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2. Classical and Fuzzy Sets 2012 1 3.1. Crisp vs. Fuzzy Sets: The concept of a set is fundamental to mathematics. How-ever, our own language uses sets extensively. For example, car indicates the set of cars. When we say a car, we mean one out of the set of cars. The classical example in fuzzy sets is tall men. The elements of the fuzzy set “tall persons” are all persons, but their degrees of membership depend on their height. 2012 2 Example: Crisp vs. Fuzzy Sets: TALL-Men set Person Height Degree of Membership Crisp Fuzzy Person 1 205 1 1.00 Person 2 182 1 0.81 Person 3 175 0 0.38 Person 4 167 0 0.10 Person 5 155 0 0.04 Person 6 152 0 0.00 2012 3 Crisp Set (clearly defined Set) 2012 4 Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It deals with degrees of membership and degrees of truth. Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true), accepting that things can be partly true and partly false at the same time. 0 0 0 1 1 1 0 0 0.2 0.4 0.6 0.8 1 1 (a) Boolean Logic. (b) Multi-valued Logic. 2012 5 Example: Fuzzy set of Tall Men The x-axis represents the Universe of Discourse (universe of all available information on a given problem/ the space where the fuzzy variables are defined). According to this representation, the universe of person’s heights consists of all ’tall’ men. The y-axis represents the Membership Value of the fuzzy set. In our case, the fuzzy set of “tall men” maps height values into corresponding membership values. 2012 6 3.2. Crisp Sets: Let X be the universe of discourse and its elements be denoted as x. In the classical set theory, a crisp set A of X is defined by a function fA(x) called the characteristic function of A: fA(x) : X {0/1}, where This set maps universe X to a set of two elements. For any element x of universe X, characteristic function fA(x) is equal to 1 if x is an element of set A, and is equal to 0 if x is not an element of A. x∈X ⇒ x belongs to X Notations: A⊂B ⇒ A is fully contained in B (if x ∈ A, then x ∈ B) A⊆B ⇒ A is contained in or is equivalent to B ⇒ A ⊆ B and B ⊆ A (A is equivalent to B) The null set, ∅, is the set containing no elements (A ↔ B) Conventional or crisp sets are binary. An element either belongs to the set (True/1) or doesn’t (False/0) . 2012 7 3.3. Fuzzy Sets: In the fuzzy theory, a fuzzy set A of universe X is defined by function µA(x) called the membership function of set A µA(x) : X {0, 1}, where µA(x) = 1 if x is totally in A; µA(x) = 0 if x is not in A; 0 < µA(x) < 1 if x is partly in A. This set allows a continuum of possible choices. For any element x of universe X, membership function µA(x) equals the degree to which x is an element of set A. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element x in set A. Thus Membership functions of fuzzy sets are continuous. An element can belong to the set to a certain degree {0 1} . 2012 8 First determine the member- 3.4. Fuzzy Set Representation: ship functions: In the “tall men” example, we can obtain Crisp set of tall, short and average men, and also of Fuzzy set. The universe of discourse – the men’s heights – consists of three sets: short, average and tall men. A man who is 184 cm tall is: Crisp: a member of the tall set but not of the others Fuzzy: he is a member of average men set with a degree of membership of 0.1, and at the same time, he is also a member of the tall men set with a degree of 0.4, and is not a member of short men set 2012 9 Fuzzy Set Representation (Contd.) There are Continuous universes comprising of an infinite collection of elements (see fuzzy set of Tall Men on slide 6) , and Discrete universes, that are composed of a finite collection of elements. In the last case when elements are discrete the grade of membership can be represented by: A = {(1,0.7), (2, 0.5),(3,1.0)} Or A = 0.7/1 + 0.5/2+1.0/3 be sure to notice that the symbol ‘+’ implies not addition but union. More Generally, we use Or 2012 10 Example: Discrete vs. Continuous Membership functions A discrete membership function for ”x is close