BERHAMPUR SOFT COMPUTING (PECS 3401)-FUZZY...

AS PER THE SYLLABUS OF BPUT FOR SEVENTH

Lecture Notes | KISHORE KUMAR SAHU

RIT, BERHAMPUR SOFT COMPUTING (PECS 3401)-FUZZY LOGIC SEMESTER OF AE&IE BRANCH.

LECTURE NOTES ON SOFT COMPUTING P a g e | 2

BY: KISHORE KUMAR SAHU, DEPT OF INFORMATION TECHNOLOGY, RIT, BERHAMPUR.

CHAPTER 01 INTRODUCTION TO SOFT COMPUTING

ORIGIN OF SOFT COMPUTING Artificial intelligence is a branch of computer science dealing with building of system that exhibits automation in intelligent behaviour. But intelligence is not very well defined due to which the tasks associated with intelligence i.e. learning, intuition, creativity and decision making also seems to partially understood. This quest of building intelligent systems gave rise to new techniques that aided a lot in building intelligent problem solvers. Some of them are expert system, neural networks, fuzzy logic, cellular automata and probabilistic reasoning. Out of this fuzzy logic, neural networks and probabilistic reasoning are called as soft computing. The term soft computing was coined by Lotfi A. Zadeh. Soft computing differs from hard computing (conventional computing) in its tolerance to imprecise, uncertain and partial truth. Hard computing basically deals with mathematical approaches that demands a great degree of precision and accuracy. But in engineering problems, it is very difficult to determine the input with great degree of precision. Hence a best estimate is made to find the solution which restricts the use of mathematical approaches, especially in inverse problems. The application of soft computing to the inverse problems is to exploit the tolerance for imprecision, uncertainty and partial truth to achieve tractability, robustness and low cost solutions.

CONSTITUENTS OF SOFT COMPUTING The basic components of soft computing are as follows: • Fuzzy Systems (models uncertainty in the system) • Neural Networks (models biological neuron of human brain) • Genetic Algorithm (selection of a good solution from a solution set)

FUZZY LOGIC It is a mathematical tool to model uncertainty in the system. This is applicable where imperfection in data is present. Basically used to represent uncertainty that arises due to generality, vagueness, ambiguity, chance or incomplete knowledge. In fuzzy sets the members of the sets are associated with a value representing its grade of membership in the fuzzy set. This is in contrast to that of classical set

called as crisp set i.e. the members are either belongs to the set or not. But each member in the fuzzy set is associated with a membership value between [0, 1]. Let A={ram, kiran, john} and their corresponding height are 6, 5.5 and 5.1 feets respectively. Now Height >6ft(90%), Height<5ft(10%), and Height=5ft(50%). So set A can be represented as {0.9/ram,0.5/kiran, 0.1/john} where the fractional values represents the membership values. ARTIFICIAL NEURAL NETWORKS ANN is based on learning and testing theory. A set of data is needed to learn things. f(x)=x2 in the range (0,7) x : 0 1 2 3 4 5 6 7 f(x) : 0 1 4 9 16 25 36 49 There are massively highly interconnected N/W of processing elements called neurons. There are unorthodox search and optimization algorithms insipid by the biological process. EVOLUTIONARY TECHNIQUES This is also called as genetic process. It is used to mimic natural evolution. GA makes a random search through a given set of alternate solution to find the best alternate solution w.r.t the given criteria of goodness. e.g. ant colony optimization, swarm intelligence etc. Some of the techniques that are combination of these basic components of soft compounding are called as hybrid techniques. Some of these are as follows: neuro-fuzzy, neuro-GA, fuzzy-GA. ADVANTAGES OF SOFT COMPUTING • Model based on human reasoning • Models can be simple and accurate • Fast computing • Good in practice

APPLICATION AREAS • Process control • Pattern recognition• Robotics • Time series forecasting • Optimization techniques • Medicine and diagnosis

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inality: The numbesented as |A| or #Aly of sets: A sets we.g. A={{1,2,3},{2,3empty sets: A set csented as ∅ or {}. Aeton set: A set wnality one. e.g. A={et: Given two set Ained in the set A, thrset: Given two setined in the set A, . r set: A power setor a set A={1,2}, thRATIONS ON Cn: Union operationsented by AUB thnt in A and B. {a,b,c,d,f}. section: Intersectiher set representedresent in A as weAnB={b,c}.

