Conventional Sets and Fuzzy Sets

Transcription

1 Conventional Sets and Conventional Sets A set is a collection of things, for example the room temperature, the set of all real numbers, etc. 2

2 Conventional Sets Such collection of things are called the Universe of Discourse, X, and is defined as the range of all possible values for a variable. Universe of Discourse can be divided into sets or subsets. For Example, consider a set A of the real numbers between 5 and 8 from the universe of discourse X. Conventional sets called crisp sets 5 8 X 3 Conventional Sets Operations on Classical Sets If we have two sets A and B consisting of a collection of elements in X universe of discourse. x Є X (x belongs to X) x Є A (x belongs to A) x Є X (x does not belong to A) A B (A is fully contained in B; if x Є A, then x Є B) A B (A is contained in or is equivalent to B) A = B (A B and B A) The null set Ø is the set with no elements, and the whole set, X, is the set of elements in the universe. 4

3 Conventional Sets Operations on Classical Sets: Union A U B ; the union represent all elements that reside in both sets A and B. This is called the logic or. A B A U B = [x x Є A or x Є B ] 5 Conventional Sets Operations on Classical Sets: Intersection A Π B ; the intersection represent all elements that simultaneously reside in both sets A and B. This is called the logic and. A A Π B = [x x Є A and x Є B ] B 6

4 Conventional Sets Operations on Classical Sets: Complement Ā ; the complement of set A is the collection of all elements on the universe that do not reside in set A. A Ā = [x x Є A and x Є X ] 7 Conventional Sets Operations on Classical Sets: Difference A B ; the collection of all elements on the universe that reside in A and do not reside in B at the same time. A A B = [x x Є A and x Є B] B 8

5 Conventional Sets Properties of Classical Sets Commutativity: A U B = B U A; also for the intersection Associativity: A U (B U C) = (A U B) U C Distributivity: A U (B Π C) = (A U B) Π (A U C) Transitivity: if A B C, then A C. Identity: A U Ø = A A Π Ø= Ø A U X = X A Π X = A 9 Conventional Sets Properties of Classical Sets Law of Excluded Middle: A U Ā = X A ΠĀ= Ø De Morgan s law: A Π B = Ā U B A U B = ĀΠB A A B B The complement of a union or an intersection is equal to the intersection or union of the respective complement

6 Conventional Sets Properties of Classical Sets: Example Consider an arch consists of two members, if either members fails then the arch will collapse. If E represents survival of member and E2 member 2. Load The survival of the arch will be represented by E Π E2. The collapse is E Π E2. Logically collapse will occur if either members fail, i.e., E U E2. Mapping Conventional Sets If an element x is contained in X and corresponds to an element y contained in Y, it is termed a mapping from X to Y, ƒ : X Y. This is called the characteristic function µ A(x) =, x Є A, x Є A 5 8 X 2

7 In classic sets, the transition of an element in the universe between being a member and non member in a given set is abrupt. In fuzzy sets, this transition occurs gradually A fuzzy set is a set containing elements that have varying degree of membership in the set. Accordingly, elements in a fuzzy sets can be members of other fuzzy set on the same universe. Elements of fuzzy sets are mapped to a universe of membership values using a function-theoretic form 3 This function maps elements of fuzzy set A to a real numbered value between and. A fuzzy set A in the universe X can be defined as set of ordered pairs A = {(x, µ A(x) x Є X} A discrete and finite fuzzy set is represented as follow A = µ A(x) /x + µ A(x2) /x2 + When x is continuous A = µ A(x) /x 4

8 Example Score High Medium Low low medium high Example Score Medium B = Medium score = {(, ), (2, ), (3,.), (4,.5), (5,.8), (6, ), (7,.8), (8,.5), (9, ), (, )} Or B = (3,.), (4,.5), (5,.8), (6, ), (7,.8), (8,.5)} Or B =./3 +.5/4 +.8/5 + /6 +.8/7 +.5/8 6

9 Operations Union: the membership functions of the union of the two fuzzy sets A and B is defined as the maximum of both µ A U B (x) = µ A (x) V µ B (x) 7 Operations Intersection: the membership functions of the intersection of the two fuzzy sets A and B is defined as the minimum of both µ A Π B (x) = µ A (x) ^ µ B (x) 8

10 Operations Complement: the membership functions of the complement of fuzzy set A is defined as 9 Operations The same operations of the classical sets are still valid for the fuzzy sets. Commutativity: A U B = B U A; also for the intersection Associativity: A U (B U C) = (A U B) U C Distributivity: A U (B Π C) = (A U B) Π (A U C) Transitivity: if A B and B C, then A C. De Morgan s law 2

11 Operations Two fuzzy sets are equal if and only if µ A (x) = µ B (x) for all x Є X. A is a sub set of B: A B, if and only if µ A (x) < µ B (x) for all x Є X. 2 Example Consider the following two fuzzy sets: A = { /2 +.5/3 +.3/4 +.2/5} B = {.5/2 +.7/3 +.2/4 +.4/5} Complement Ā = { / + /2 +.5/3 +.7/4 +.8/5} Complement B = { / +.5/2 +.3/3 +.8/4 +.6/5} Union: A U B = {/2 +.7/3 +.3/4 +.4/5} Intersection: A Π B = {.5/2 +.5/3 +.2/4 +.2/5} 22

12 Example Consider the following two fuzzy sets: A = { /2 +.5/3 +.3/4 +.2/5} B = {.5/2 +.7/3 +.2/4 +.4/5} Difference A B = A Π B = {.5/2 +.3/3 +.3/4 +.2/5} De Mogan s law = A U B = ĀΠB = { / + /2 +.3/3 +.7/4 +.6/5} 23 Normal Fuzzy Set A fuzzy set A is normal if its maximal degree of membership is unity (i.e., there must exist at least one x for which µ A (x) =. On the other hand, non-normal fuzzy sets have maximum degree of membership less than one Degree of Membership Universe of Discourse 24