to 1” A Continuous membership function for ”x is close to 1” 2012 11 3.5. Standard Operations on Fuzzy Sets: Consider a universal set X which is defined on the age domain. X = {5, 15, 25, 35, 45, 55, 65, 75, 85} age(element) infant young adult senior 5 0 0 0 0 15 0 0.2 0.1 0 25 0 1 0,9 0 35 0 0.8 1 0 45 0 0.4 1 0.1 55 0 0.1 1 0.2 65 0 0 1 0.6 75 0 0 1 1 85 0 0 1 1 2012 12 2012 13 1) Complement: Definition: Crisp Sets: Who does not belong to the set? Fuzzy Sets: How much do elements not belong to the set? The complement of a set is an opposite of this set. For example, if we have the set of tall men, its complement is the set of NOT tall men. When we remove the tall men set from the universe of discourse, we obtain the complement. 2012 14 The complement set of a fuzzy set A is found as follows: The degreee of Membership is given by: In our example A is the “adult” set and is given by: A = {(5, 0), (15, 0.1), (25, 0.9),(35,1)}. Then: = {(5, 1), (15, 0.9), (25, 0.1),(35,0}. 2012 15 Not Adult 2012 16 2. Union ( ) Definition Crisp Sets: Which element belongs to either set? Fuzzy Sets: How much of the element is in either set? The union of two crisp sets consists of every element that falls into either set. For example, the union of tall men and fat men contains all men who are tall OR fat. In fuzzy sets, the union is the largest membership value of the element in either set. 2012 17 Membership value of member x in the union takes the greater value of membership between A and B Of course, A and B are subsets of A B, now if B represents “young” and is given by: B= {(5, 0), (15, 0.2), (25, 1),(35,0.8 ), (45, 0.4),(55,0.2)} and as before adult A={(5, 0), (15, 0.1), (25, 0.9),(35,1)} Then the union of “young” and “adult” is A B = {(15,0.2), (25,1), (35,1), (45,1), (55,1), (65,1),(75,1), (85,1)} 2012 18 Young Adult 2012 19 3. Intersection ( ( ) Definition Crisp Sets: Which element belongs to both sets? Fuzzy Sets: How much of the element is in both sets? In classical set theory, an intersection between two sets contains the elements shared by these sets. For example, the intersection of the set of tall men and the set of fat men is the area where these sets overlap. In fuzzy sets, an element may partly belong to both sets with different memberships. A fuzzy intersection is the lower membership in both sets of each element. 2012 20 Intersection of fuzzy sets A and B takes the smaller value of membership function between A and B. Intersection A B is a subset of A or B. For instance, the intersection of “young” and “adult” is A B= {(15, 0.1), (25, 0.9), (35, 0.8), (45, 0.4), (55, 0.1)}. Note that complement, union and intersection sets are applicable even if membership function is restricted to 0 or 1 (i.e. crisp set) 2012 21 Young Adult 2012 22 Another Example: Let A be a fuzzy interval “between 5 and 8” and B be a fuzzy number “about 4”. 2012 23 The following figure shows the fuzzy set “between 5 and 8” AND “about 4” (Intersection) 2012 24 The Fuzzy set “between 5 and 8” OR “about 4” is shown in the next figure (Union) 2012 25 This figure gives an example for a negation. The blue line is the NEGATION (Complement) of the fuzzy set A. i.e. “not a fuzzy interval between 5 and 8” 2012 26 Example: Consider the fuzzy set of the points included in a circle Distance membership function Representation of the grade of membership of the inclusion within the circle with black representing the points not in the set i.e. μP1=0 2012 2010 Hany Selim 27 Representation of the grade of membership of the inclusion within a shifted circle with black representing the points not in the set i.e. μP2 Union of the 2 sets μP1 μP2 2012 2010 28 The intersection of the two sets μP1 μP2 The complement of the first set above: μP1 2012 2010 29 The intersection of the first set and its complement is not empty (remember by Crisp A.A=0) The union of the first set and its complement is not 1 (remember by Crisp A+A=1) 2012 2010 30 2012 31 3.6. Properties of Fuzzy Sets: Equality of two fuzzy sets Inclusion of one set into another fuzzy set Cardinality