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er of elements in aA. e.g. A={1,2,3,4} twhose members ar,4,5},{1,3,5}}. containing no elemAnd |∅|=0 as it conwith a single elea} is a singleton seA and B over the uhen B is said to bets A and B over thethen A is said tot is a collection ofhe power set is2A={

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Covering: Division of set into many subset such that they have common element. Rule of Addition: Given a partition on A where Ai, i=1,2,...,n are its non-empty subsets then, | | = | ⋃ | = ∑ | |=1=0 . Rules of Inclusion and Exclusion: Rules of addition cannot be applied to covering of a set A. In this case we use principle of inclusion and exclusion which is as follows, | | = ⋃ | = ∑ | | − ∑ ∑ ∩ + ∑ ∑ ∑ ∩ ∩…−1 +1 =1 |. Problems 1: Assume that |E|=600, |A|=300, |B|=225, |C|=160. Where A is the set of male students, B is the set of bowlers and C is the set of batsmen. Also given AnB be 100, 25 of whom are batsmen too i.e. AnBnC. And the total number of male batesmen i.e. AnC is 85. Determine the number of students who are i. Females, ii. Not Bowlers, iii. Not Batsmen and iv. female students who can bowl.

Solution: i. No. of females = |E|-|A|=600-300=300. ii. Not Bowlers= |E|-|B|=600-225=375. iii. Not Batsmen=|E|-|C|=600-160=440. iv. Female who can bowl=|A’|n|B|=225-100=125.

Problem 2: Given |E|=100, where E indicates a set of students who have chosen subjects form different streams in the computer science discipline, it is found that 32 study subjects chosen from CN stream, 20 from MMT stream, 45 from the SS stream. Also 15 study subjects CN and SS streams, 10 study subjects CN and MMT and 7 from both MMT and SS streams, and 30 do not study any subjects chosen from either of the three streams.

Solution: |A’ n B’ n C’|=30. Given. Then |A U B U C|’=30 by De Morgan’s Law. So we have |AUBUC|=|E|-|AUBUC|’

=100-30=70. From the Principle of inclusion and exclusion |AUBUC|=|A|+|B|+|C|-|AnB|-|BnC|-|CnA|+|AnBnC| =>|AnBnC|=-|A|-|B|-|C|+|AnB|+|BnC|+|CnA|+|AUBUC|

-32-20-45+15+7+10+70= -97+102=5.

PROPERTIES OF FUZZY SETS Commutative ∪ = ∪ ∩ = ∩

Associative ∪ ( ∪ ) = ( ∪ ) ∪ ∩ ( ∩ ) = ( ∩ ) ∩ Distributed ∪ ( ∩ ) = ( ∪ ) ∩ ( ∪ )∩ ( ∪ ) = ( ∩ ) ∪ ( ∩ ) Idempotent ∪ = ∩ =

Identity ∪ ∅ = ; ∪ = ; ∩ ∅ = ∅; ∩ = Low of Absorption ∪ ( ∪ ) = ∩ ( ∪ ) =De Morgan’s Law ∪ = ̅ ∩ ∩ = ̅ ∪

