# Introduction To Fuzzy Logic

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Introduction toFuzzy LogicIJCAI2018Tutorial1

Crisp set vs. Fuzzy setA traditional crisp setA fuzzy set2

Crisp set vs. Fuzzy set3

Crisp Logic Example I Crisp logic is concerned with absolutes-true or false,there is no in-between. Example:Rule:If the temperature is higher than 80F, it is hot; otherwise, it isnot hot.Cases:Temperature 100FHotTemperature 80.1FHotTemperature 79.9FNot hotTemperature 50FNot hot––––4

Crisp Logic Example I cont.Membership function of crisp logicTrue1HOTFalse080FTemperatureIf temperature 80F, it is hot (1 or true);If temperature 80F, it is not hot (0 or false).5

Example II : Crisp set1015ft 11ins7ft heightA crisp way of modellingtallness05ft 11ins7ft heightA crisp version of short6

Example II : Crisp set1tallquite tall0very tall5ft 11ins7ftheightcrisp definitions for tallness7

Example II : Fuzzy set10height8

Example II : Fuzzy set105ft 11ins7 ftheightA possible fuzzy set tall9

Example II : Fuzzy set105ft 11ins7 ftheightA possible fuzzy set short10

Example II : Fuzzy set1tallshort0.750.4005ft 11ins7 ftheightMembership functions thatrepresent tallness and short11

Fuzzy Sets Fuzzy sets is fully defined by its membershipfunctions. Membership function is a function in [0,1]that represents the degree of belonging.Fuzzy valuesDOMDegree ofMembership 1YoungMiddleOld0.50254055Age12

Some maths!Formal definitions of a fuzzy set For any fuzzy set, (let’s say) A, the function µArepresents the membership function for which µA(x)indicates the degree of membership that x, of theuniversal set X, belongs to set A and is, usually,expressed as a number between 0 and 1µA(x) : X [0,1] Fuzzy sets can be either discrete or continuous13

The notation for fuzzy sets: for the member, x, ofa discrete set with membership µ, we use thenotation µ/x . In other words, x is a member ofthe set to degree µ. Discrete sets are defined as:A µ1 /x1 µ2/x2 . µn/xn or (in a more compact form)x1 ,x2 , . xn : members of the set Aµ1, µ2, .µn : x1 , x2 . xn ’s degree of membership.14

A continuous fuzzy set A can be defined as:µExample:Discrete andContinuousfuzzy sets torepresent the setof numbers“close to 1”numbers15

Example: describing people as “young”, “middle-aged”,and “old”µ1Middle-agedyoungold0Age Fuzzy Logic allows modelling of linguistic terms usinglinguistic variables and linguistic values. The fuzzy sets“young”, “middle-aged”, and “old” are fully defined bytheir membership functions. The linguistic variable “Age”can then take linguistic values.16

Key Points for a fuzzy set:– The members of a fuzzy set are members to somedegree, known as a membership grade or degree ofmembership– A fuzzy set is fully determined by the membershipfunction– The membership grade is the degree of belongingto the fuzzy set. The larger the number (in [0,1]) themore the degree of belonging.– The translation from x to µA(x) is known asfuzzification– A fuzzy set is either continuous or discrete.– Fuzzy sets are NOT probabilities– Graphical representation of membership functionsis very useful.17

Fuzzy Sets18

Fuzzy Sets – definition19

Membership Degree20

Fuzzy Sets Representation I21

Fuzzy Sets Representation II22

Example of Membership Functions23

Unimodal Functions24

Singletons25

Classical Set Function26

Linear Functions27

Triangular Functions28

Triangular Functions29

Trapezoidal Functions30

Trapezoidal Functions31

Support of Fuzzy Sets32

Empty Fuzzy SetAn empty fuzzy set has an empty support33

Compact Support34

Alpha-cuts Sets35

Level Set36

Resolution Principle37

Cardinality38

Cardinality – cont.39

Height of a fuzzy set40

Fuzzy subsets41

Equal Fuzzy Sets42

Fuzzy Membership Functions One of the key issues in all fuzzy sets is how todetermine fuzzy membership functions The membership function fully defines the fuzzy set A membership function provides a measure of thedegree of similarity of an element to a fuzzy set Membership functions can take any form, but thereare some common examples that appear in realapplications43