of a fuzzy set An empty fuzzy set -cuts (alpha-cuts) Fuzzy set Normality Height of the Fuzzy set Fuzzy set Core Fuzzy set Support 2012 32 1. Equality: Fuzzy set A is considered equal to a fuzzy set B, IF AND ONLY IF (iff): A(x) = B(x), xX A = 0.3/1 + 0.5/2 + 1/3 B = 0.3/1 + 0.5/2 + 1/3 therefore A = B 2012 33 2. Inclusion: Inclusion of one fuzzy set into another fuzzy set. Fuzzy set A X is included in (is a subset of) another fuzzy set, B X: A(x) B(x), xX Consider X = {1, 2, 3} and sets A and B A = 0.3/1 + 0.5/2 + 1/3; B = 0.5/1 + 0.55/2 + 1/3 then A is a subset of B, or A B 2012 34 3. Cardinality: Cardinality of a non-fuzzy set, Z, is the number of elements in Z. BUT the cardinality of a fuzzy set A, the so-called SIGMA COUNT, is expressed as a SUM of the values of the membership function of A, A(x): cardA = A(x1) + A(x2) + … A(xn) = ΣA(xi), for i=1..n Consider X = {1, 2, 3} and sets A and B A = 0.3/1 + 0.5/2 + 1/3; B = 0.5/1 + 0.55/2 + 1/3 then cardA = 1.8 cardB = 2.05 2012 35 4. Empty Fuzzy Set: A fuzzy set A is empty, IF AND ONLY IF: A(x) = 0, xX Consider X = {1, 2, 3} and set A A = 0/1 + 0/2 + 0/3 then A is empty 2012 36 5. Alpha-cut: An -cut or -level set of a fuzzy set A X is a Crisp set A X, such that: A={A(x), xX} Consider X = {15,25,35,45,55} representing age and set A representing young A = 0.2/15 + 1/25 + 0.8/35+0.3/45+0/55 then A0.5 = {25, 35}, A0.1 = {15, 25, 35, 45}, A1 = {25} A0.5 means “the age that we can say of being young with possibility not less than 0.5”. 2012 37 6. Fuzzy set Normality: A fuzzy subset of X is called normal if there exists at least one element xX such that A(x) = 1 • A fuzzy subset that is not normal is called subnormal. • All crisp subsets except for the null set are normal. •Example: A = 0.2/15 + 1/25 + 0.8/35+0.3/45+0/55 is a Normal set. But A = 0.2/15 + 0.9/25 + 0.8/35+0.3/45+0/55 Is a Subnormal set. 2012 38 7. Height of the Fuzzy set: The height of a fuzzy subset A is the large membership grade of an element in A height(A) = maxx (A(x)) Example: If A = 0.2/15 + 0.6/25 + 0.8/35+0.3/45+0/55 then height(A)=0.8 2012 39 8. Fuzzy set Core: the core of A is the crisp subset of X consisting of all elements with membership grade = 1: core(A) = {x A(x) = 1 and xX} Example: If A = 0.2/15 + 1/25 + 0.8/35+1/45+0/55 then core(A)={25,45} 2012 40 9. Fuzzy set Support: the support of A is the crisp subset of X consisting of all elements with a degree of membership differing from zero: supp(A) = {x A(x) 0 and xX} Example: If A = 0.2/15 + 0/25 + 0.8/35+1/45+0/55 then supp(A)={15, 35,45} 2012 41 3.7. Fuzzy Set Math Operations: kA = {kA(x), xX} Let k =0.5, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then kA = {0.25/a, 0.15/b, 0.1/c, 0.5/d} Ak = {A(x)k, xX} Let k =2, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then Ak = {0.25/a, 0.09/b, 0.04/c, 1/d} 2012 42 3.8. Properties of Fuzzy: 2012 43 Quiz: Consider two fuzzy subsets of the set X, X = {a, b, c, d, e } referred to as A and B A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e} and B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e} 2012 44 1. Cardinality: card(A) = A = {1/a, 0.3/b, 0.3/c ,0.8/d, 0/e} B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e} card(B) = 2. Complement of A and of B A= B= 3. Union: AB= 2012 45 4. Intersection: A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e} AB= B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e} 5. a-cut: A0.2 = A0.3 = A0.8 = A1 = 2012 46 6. Support: supp(A) = A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e} B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e supp(B) = 7. Core: core(A) = core(B) = 8. aA for a=0.5 9. Aa: for a=2 2012 47 Example: Where the possibility for X=3 is 1, the probability is only 0.1 The example shows, that a possible event does not imply that it is probable. However, if it is probable it must also be possible. You might view a fuzzy membership function as your personal distribution, in contrast with a statistical distribution based on observations 2012 48

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