OPERATIONS ON FUZZY SETS The set operations on fuzzy sets are similar to that of crisp sets. Whenever set operations are applied on one or more fuzzy sets it always results in a fuzzy sets. Let = . , . , . , . , = . , . , . , . and = . , . , . , . . Then the set operation are as follows: Union: Let A and B be two fuzzy sets then the union operation results in a fuzzy set whose membership function is defined as follows ∪ ( ) = max { ( ), ( )}. ∪ ( ) = max{0.2,0.6} = 0.6, ∪ ( ) = max{0.3,0.1} = 0.3, ∪ ( ) = max{0.7,0.6} = 0.7, ∪ ( ) = max{0.5,0.4} = 0.5, ∪ = . , . , . , . Intersection: Let A and B be two fuzzy sets then the intersection operation results in a fuzzy set whose membership function is defined as follows ∩ ( ) =min { ( ), ( )}. ∩ ( 1) = min{0.2,0.6} = 0.2, ∩ ( ) = min{0.3,0.1} = 0.1, ∩ ( ) = min{0.7,0.6} = 0.6, ∩ ( ) = min{0.5,0.4} = 0.4, ∩ = . , . , . , . Complementation: Let A be a fuzzy sets then the complementation operation results in a fuzzy set whose membership function is defined as follows ( ) = {1 − ( )}. ( ) = {1 − 0.2} = 0.8, ( ) = {1 − 0.3} = 0.7, ( ) = {1 − 0.7} = 0.3,

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= {1 − 0.5} = 0.5,, . , . , . y: Let A and B be onding member of= ( ) = 0.2, = ( ) = 0.3, = ( ) = 0.7, = ( ) = 0.5, Het of a fuzzy set wihen the product oped as follows .= 2 ∗ 0.2 = 0.4 = 2 ∗ 0.3 = 0.6 = 2 ∗ 0.7 = 1.4 == 2 ∗ 0.5 = 1 . , . , , . t of two fuzzy set:in a fuzzy set = ( ). ( ). ) = 0.2 ∗ 0.6 = 0.12) = 0.3 ∗ 0.1 = 0.03) = 0.7 ∗ 0.6 = 0.35) = 0.5 ∗ 0.4 = 0.2.. , . , . , . of a fuzzy set: Leton results in a fuz= { ( )} . Let == {0.2} = 0.04, = {0.3} = 0.09, = {0.7} = 0.49, = {0.5} = 0.25. , . , . , . nce: Let A and B beet whose membementation operatio) = min{0.2,1 − 0.) = min{0.3,1 − 0.) = min{0.7,1 − 0.) = min{0.5,1 − 0. B

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LECTURE NOTES

MAR SAHU, DEPT O

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= . , . , . , .nctive sum: Let A ts in a fuzzy set wh= ( ∩ ) ∪ ( ∩lows ⊕ ( ) = m( ) = max {min{1( ) = max{min{1min{0.8,0.6} , min{( ) = max{min{1min{0.7,0.1} , min{( ) = max{min{1min{0.3,0.6} , min{( ) = max{min{1min{0.5,0.4} , min{

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and B be two fuzzhose membership f∩ ). And the memmax {min{1 − (− ( ), ( )} , m− 0.2,0.6} , min{0.20.2,0.4}} = max{0− 0.3,0.1} , min{0.30.3,0.9}} = max{0− 0.7,0.6} , min{0.70.7,0.4}} = max{0− 0.5,0.4} , min{0.50.5,0.6}} = max{0NCTIONS t is characterized bMembership functrom U to [0,1], whand the range ship value. memberepresented by ∈[0,1]. Since (able x, so we drawn U.

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∪ ∩ ( ∪ ) tion: ( ) = ; =( ) = 1 − =( ) = 2 ; =( ) = 1 − =

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RELATIONS IN CRISP LOGIC A relation R in crisp logic is defined as a subset of Cartesian product of two crisp set A and B i.e. ⊆ × . The elements of a relations are ordered pairs i.e. (xi, yi) such that xi∈A and yi∈B. Let A={1, 2, 3} and B={2, 3, 4}, then A×B={(1,2), (2,2), (3,2), (1,3), (2,3), (3,3), (1,4), (2,4), (3,4)} and let R={y=x+1|x∈A and y∈B}. Then R={(1,2),{2,3),(3,4)}. OPERATIONS ON RELATIONS Relations are nothing but sets intern, so the operations that can be applied to set can be also applied to relations also. Let R and S be the two relations. R={y=x+1|x∈A and y∈B}. Then R={(1,2),{2,3),(3,4)}. S={y=x|x∈A and y∈B}. Then s={{2,2),(3,3)}. The operations on sets are as follows: i. Union: The union operation on two relation R and S is defined as follows: ∪ ( , ) = max { ( , ), ( , )}.

