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Fuzzy Set and Fuzzy Logic

Fuzzy Set and Fuzzy Logic. Basic ideas and entities in fuzzy set theory Operations and relations on fuzzy sets How to compute with fuzzy sets and numbers - arithmetic, unions, intersections, complements. Fuzzy Logic. IIIT Allahabad. 11/09/11. 1. Books.

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Fuzzy Set and Fuzzy Logic

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  1. Fuzzy Set and Fuzzy Logic • Basic ideas and entities in fuzzy set theory • Operations and relations on fuzzy sets • How to compute with fuzzy sets and numbers - arithmetic, unions, intersections, complements. • Fuzzy Logic. IIIT Allahabad 11/09/11 1

  2. Books • ‘Fuzzy Set Theory and its Applications’ by zimmermann . • ‘Fuzzy Set and Fuzzy Logic’ by George J. Klir and Bo Yuann . • Fuzzy Logic with Engineering Applications by J. Ross. IIIT Allahabad 11/09/11 2

  3. OUTLINEII. BASICS A. Definitions and examples 1. Sets 2. Fuzzy numbers B. Operations on fuzzy sets – union, intersection, complement C. Operations on fuzzy numbers – arithmetic, equations, functions and the extension principle IIIT Allahabad 11/09/11 3

  4. DEFINITIONS A. Definitions 1. Sets a. Classical sets – either an element belongs to the set or it does not. For example, for the set of integers, either an integer is even or it is not (it is odd). However, either you are in the INDIA or you are not. What about flying into INDIA, what happens as you are crossing? Another example is for black and white photographs, one cannot say either a pixel is white or it is black. However, when you digitize a b/w figure, you turn all the b/w and gray scales into 256 discrete tones. IIIT Allahabad 11/09/11 4

  5. Classical sets Classical sets are also called crisp (sets). Lists: A = {apples, oranges, cherries, mangoes} A = {a1,a2,a3 } A = {2, 4, 6, 8, …} Formulas: A = {x | x is an even natural number} A = {x | x = 2n, n is a natural number} Membership or characteristic function IIIT Allahabad 11/09/11 5

  6. Fuzzy Sets: Why do we need them? • Situations in which boundaries of the set are not sharply defined or all elements of the set cannot be assigned equal importance. • Example- Suppose we want to represent the set of intelligent students in a class in the classical set theory. How we define set of ‘tall persons’, ‘healthy persons’ comfortable houses? IIIT Allahabad 11/09/11 6

  7. Definitions – fuzzy sets Fuzzy sets – If X is a collection of objects denoted generically by x, then a fuzzy set A in the set X is a set of ordered pairs {x, (x)} where A (x) is the membership function of x. Symbolically we may write: A fuzzy set has a graphical description that expresses how the transition from one to another takes place. This graphical description is called a membership function. IIIT Allahabad 11/09/11 7

  8. Examples • Example 1: Consider the statement: a ‘suitable’ have for a three member family, when available houses have up to sight rooms.The above statement can be represented by the following fuzzy set A. • Example 2: A real number close to 10 may be represented as IIIT Allahabad 11/09/11 8

  9. There is nothing fuzzy about Fuzzy Sets In fuzzy set literature one finds different methods of describing a fuzzy set. All methods, however, lead to a clear definition of an ambiguous statement. For instance in the example of a suitable on acceptable house for a family of three persons, the degree of acceptability is well defined from the concerned family’s point of view. Similarly once the membership function Ã(x) has been defined as in previous slide there is no ambiguity about user’s view of real numbers close to 10. Thus once an ambiguous statement has been precisely defined by the user; it is not longer ambiguous, at least for the user. Or as they say these is nothing fuzzy about the fuzzy sets. IIIT Allahabad 11/09/11 9

  10. Definitions – fuzzy sets (figure from Klir&Yuan) IIIT Allahabad 11/09/11 10

  11. Definitions: Fuzzy Sets (figure from Klir&Yuan) IIIT Allahabad 11/09/11 11

  12. Membership functions (figure from Klir&Yuan) IIIT Allahabad 11/09/11 12

  13. Fuzzy set (figure from Earl Cox) IIIT Allahabad 11/09/11 13

  14. Properties of Fuzzy Sets: • Cardinality and Relative Cardinality In crisp so theory, the cardinality of a set is the total number of elements in that set. The cardinality of a fuzzy set A denoted by card (A) or  A is defined as Cardinality of a fuzzy set plays an important role in fuzzy data bases and information systems Relative cardinality of a fuzzy set A denoted by A is defined as Where U is the cardinality of the universe U of discourse IIIT Allahabad 11/09/11 14

  15. Height and Suppot of a Fuzzy Set • Height of a fuzzy set is the highest membership value of its membership function. • A fuzzy set with height 1 is called a Normal Fuzzy Set. • The support of a fuzzy set A is the set of elements whose membership function is non zero. Let a fuzzy set A be defined on the universe of discourse U. Then we may define support of fuzzy set A as IIIT Allahabad 11/09/11 15

