associated degrees of mem)ersip in te set.
-
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
63/96
+'
5e(bers9ip Fun2tions o# t9e e!(ple on !,e
% 89en = -
A%&' = &G$;'@% 89en - G$
$ 89en H=G$
$ 89en eit9er = - or H=
A-&' = &;-'@% 89en - G$ &;'@% 89en G$
$ 89en = G$
A3&' = &;G$'@% 89en G$
% 89en H=
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
64/96
+*
5e(bers9ip Fun2tions usin, Fuzzy Sets 0Z!de97s Not!tion
@ .-Q- L .-Q2J L -.JQ'2.J L -.-Q*-
& @ -.-Q2J L -.JQ'2.J L .-Q*- L -.JQ*H.J L -.-QJJ
# @ -.-Q*- L -.JQ *H.J L .-QJJ
Linguistic Values
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
65/96
Linguistic Values
>
embership
0
1
Heightshort medium tall
180 1?0 1@0 (cm)
>embership
0
1
Weightli7ht medium heay
0 ?0 90 (7)
Definitions7 Fu! Sets $figure from UlirVuan%
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
66/96
++
Definitions7 Fu! Sets $figure from UlirVuan%
&em)ersip functions $figure from UlirVuan%
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
67/96
+H
p $ g %
F set $fig re from Earl Co %
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
68/96
+G
Fu! set $figure from Earl Co6%
Fu! Set $figure from Earl Co6%
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
69/96
+;
Fu! Set $figure from Earl Co6%
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
70/96
H-
Fuzzy Set 1per!tions
1nion 7 )()(BBBB
xx BABA =
Intersection 7 )()(BBBB
xxBABA
=
Complement 7 )(1)(BB
xxAA
=
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
71/96
?1
Operations of Fuzzy SetsOperations of Fuzzy Sets
.ntersection /nion
Complement
ot A
A
Containment
A
B
BA A B
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
72/96
?!
ComplementComplement
Crisp #ets$ ho does not belon7 to the setD -uzzy #ets$ Eo6 much do elements not belon7 to the setD
%he complement of a set is an opposite of this setF -or
e2ample, if 6e hae the set of tall men, its complement is
the set of G% tall menF hen 6e remoe the tall men set
from the unierse of discourse, 6e obtain the complementF
.f A is the fuzzy set, its complement Acan be found asfollo6s$
A(x) 1 A(x)
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
73/96
?3
ContainmentContainment
Crisp #ets$ hich sets belon7 to 6hich other setsD -uzzy #ets$ hich sets belon7 to other setsD
#imilar to a Chinese bo2, a set can contain other setsF %he
smaller set is called the subsetF -or e2ample, the set of tall
men contains all tall menH ery tall men is a subset of tall
menF Eo6eer, the tall men set is Iust a subset of the set of
menF .n crisp sets, all elements of a subset entirely belon7
to a lar7er setF .n fuzzy sets, ho6eer, each element canbelon7 less to the subset than to the lar7er setF lements of
the fuzzy subset hae smaller memberships in it than in the
lar7er setF
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
74/96
?5
IntersectionIntersection
Crisp #ets$ hich element belon7s to both setsD -uzzy #ets$ Eo6 much of the element is in both setsD
.n classical set theory, an intersection bet6een t6o sets contains the
elements shared by these setsF -or e2ample, the intersection of the set
of tall men and the set of fat men is the area 6here these sets oerlapF
.n fuzzy sets, an element may partly belon7 to both sets 6ith different
membershipsF
A fuzzy intersection is the lower membershipin both sets of each
elementF %he fuzzy intersection of t6o fuzzy setsAandBon unierse
of discourse X$
AB(x) min
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
75/96
?
UnionUnion
Crisp #ets$ hich element belon7s to either setD -uzzy #ets$ Eo6 much of the element is in either setD
%he union of t6o crisp sets consists of eery element that falls into
either setF -or e2ample, the union of tall men and fat men contains all
men 6ho are tall & fatF
.n fuzzy sets, the union is the reerse of the intersectionF %hat is, the
union is the largest membership alue of the element in either setF
%he fuzzy operation for formin7 the union of t6o fuzzy sets A and "
on unierse X can be 7ien as$
A"(2) ma2
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
76/96
?8
Operations of Fuzzy SetsOperations of Fuzzy Sets
Complement
0x
1
(x)
0x
1
Containment
0x
1
0x
1
A B
Not A
A
.ntersection
0x
1
0x
A B
/nion
0
1
A B
A B
0x
1
0x
1
BA
B
A
(x)
(x) (x)
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
77/96
HH
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
78/96
HG
De 5or,!n7s L!8s
BBBB
BABA =
BBBB
BABA =
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
79/96
H;
6roperties o# Fuzzy Set
Commutati/it! BBBB ABBA =
BBBBABBA =
ssociati/it!BBBBBB
CBACBA =
BBBBBB CBACBA =
CABACBA
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
80/96
G-
Distri)uti/it! BBBBBBBCABACBA =
BBBBBBBCABACBA =
Idempotenc!BBB
AAA =BBBAAA =
Identit! BBAA =
=B
A
XXA =B
BBAXA =
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
81/96
G
Transiti/it!BBBBB
CAthenCBAIf
In/olutionBB
AA =
Artificial .ntelli7ence ' C#385Artificial .ntelli7ence ' C#385-uzzy Lo7ic-uzzy Lo7ic
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
82/96
@!