ii. Intersection: The intersection operation on two relation R and S is defined as follows: ∩ ( , ) = min { ( , ), ( , )}.

iii. Complementation: The complementation operation on a relation R and S is defined as follows: ( , ) = 1 − ( , ).

iv. Composition: The composition operation on two relation R and S is defined as follows: ( , ) = max { ( , ), ( , )}. RELATIONS IN FUZZY LOGIC A relation R in fuzzy logic is defined in a similar manner like that of crisp logic as a sunset of Cartesian product of two fuzzy set A and B i.e. ⊆ × . The elements of a fuzzy relations are represented as ( , ), where 'R' represent the relation. Let us take two sets: = . , . , . and = . , . , Then × is defined as a set whose membership function is given by the relation × ( , ) = min { ( ), ( )}. × ={ ( , ), 0.2 , ( , ), 0.2 , ( , ), 0.5 , ( , ), 0.6 , ( , ), 0.4 , ( , ), 0.4 } The same thing be represented in the form of a table as above. OPERATIONS ON FUZZY RELATIONS Similar to crisp logic we also have a similar set of operations on fuzzy logic. To understand the operations let us take two fuzzy relations R and S as follows. Union, intersection requires two relations to union compatible i.e. R and S should be same order =mxn in both R and S. The operations on fuzzy logic are as follows:

i. Union: The union operation on two fuzzy relation R and S is defined as follows: ∪ ( , ) = max{ ( , ), ( , )}. ∪ ( , ) = max{ ( , ), ( , )} =max{0.7,0.8} =0.8, ∪ ( , ) = max{ ( , ), ( , )} =max{0.6,0.5} =0.6, ∪ ( , ) = max{ ( , ), ( , )} =max{0.8,0.1} =0.8, ∪ ( , ) = max{ ( , ), ( , )} =max{0.3,0.6} =0.6. ii. Intersection: The intersection operation on two fuzzy relation R and S is defined as follows: ∩ ( , ) = min{ ( , ), ( , )}. ∩ ( , ) = min{ ( , ), ( , )} =min{0.7,0.8} =0.7, ∩ ( , ) = min{ ( , ), ( , )} =min{0.6,0.5} =0.5, ∩ ( , ) = min{ ( , ), ( , )} =min{0.8,0.1} =0.1, ∩ ( , ) = min{ ( , ), ( , )} =min{0.3,0.6} =0.3. iii. Complementation: The complementation operation on a fuzzy relation R is

R 1 2 32 1 0 03 0 1 04 0 0 1

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R 1 2 3 R' 1 2 3 S 1 2 3 S' 1 2 32 1 0 0 2 0 1 1 2 0 1 0 2 1 0 13 0 1 0 3 1 0 1 3 0 0 1 3 1 1 04 0 0 1 4 1 1 0 4 0 0 0 4 1 1 1

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R 1 2 S 1 2 3 1 2 32 1 0 1 1 0 0 2 1 0 03 0 1 2 0 1 0 3 0 1 04 0 0 4 0 0 0

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x1 0.7 0.5 x2 0.1 0.3



defined as follows: ( , ) = 1 − ( , ). ( , ) = 1 − ( , ) =1-0.7 =0.3, ( , ) = 1 − ( , ) =1-0.6 =0.4, ( , ) = 1 − ( , ) =1-0.8 =0.2, ( , ) = 1 − ( , ) =1- 0.3=0.7. iv. Composition: For composition operation R and S may not be necessary but they have number of columns of first relation equals to numbers of rows of second relations. i.e. if order of R(x, y) and S(y, z) are m x n and n x p respectively, the composition is possible and the resultant relation is of order m x p. Let the R and S be represented as follows:

a. Max-Min: The Max-Min composition on two fuzzy relation R and S is defined as follows: ( , ) = max{min{ ( , ), ( , )}| ∈ , ∈, ∈ . ( , ) =max min{ ( , ), ( , )},min{ ( , ), ( , )}, min{ ( , ), ( , )}. = min{0.7, 0.8},min{0.6, 0.1}, min{0.2, 0.7}.