  16.  - Cut: • The notion of -cut (also called -level cut) is more general than that of support. Let  be a number between 0 and 1. The -cut of fuzzy set A at level  is the set of those elements of A where membership function is greater than or equal to . Mathematically the -cut of a fuzzy set A defined over a universe of discourse U is • Based on the notion of  - cuts, a fuzzy set can be decomposed in to multiple crisp sets using different  - levels. Intuitively each  - level specifies a slice of the membership function. The original member ship function can be reconstructed by piling up these slices in order. IIIT Allahabad 11/09/11 16

  17. Alpha levels, core, support, normal IIIT Allahabad 11/09/11 17

  18. Compliment of a Fuzzy Set: • Let à be a fuzzy set defined over the universe of discourse U. Then the compliment of fuzzy set à denoted by C(Ã) or – à is a fuzzy set whose elements are same as that of à with membership function. • In other words if then its complement • Example: If A = {(2, .2), (3, .6), (4, .9), (5, 1), (6, .8)} is a fuzzy set defined over the universe of discourse U = {1, 2, 3, ---8}, then C (A) = {(1, 1), (2, .8), (3, .2), (4, .1), (6, .2), (7, 1), (8,11)}. IIIT Allahabad 11/09/11 18

  19. Subset of a Fuzzy Set • A fuzzy set B is called a subset of fuzzy set A (B A). If • In other words for every element x in the universe of discourse U, the membership degree in B is less than membership degree in A. • Example: Let defined over the universe of discourse then is subset of A. Is not a subset of A. IIIT Allahabad 11/09/11 19

  20. Convex Fuzzy Sets: • Let A be a fuzzy set defined the universe of discourse U. Then set A is said to be a convex fuzzy set if and only if For each x1, x2 U and 0  1. Geometrically it implies that a convex fuzzy set will not have any valley in the interval of discourse. IIIT Allahabad 11/09/11 20

  21. Union and Intersection of Fuzzy sets: • Let A and B be two fuzzy sets with membership functions A(x) and B(x) respectively defined on the same universe of discourse X. • Then A U B is a fuzzy set C whose membership function C (x) is maximum of A (x) and B(x) for each x X. • Similarly A ∩ B is a fuzzy set D whose membership function D (x) is minimum of  A (x) and B (x) for each x X. IIIT Allahabad 11/09/11 21

  22. Example: • Let fuzzy Sets Be defined over the same universe X = {1, 2, 3, ----10} then • (we do not normally write elements which zero membership functions) IIIT Allahabad 11/09/11 22

  23. Let A and B are two fuzzy set with membership functions A(x) and B(x) defined on the same universe X . • Algebraic Sum where Bounded sum Bounded difference Algebraic Product where IIIT Allahabad 11/09/11 23

  24. Power of a Fuzzy Set: Let a fuzzy set A is defined as Then the nth power of a fuzzy set is defined as Example: Let then Operations of concentration and dilation are special cases of Ãn for n = 2 and n = ½ respectively. Concentration reduces the value of Ã (x) for all x except when Ã (x) = 1. Therefore makes the set less fuzzy. Dilation increases the value of Ã (x) for all x except where Ã (x) = 1 or zero. It, therefore, makes the set more fuzzy. IIIT Allahabad 11/09/11 24

  25. 2. Fuzzy Number A fuzzy number A must possess the following three properties: 1. A must must be a normal fuzzy set, 2. The alpha levels must be closed for every , 3. The support of A, , must be bounded. IIIT Allahabad 11/09/11 25

  26. Fuzzy Number (from Jorge dos Santos) is the suport of z1 is the modal value Membership function 1 ’ is an -level of ,(0,1]  IIIT Allahabad 11/09/11 26

  27. Fuzzy numbers defined by its -levels (from Jorge dos Santos) A fuzzy number can be given by a set of nested intervals, the -levels: 1 .7 .5 .2 0 IIIT Allahabad 11/09/11 27

  28. Triangular fuzzy numbers 1 IIIT Allahabad 11/09/11 28

  29. Fuzzy Number (figure from Klir&Yuan) IIIT Allahabad 11/09/11 29

  30. B. Operations on Fuzzy Sets: Union and Intersection (figure from Klir&Yuan) IIIT Allahabad 11/09/11 30

  31. Operations on Fuzzy Sets: Intersection (figure from Klir&Yuan) IIIT Allahabad 11/09/11 31

  32. Operations on Fuzzy Sets: Union and Complement (figure from Klir&Yuan) IIIT Allahabad 11/09/11 32

  33. C. Operations on Fuzzy Numbers: Addition and Subtraction (figure from Klir&Yuan) IIIT Allahabad 11/09/11 33

  34. Operations on Fuzzy Numbers: Multiplication and Division (figure from Klir&Yuan) IIIT Allahabad 11/09/11 34

  35. Fuzzy Equations IIIT Allahabad 11/09/11 35

  36. Example of a Fuzzy Equation (figure from Klir&Yuan) IIIT Allahabad 11/09/11 36

  37. The Extension Principle of Zadeh Given a formula f(x) and a fuzzy set A defined by, how do we compute the membership function of f(A) ? How this is done is what is called the extension principle (of professor Zadeh). What the extension principle says is that f (A) =f(A( )). The formal definition is: [f(A)](y)=supx|y=f(x){ } IIIT Allahabad 11/09/11 37

  38. Extension Principle - Example Let f(x) = ax+b, IIIT Allahabad 11/09/11 38

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