1. Fuzzy Relations( 關係 ), Fuzzy Graphs( 圖 形 ), and Fuzzy Arithmetic( 運算 ) Chapter 4

2. 1. Fuzzy Relations A relationship between two objects is represented by a binary relation, a relation with two arguments ( 引數 ). Binary relation A fuzzy relation generalizes the classical notion of relation into a matter of degree

3. Petite( 嬌小 ) person fuzzy relation Fuzzy Relations (cont.)

4. 2. The Composition of Fuzzy Relations Definition 6: Let R be a fuzzy relation that maps X  Y to [0,1] and the possibility distribution of X is known to be  x (x i ). The compositional ( 組合 )rule of inference ( 推論 ) infers the possibility distribution of Y as follows:

5. Max-min composition Max-product composition e.g. About- 5’4’’ 5’4’’={0/5’,0/5’1’’,0.4/5’2’’,0.8/5’3’’,1/5’4’’,0.8/5’5’’,0.4/5’6’’} The composition of Fuzzy Relations(cont.)

6. Using the max-min composition rule of inference, we can compute the weight possibility distribution of a petite person about 5’4’’ tall The composition of Fuzzy Relations(cont.)

7. We can compute the possibility degree for other weights The composition of Fuzzy Relations(cont.)

8. Cylindrical Extension Definition 7: Let R be a fuzzy subset of U i1  U i2  …  U ik where (i1, i2, …, ik), is a subsequence of (1,2,…, n). The cylindrical( 圓柱 ) extension of R in U 1  U 2  …  U n is a fuzzy subset of U 1  U 2  …  U n, denoted as, whose membership function is defined as:

9. Cylindrical Extension(cont.)

10. Fuzzy relation

11. Projection The projection( 投射 ) operation projects a fuzzy relation to a subset of selected dimensions. This operation is often used to exact the possibility distribution of a few selected variables from given fuzzy relation.

12. Projection(conts.) Definition 8: The projection of R on U i1  U i2  …  U ik, denoted as ProjR, is defined as where  denotes applying fuzzy disjunction.

13. Projection(conts.) e.g.

14. Projection(conts.) Cylindrical extension -Extends the dimension of a fuzzy relation Projection -Reduces the dimension of a fuzzy relation

15. A formal Definition of the Composition of Fuzzy Relation A composition of two fuzzy relations is the result of three operations -cylindrically extending each relation so that their dimensions are identical. -intersection the two extended relations. -projecting the intersection to the dimensions not shared by the two original relations.

16. A formal Definition of the Composition of Fuzzy Relation Definition 9: Let R and S be two binary fuzzy relations in U 1  U 2 and U 2  U 3. The composition of the two relations is where and are cylindrical extensions of R and S in U 1  U 2  U 3.

17. 3. Fuzzy Graphs Most fuzzy relations used in real-world applications do not represent a concept( 概念 ), rather they represent a functional mapping from a set of input variables to one or more output variables. A set of fuzzy rules used in a fuzzy logic controller describes a fuzzy relation from the observed state variables to a control decision

18. Fuzzy Graphs(cont.) A fuzzy relation underlying a fuzzy logic controller can be constructed by a set of if-then fuzzy rules. A fuzzy graph describes a functional mapping between a set of input linguistic variables and output linguistic variable

19. Fuzzy Graphs(cont.) e.g. f : If X is small THEN Y is small If X is medium THEN Y is large If X is large THEN Y is small A fuzzy graph f * is f * = small  small + medium  large + large  small where  : product, + : disjunction

20. Fuzzy Graphs(cont.) Fuzzy graph approximation by a disjunction of Cartesian ( 卡笛兒 ) product

21. Fuzzy Graphs(cont.) The Cartesian product of A and B, denoted by A  B is defined as A fuzzy relation formed by a Cartesian product using min as the fuzzy conjunction operator A fuzzy graph f * from X to Y is thus a union of Cartesian products involving linguistic input- output associations

22. 4. Fuzzy Number A fuzzy number is a fuzzy subset of the universe of a numerical numbers -A fuzzy real number is a fuzzy subset of the domain of real numbers -A fuzzy real number, about-10

23. 5. Function with Fuzzy Arguments If we are given a precise function and would like to apply the function to fuzzy numbers, we need to use a technique in fuzzy logic called the extension principle. A fuzzy argument describes a possibility distribution of the argument.

24. Function with Fuzzy Arguments Applying a function to fuzzy arguments generalizes the notion of applying a function to intervals

25. Function with Fuzzy Arguments Monotonic functions -For each point that is mapped, the corresponding possibility degree is mapped along with it.

26. Function with Fuzzy Arguments Non-monotonic functions -Multiple points in its domain may map to the same point in its range.

27. Function with Fuzzy Arguments Two major concepts in extension principle -The possibility of an input value is directly propagated( 傳播 ) to the possibility of its image. -When multiple input combinations map to the same output, the possibility of the output is obtained by combining the possibility of these inputs through fuzzy disjunction.

28. Function with Fuzzy Arguments Ex3. A fuzzy integer number Around-4 for x

29. Function with Fuzzy Arguments Apply the extension principle

30. Function with Fuzzy Arguments If we choose the “max” fuzzy disjunction operator.

31. Function with Fuzzy Arguments Definition 10: Suppose f is a function with n arguments. is replaced by a fuzzy conjunction of

32. Function with Fuzzy Arguments Ex 4. Real number x is a fuzzy number Around-3 We use B to denote the image of A under f (i.e., f(A)=B).

33. Function with Fuzzy Arguments Case 1: Because f is monotonic, the interval maps to Taking the inverse of f, we get We get

34. Function with Fuzzy Arguments Case 2: Combining the two cases

35. Function with Fuzzy Arguments The image of a fuzzy number under the function

36. Function with Fuzzy Arguments Finding the image of fuzzy arguments to a real function involves finding the inverse of the function. We can rewrite the extension principle(if the inverse of f exists).

37. 6. Arithmetic Operations on Fuzzy Numbers Applying the extension principle to arithmetic operations. Fuzzy addition

38. Arithmetic Operations on Fuzzy Numbers Fuzzy subtraction Fuzzy multiplication

39. Arithmetic Operations on Fuzzy Numbers Fuzzy division

40. Arithmetic Operations on Fuzzy Numbers Ex5. A=0.3/1+0.6/2+1/3+0.9/4+0.2/5 B=0.5/10+1/11+0.5/12 To calculate the sum of these two fuzzy integers. f(A+B)=0.3/11+0.5/12+0.5/13+0.5/14 +0.2/15+0.5/12+0.6/13+1/14+0.9/15 +0.2/16+0.3/13+0.5/14+0.5/15+0.5/16 +0.2/17

41. Arithmetic Operations on Fuzzy Numbers Applying max operation f(A+B)=0.3/11+0.5/12+0.6/13+1/14+0.7/15 +0.5/16+0.2/17

42. HW#2 Page 106 -4.3 Page 108 -4.4