= 0.70.10.2 = 0.7 ( , )= max min{ ( , ), ( , )},min{ ( , ), ( , )}, min{ ( , ), ( , )}. = min{0.7, 0.5},min{0.6, 0.6}, min{0.2, 0.2}.= 0.50.60.2 = 0.5 ( , )= max min{ ( , ), ( , )},min{ ( , ), ( , )}, min{ ( , ), ( , )}. = min{0.8, 0.8},min{0.3, 0.1}, min{0.4, 0.7}.= 0.80.10.4 = 0.8

( , )= max min{ ( , ), ( , )},min{ ( , ), ( , )}, min{ ( , ), ( , )}. = min{0.8, 0.5},min{0.3, 0.6}, min{0.4, 0.2}.= 0.50.30.2 = 0.5 b. Max-Product: The Max-Product composition on two fuzzy relation R and S is defined as follows: ( , ) =max{ ( , ) ∗ ( , )| ∈ , ∈ , ∈ . ( , ) = max { ( , ) ∗ ( , )},{ ( , ) ∗ ( , )},{ ( , ) ∗ ( , )}. = {0.7 ∗ 0.8},{0.6 ∗ 0.1}, {0.2 ∗ 0.7}.= 0.560.060.14 = 0.56 ( , ) = max { ( , ) ∗ ( , )},{ ( , ) ∗ ( , )}, { ( , ) ∗ ( , )}. = {0.7 ∗ 0.5},{0.6 ∗ 0.6}, {0.2 ∗ 0.2}.= 0.350.360.04 = 0.36 ( , ) = max { ( , ) ∗ ( , )},{ ( , ) ∗ ( , )}, { ( , ) ∗ ( , )}. = {0.8 ∗ 0.8},{0.3 ∗ 0.1}, {0.4 ∗ 0.7}.= 0.640.030.28 = 0.64

Rc y1 y2 x1 0.3 0.4 x2 0.2 0.7

S z1 z2

y1 0.8 0.5y2 0.1 0.6y2 0.7 0.2

R y1 y2 y3

x1 0.7 0.6 0.2x2 0.8 0.3 0.4

RoS z1 z2 x1 0.56 0.36 x2 0.64 0.40

RoS z1 z2 x1 0.7 0.5 x2 0.8 0.5

c.

( , ) = Max-Average: Tand S is defined, ∈ , ∈ . ( ,= max {{{= 0.0.0. ( , )= max {{ {= 0.60.60.2 ( ,= max {{ {= 0.0.0. ( ,= max {{ {= 0.0.40.

B

= max { ( , ){ ( , ) { ( , )= 0.400.180.08The Max-Average d as follows: () ( , ) + ( ,( , ) + ( ,( , ) + ( ,.75.35.45 = 0.75

) ( , ) + ( ,( , ) + ( ,( , ) + ( ,606020 = 0.60 ) ( , ) + ( ,( , ) + ( ,( , ) + (802055 = 0.80 ) ( , ) + ( ,( , ) + ( ,( , ) + (654530 = 0.65

L

BY: KISHORE KUM

) ∗ ( , )},) ∗ ( , )},) ∗ ( , )}. == 0.40 composition on , ) = max{( ( ,

, )}/2,, )}/2,, )}/2. =

)}/2,)}/2,)}/2. =

, )}/2,, )}/2,, )}/2. =

, )}/2,, )}/2,, )}/2. =RoSx1

x2

LECTURE NOTES O

MAR SAHU, DEPT O

{0.8 ∗ 0.5},{0.3 ∗ 0.6}, {0.4 ∗ 0.2}.two fuzzy relatio, ) + ( , ))/2|{0.7 + 0.8}/2,{0.6 + 0.1}/2, {0.2 + 0.7}/2.{0.7 + 0.5}/2,{0.6 + 0.6}/2,{0.2 + 0.2}/2.{0.8 + 0.8}/2,{0.3 + 0.1}/2, {0.4 + 0.7}/2.{0.8 + 0.5}/2,{0.3 + 0.6}/2, {0.4 + 0.2}/2.