!uality!uality
-uzzy setAis considered e4ual to a fuzzy setB, .- AGJGLK .- (iff)$
A(x) B(x), xX
A 0F31 M 0F! M 13
B 0F31 M 0F! M 13
thereforeAB
Artificial .ntelli7ence ' C#385Artificial .ntelli7ence ' C#385-uzzy Lo7ic-uzzy Lo7ic
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
83/96
@3
InclusionInclusion
.nclusion of one fuzzy set into another fuzzy setF -uzzy setA Xis included in (is a subset of) another fuzzy set,B X$
A(x) B(x), xX
ConsiderX 1, !, 3* and setsAandB
A 0F31 M 0F! M 13H
B 0F1 M 0F! M 13
thenAis a subset ofB, or AB
Artificial .ntelli7ence ' C#385Artificial .ntelli7ence ' C#385-uzzy Lo7ic-uzzy Lo7ic
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
84/96
@5
Car"inalityCar"inality
Cardinality of a non;fuzzy set, N, is the number of elements in NF"/% the cardinality of a fuzzy set A, the so;called #.O>A C/G%, is
e2pressed as a #/> of the alues of the membership function of A,
A(x)$
cardA A(x1) M A(x!) M P A(xn) QA(xi), for i1FFn
ConsiderX 1, !, 3* and setsAandB
A 0F31 M 0F! M 13H
B 0F1 M 0F! M 13
cardA 1F@
cardB !F0
Artificial .ntelli7ence ' C#385Artificial .ntelli7ence ' C#385-uzzy Lo7ic-uzzy Lo7ic
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
85/96
@
mpty Fuzzy Setmpty Fuzzy Set
A fuzzy setAis empty, .- AGJ GLK .-$A(x) 0, xX
ConsiderX 1, !, 3* and setA
A 01 M 0! M 03
thenAis empty
E6ample7 n uni/erse as tree elements, 4@ a,),c . We desire to
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
86/96
G+
map te elements of te po5er set of 4, i.e., P$4%, to a uni/erse, ,
consisting of onl! t5o elements $ te caracteristic function% @ -,
Te elements of te po5er set are enumerated as follo5s7
P$4% @
" a, ), c, a,), ),c, $a,c, $a,),c
Tus, te elements in te /alue set $% as determined from te
mappings are
P$4% @ -,-,-, ,-,-, -,,-, -,-,, ,,-, -,,, ,-,,,,
For e6ample, te tird su)set in te po5er set P$4% is te element ).
For tis su)set tere is no a, so a /alue of - goes in te first
position of te data tripletA tere is a ), so a /alue of goes in te
second position of te data tripletA and tere is no c, so a /alue of -goes in te tird position of te data triplet. 9ence, te tird su)set
of te /alue set is te data triplet -,,-, as alread! seen.
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
87/96
GH
E!(ple
+++=
+++=
5F
5!F
3?F
!F
!F
53F
3F
!1
BBBA
++++=
++++=
8F
5
@F
3
3F
!
F
1
1
@F
5
?F
3
F
!
0
1
1
BB
BA
+++=
5F
5
3F
3
?F
!
1
BBBA
+++=
!F
5
!F
3
F
!
F
BBBA
!33
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
88/96
GG
+++==
!F
5
3F
3
3F
!
F+
BBBBBABA
+++==
5F
5
!F
3
F
!
0+BBBB
ABAB
++++==
8F
5
?F
3
3F
!
0
1
1
BBBB BABA
++++==
@F
5
@F
3
F
!
F
1
1
BBBBBABA
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
89/96
G;
++++=
@F
5
?F
3
F
!
1
1
1
BBAA
+++=
5F
5
!F
3
3F
!
F
BBBB
E l f t ti
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
90/96
;-
E6ample fu! set operationsA:
A" A"
A "
A
#perations on Fu! Sets
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
91/96
;
( Sligtl! differs from regular set operations
Fu! inter/al )et5een J V G Fu! num)er a)out *
AND" 14" NEATI1N
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
92/96
;2
" "
Canalso )e called intersect, unif!, and negate
ND #3 NETI#N
oung emplo!ees salar!
7/26/2019 CLASSICAL AND FUZZY SETS-VTU.ppt
93/96
;'
g p ! !( Xuestion7 5at is a !oung emplo!ees salar!
Top Related