13 Support of a Fuzzy Set Support of a fuzzy set A (written as supp(a)) is a (crisp) set of points in X for which µa is positive supp(a) = { x Є X µ A (x)>} Score Medium Support (B) = Medium score = {3, 4, 5, 6, 7, 8} 25 Convex Fuzzy Set A fuzzy set A is convex if and only if it satisfies the following µ A ( λx + ( λ ) x2 ) min ( µ A ( x), µ A ( x2 )), where λ is in the interval [,], and x < x2 26

14 α-cut of a Fuzzy Set α-cut is defined as a crisp set A α (or a crisp interval) for a particular degree of membership, α: A α = [a α, b α ], where α can take on values between [,] 27 α-cut of a Fuzzy Set: Example Consider the score example Score Medium B.8 = Medium score.8 = { 5, 6, 7} 28

15 Fuzzy Numbers Fuzzy number is a fuzzy set which is both normal and convex. In addition, the membership function of a fuzzy number must be piecewise continuous. Most common types of fuzzy numbers are triangular and trapezoidal. Other types of fuzzy numbers are possible, such as bell-shaped or gaussian fuzzy numbers, as well as a variety of one sided fuzzy numbers. Triangular fuzzy numbers are defined by three parameters, while trapezoidal require four parameters 29 Fuzzy Numbers 3

16 Resolution Principle A fuzzy set A can be expanded in terms of its α-cuts. µ A (x) = α ^ µ A α (x); x Є X This means that a fuzzy set can be decomposed into αa α, α Є [, ]. µ A (x) α 2 α α 2 Aα 2 α Aα Aα 2 Aα X 3 Resolution Principle: Example Consider the following fuzzy set: A = {./5 +.3/6 +.5/7 +.8/8 + /9 + /} Using the resolution principle: A =. {/5 + /6 + /7 + /8 +/9 + /} +.3 {/6 + /7 + /8 +/9 + /} +.5 {/7 + /8 +/9 + /} +.8 {/8 +/9 + /} + {/9 + /} =. A. +.3 A A A.8 + A 32

17 Representation Theorem As opposed to the resolution principle, a fuzzy set A can be represent in terms of its α-cuts. i.e., A fuzzy set can be retrieved as a union of its αa α. µ A (x) A = U αa α α 2 α Aα 2 Aα X 33 Representation Theorem: Example If we are given: A. = {, 2, 3, 4, 5}, A.4 = {2, 3, 5}, A.8 = {2, 3}, and A = {3} Then, fuzzy set A can be expressed as: A = U αa α for αє[, ]. A =. A. +.4 A A.8 + A =. {/ + /2 + /3 + /4 +/5} +.4 {/2 + /3 + /5} +.8 {/2 + /3} + {/3} =./ +.8/2 + /3 +./4 +.4/5 34

18 Extension Principle Consider a single relationship between one independent variable x and one dependent variable y. x ƒ(x) y The function ƒ(x) represents the mapping of x on y. y = ƒ(x) The function y = ax + b, are mapping from one universe X to another universe Y and is written as: ƒ : X Y Sometimes it is called the image of x under ƒ for y=ƒ(x) 35 Extension Principle The extension principle can be also applied to fuzzy sets. Given a function f : U V, and a set A in U for x Є U, then its image, set B, in the universe V is found from the mapping, B = ƒ(a) x B (y) = x f(a) (y) µ B (y) = V µ f(a) (y); y=f(x) 36

19 Extension Principle: Example Consider a crisp set A = [, ] defined in the universe X = {-2, -,,, 2}, where A = {/-2 + /- + / + / +/2} and mapping function y= 4x +2. Find the set B on an output universe Y using the extension principle. The universe Y = f(x) for x Є X Then Y = {2, 6, }, the mapping for membership µ B (2) = V [µ A ()] = µ B (6) = V [µ A (-), µ A ()] = µ B () = V [µ A (-2), µ A (2)] = Then B = {/2 + /6 + /} or B = [2, 6 ] 37 Extension Principle The same operations of the classical sets are still valid for the fuzzy sets. Given a function ƒ : U V and a fuzzy set A in U, where A = µ /x + µ 2 /x 2 + µ 3 /x 3 +., the extension principle states: ƒ(a) = ƒ(µ /x + µ 2 /x 2 + µ 3 /x 3 +.) = µ /ƒ (x )+ µ 2 /ƒ(x 2 ) + µ 3 /ƒ(x 3 )+. Or the resulting set B = µ A (x )/y + µ A (x 2 )/y 2 + If more than one element of U is mapped to the same element y of V, then the max membership is taken 38

20 Extension Principle: Example Consider X = {, 2, 3, 4, 5, 6, 7, 8, 9, }; a fuzzy set A = large is given as = {.5/6 +.7/7 +.8/8 +.9/9 +/} Given a function ƒ : y=f(x) = x 2, find the fuzzy set B = large 2 B = {.5/36 +.7/49 +.8/64 +.9/8 + /} One to one mapping always reserve the membership values 39 Extension Principle: Example Consider A = {./-2 +.4/- +.8/ +.9/ +.3/2} ƒ(x) = x 2 3, using extension principle to find B = ƒ(x) B ={./(4-3)+.4/(-3)+.8/(-3)+.9/(-3)+.3/(4-3)} B = {./ +.4/-2 +.8/-3 +.9/-2 +.3/} B = {(. V.3)/ + (.4 V.9)/-2 +.8/-3} B = {.3/ +.9/-2 +.8/-3} 4

21 Extension Principle: Example ƒ(x) = x