S z1 z20.75 0.600.80 0.65

ON SOFT COMPUT

OF INFORMATION

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FUZZY INFERE

uzzy inference systDatabase: The linand this is the placRulebase: The linand inferences to form of "if then" ruDecision making sunit and processeoutput to the defuzFuzzification unitreal world that aredifferent fuzzificatsystem. Defuzzification unthe real world thatdecision making sconvert the same inSP LOGIC OR BOrisp logic we repectives and inferenposition: A fact in t

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tem basically consnguistic variable ae where the membguistic variables aform rules and thules. system: It receives it with the relazzification unit.

t: It is the unit, whe crisp in nature. tion techniques anit: It is the unit, wt, are crisp in natusystem and appliento crisp output. OOLEAN LOGICpresent the fact nce rules. the real world is rted as P(x) whereatomic or simpleg use of connectivere used to build he connectives are

sists of five functioare represented abership functions aare combined withis is where these es the fuzzy inputtions stored in thhich actually receIt the converts theand forwards it twhich is responsiure. It receives thees different defuzzC OR CLASSICAof real world inrepresented as a pe x represents Rame proposition. Wes. complex proposie as follows:

nal units as follows membership funare stored. th different connerules are stored ts from the fuzzifihe system to give ives the input froem into fuzzy inputo the decision mble for giving outpe fuzzy output forzification techniquAL LOGIC nterms of propo

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P Q ~P ~Q ⋁ ∧ ⟹ ~ ⋁T T F F T T T TT F F T T F F FF T T F T F T TF F T T F F T TInference Rules: By making use of given propositions we can infer some unknown fact. This mechanism is called as inferenceing. There are two kinds of inference rules that are as follows: i. Modus Ponens: It states that if P is true and P=>Q is true, then we infer Q is true. Here we infer P as a fact and P=>Q is a rule and Q as inference.

P is true is a fact P=>Q is true is a rule Q is true is inferred.

ii. Modus Tollens: It states that if ~Q is true and P=>Q is true, then we infer ~P is true. Here we infer ~Q as a fact and P=>Q is a rule and ~P as inference. ~Q is true is a fact P=>Q is true is a rule ~P is true is inferred.

FUZZY LOGIC In crisp logic we consider a proposition to be either true or false i.e. T/F, but in fuzzy logic we consider a proposition to take fuzzy truth-values i.e. values between 0 and 1. Fuzzy Proposition: A fuzzy proposition is represented as T(P) which takes truth values between 0 and 1. A fuzzy proposition similar to fuzzy sets are represented by a membership functions i.e. = ( ) ℎ ℎ 0 ≤ ( ) ≤ 1. E.g. Let =Ram is good, then T( )=0.8 implies the statement is partially true, where as T( )=1 implies the statement is absolutely true. Connectives: Complex or compound fuzzy propositions are constructed by making use of connectives. The use of connectives in fuzzy logic is as follows: Inference Procedure: In fuzzy logic we represent the rules in terms of "if then" rules. i.e. if we have two fuzzy proposition A and B and we have A=>B, then this is interpreted as "if x is in A then y is in B". Another form rule i.e. "if x in A then y in B else y in C" is also possible. We represent these rules in terms of a fuzzy relation R,

that takes different form depending on the rule for which is designed i.e. if we are making use of "If x in A then y in B" then = ( × ) ∪ ( × ), and "if x in A then y in B else y in C" then = ( × ) ∪ ( × ). Example: Let = { , , , }, = {1, 2, 3, 4} be two sets. Let A be a fuzzy proposition defined on the set X as = . , . , . , . , Let B, C be fuzzy propositions defined on the set Y as = . , . , . , . and = . , . , . , . respectively. Then for the rule "if x in A then y in B", we have = × ∪ ( × ), where is the fuzzy set representing the set Y such that each element has membership function equal to 1, i.e. = . , . , . , . . So ( , ) = max {min{ ( ), ( )} , min{1 − ( ), ( )} } , this can be rewritten as ( , ) = max {min{ ( ), ( )} , 1 − ( ) } as ( ) is always 1 and will be greater than 1- ( ) so min{1 − ( ), ( )} is equivalent to 1- ( ).