Membership functions can– either be chosen by the user arbitrarily, based onthe user’s experience (MF chosen by two userscould be different depending upon theirexperiences, perspectives, etc.)– Or be designed using machine learning methods(e.g., artificial neural networks, geneticalgorithms, etc.) There are different shapes of membershipfunctions; triangular, trapezoidal, piecewiselinear, Gaussian, bell-shaped, etc.44

Fuzzy Operations(Fuzzy Union, Intersection, and Complement) Fuzzy logic begins by borrowing notions from crisp logic, just asfuzzy set theory borrows from crisp set theory. As in our extensionof crisp set theory to fuzzy set theory, our extension of crisp logic tofuzzy logic is made by replacing membership functions of crisplogic with fuzzy membership functions [J.M. Mendel, Uncertain RuleBased Fuzzy Logic Systems, 2001] In Fuzzy Logic, intersection, union and complement are defined interms of their membership functions This section concentrates on providing enough of a theoretical basefor you to be able to implement computer systems that use fuzzylogic Fuzzy intersection and union correspond to ‘AND’ and ‘OR’,respectively, in classic/crisp/Boolean logic These two operators will become important later as they are thebuilding blocks for us to be able to compute with fuzzy if-then rules45

Classic/Crisp/Boolean Logic Logical AND ( ) Logical OR (U)Truth TableA0011B0101A B0001ABCrisp IntersectionA0011Truth TableBAUB00110111ABCrisp Union46

Fuzzy Union The union (OR) is calculated using t-conorms t-conorm operator is a function s(.,.) Its features are––––s(1,1) 1, s(a,0) s(0,a) a (boundary)s(a,b) s(c,d) if a c and b d (monotonicity)s(a,b) s(b,a) (commutativity)s(a,s(b,c)) s(s(a,b),c) (associativity) The most commonly used method for fuzzy union is totake the maximum. That is, given two fuzzy sets A and Bwith membership functions µA(x) and µB(x)µ AUB ( x) max(µ A ( x), µ B ( x))47

Fuzzy Intersection The intersection (AND) is calculated using t-norms. t-norm operator is a function t(.,.) Its features––––t(0,0) 0, t(a,1) t(1,a) a (boundary)t(a,b) t(c,d) if a c and b d (monotonicity)t(a,b) t(b,a) (commutativity)t(a, t(b,c)) t(t(a,b),c) (associativity) The most commonly adopted t-norm is the minimum.That is, given two fuzzy sets A and B with membershipfunctions µA(x) and µB(x)µ AÇB ( x) min(µ A ( x), µ B ( x))48

Fuzzy Complement To be able to develop fuzzy systems we also haveto deal with NOT or complement. This is the same in fuzzy logic as for Booleanlogic For a fuzzy set A, A denotes the fuzzycomplement of A Membership function for fuzzy complement isµ ( x) 1 - µ A ( x)A49

Product of Two Fuzzy Sets50

Sum of Two Fuzzy Sets51

Linguistics Hedges and Operators A fuzzy set can be regarded as corresponding to a linguistic value suchas “tall”, and a linguistic variable “height” can be regarded as rangingover such linguistic values. One powerful aspect of fuzzy sets in this context is the ability to dealwith linguistic quantifiers or “hedges”. Hedges such as more or “less”, “very”, “not very”, “slightly” etccorrespond to modifications in the membership function of the fuzzy setinvolved. The fuzzy set operations such CON, DIL, INT etc (see the table ofhedges and Operators) can be used to modify the fuzzy set.52

Hedges and Operators53

Fuzzy Logic IJCAI2018 Tutorial 1. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2. . A possible fuzzy set short 10. Example II : Fuzzy set 0 1 5ft 11ins 7 ft height . Fuzzy logic begins by borrowing notions from crisp logic, just as

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