For the rule "if x in A then y in B else y in C", we have = × ∪ ( × ). So ( , ) = max {min{ ( ), ( )} , min{1 − ( ), ( )} }. = . , . , . , . , = . , . , . , .

In Fuzzy we have two inference processes i.e. Generalized Modus ponens (GMP) and Generalized Modus Tollens (GMT). The inference process in fuzzy logic is as follows:

Connectives Membership Functions~P 1-T(P)⋁ max(T(P),T(Q))∧ min(T(P),T(Q))⟹ or ~ ⋁ max(1-T(P),T(Q))

AxB 1 2 3 4a 0.0 0.0 0.0 0.0b 0.2 0.8 0.8 0.0c 0.2 0.6 0.6 0.0d 0.2 1.0 0.8 0.0AcxY 1 2 3 4A 1.0 1.0 1.0 1.0B 0.2 0.2 0.2 0.2C 0.4 0.4 0.4 0.4D 0.0 0.0 0.0 0.0

R 1 2 3 4 a 1.0 1.0 1.0 1.0 b 0.2 0.8 0.8 0.2 c 0.4 0.6 0.6 0.4 d 0.2 1.0 0.8 0.0

AxB 1 2 3 4a 0.0 0.0 0.0 0.0b 0.2 0.8 0.8 0.0c 0.2 0.6 0.6 0.0d 0.2 1.0 0.8 0.0AcxC 1 2 3 4a 0.0 0.4 1.0 0.8b 0.2 0.4 0.2 0.2c 0.0 0.4 0.4 0.4d 0.0 0.0 0.0 0.0

R 1 2 3 4 a 0.0 0.4 1.0 0.8 b 0.2 0.8 0.8 0.2 c 0.2 0.6 0.6 0.4 d 0.2 1.0 0.8 0.0



i. Generalized Modus Ponens (GMP): It works when a fuzzy proposition P and a rule P=>Q is given and we need to infer Q. Example: P="x in A" Given fact if "x in A" then "y in B" Given rule Q="y in B" Inference This can be also shown by composition operation also i.e. = where = × ∪ ( × ). Example: GMP can also be applied to problem where we have more than one variable i.e. "if old and car is high power then risk is high". So here GMP take the form "if x in A and y in B then z in C" P="x in A" and Q="y in B" Given fact if "x in A" and "y in B" then "z in C" Given rule S="z in C" Inference Here we make use of = ( ∪ ) ii. Generalized Modus Tollens (GMT): It works when a fuzzy proposition ~Q and a rule P=>Q is given and we need to infer ~P. Example: ~Q="x not in A" Given fact if "x in A" then "y in B" Given rule ~P="y not in B" Inference This can be also shown by composition operation also i.e. ~ = ~ where = × ∪ ( × ). Example: GMP can also be applied to problem where we have more than one variable i.e. "if risk not high then not old or car is not high power ". So here GMP take the form "if x in A and y in B then z in C" S="z not in C" Given fact if "x in A" and "y in B" then "z in C" Given rule P="x not in A" or Q="y not in B" Inference Here we make use of ~ ∩ ~ = ~ FUZZY UNION AND INTERSECTION The intersection and union of two fuzzy sets can also be performed with T-norm and T-conorm or S-norm respectively. Let us discuss these operations in detail. T-NORMS T-norm operator is a two place function i.e. T( . , . ) satisfying the following properties:

Properties: i. Boundary: T(0,0)=0, T(a,1)=T(1,a)=a i.e. it imposes correct generalization to crisp sets. ii. Monotonicity: T(a, b)<=T(c, d) if a<c and b<d i.e. increase in values of a and b also increased the value in T(a, b). iii. Commutative: T(a, b)=T(b, a) i.e. the operator is indifferent to the order of fuzzy sets to be combined. iv. Associativity: T(a, T(b, c))=T(T(a, b), c) i.e. any number of fuzzy sets and in any order can be combined to form pair-wise grouping since T-norm is a two place function Operation using T-norms: The four operations that are allowed in T-norms are as follows: i. Minimum: ( , ) = min{ , } = ⋀ . ii. Algebraic product: ( , ) = . iii. Bounded product: ( , ) = 0 ∨ (a + b − 1). iv. Drastic product: ( , ) = , = 1., = 1.0, , < 1. Relationship between the different operations in T-Norms ( , ) ≤ ( , ) ≤ ( , ) ≤ ( , ) S-NORM S-norm operator also called as T-conorm is a two place function i.e. S( ., . ) satisfying the following properties: Properties: i. Boundary: S(1,1)=1, S(a,0)=S(0,a)=a i.e. it imposes correct generalization to crisp sets. ii. Monotonicity: S(a, b)<=S(c, d) if a<c and b<d i.e. increase in values of a and b also increased the value in T(a, b). iii. Commutative: S(a, b)=S(b, a) i.e. the operator is indifferent to the order of fuzzy sets to be combined. iv. Associativity: S(a, S(b, c))=S(S(a, b), c) i.e. any number of fuzzy sets and in any order can be combined to form pair-wise grouping since T-norm is a two place function

OperatiThe fouri. ii. iii. iv. Relation

DECOMThe outfuzzy ouconversbroughtinferencprocess out the di. Cfumbse S

ion using S-normsr operations that aMaximum: Algebraic sum: Bounded sum: Drastic sum:

nship between the( ,MPOSITION put of decision-mautputs. These fuzzion of fuzzy output about with the dce system this is is the reverse of fdefuzzification pro

Centroid Method:unction is createdmembership functeing called as a seegment the area anSegment 1 12 x2 2.63 12 x 04 0.45 1.56 0.57 12 x 0 B

s: are allowed in T-no( , )( , ) =( , ) =( , ) =e different operat) ≤ ( , ) ≤aking system on azy outputs cannot ut to crisp output defuzzification prresponsible for gfuzzification proceocess which are asFrom the givend that contain all ion is then dividegment. The segmnd its centroid w.rArea 1 x 0.3 = 0.15 6 x 0.3 = 0.78

0.4 x 0.2 = 0.04 4 x 0.3 = 0.12 5 x 0.5 = 0.75 5 x 0.5 = 0.25 .5 x 0.5 = 0.125

L

BY: KISHORE KUM

orms are as follow) = max{ , } == + − . = 1 ∧ (a + b). = , = 0., = 0.1, , 0.tions in T-Norms ( , ) ≤ (application of fuzzybe interpreted byis highly necessarrocess. It acts as tgiving crisp outpess. There are diffe follows: n membership fu the functions toded into triangles ments are numberer.t the origin is calc

0.673.6 − 12 + 13.6 + 23 x 0.40.42 + 3.61.52 + 4 =0.52 + 5.5 =13 x 0.5 + 5.5

LECTURE NOTES O

MAR SAHU, DEPT O

ws: ∨ .

, ) y inference rule giy the real world, sry. This conversiothe last unit of fuut to the users. erent method to canctions an envel scale. The enveland rectangle, eed and for each of culated. Area0.10

1 = 2.3 1.794 = 3.8 0.15= 3.8 0.454.75 3.56

= 5.75 1.435 = 5.66 0.70

ON SOFT COMPUT

OF INFORMATION

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ING P a g e | 13

TECHNOLOGY, RI

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T, BERHAMPUR.

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L

BY: KISHORE KUM

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d the maxima arend their mean is = 6 + 72

ENCE SYSTEMSem, they are as follni fuzzy inference These fuzzy output

LECTURE NOTES O

MAR SAHU, DEPT O

re form the origins: m the origin Area5 3.00 7.5 13lculated, the centr5.0

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ON SOFT COMPUT

OF INFORMATION

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ING P a g e | 14

TECHNOLOGY, RI

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BERHAMPUR SOFT COMPUTING (PECS 3401)-FUZZY...

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