1. Fuzzy Set and Opertion

2. Outline Fuzzy Set and Crisp Set Expanding concepts Standard operation of fuzzy set Fuzzy relations Operations on fuzzy relations

3. Set

4. Crisp set and Fuzzy set Crisp set – membership function – membership degree: {0,1} Fuzzy set – membership function: user specify – membership degree: [0,1] 4

5. Crisp set and Fuzzy set

6. Expression of fuzzy set 6

7. Example of fuzzy set 7

8. Support Support of A example 8

9. Height The maximum value of the membership degree 9

10. Normalized fuzzy set – height is 1 – young, adult, and senior are normalized fuzzy sets 10

11. -cut -cut set Example If, 11 5

12. -cut 12

13. Level set Example 13

14. Convex fuzzy set 14

15. Non-convex fuzzy set 15

16. Relation of fuzzy sets A and B are equivalent A is a subset of B A is a proper subset of B 16

17. Subset of fuzzy set 17

18. Standard operation of fuzzy set Complement Union Intersection 18

19. Example Complement 19

20. Union Example 20

21. Intersection Example 21

22. Other operations Disjunctive sum (exclusive OR)

23. Other operations

24. Other operations

25. 25 Product set

26. 26 Product set

27. A={a1,a2} B={b1,b2} C={c1,c2} AxBxC = {(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a 2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)} 27

28. 28 Crisp relation A relation among crisp sets is a subset of the Cartesian product. It is denoted by. Using the membership function defines the crisp relation R :

29. 29 Fuzzy relation A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets A 1, A 2,..., A n where tuples (x 1, x 2,..., x n ) may have varying degrees of membership within the relation. The membership grade indicates the strength of the relation present between the elements of the tuple.

30. 30 Representation methods Matrix (Crisp)(Fuzzy)

31. 31 Representation methods Digraph (Crisp)(Fuzzy)

32. 32 α-cut of fuzzy relation Example

33. 33 α-cut of fuzzy relation

34. 34 Operations on fuzzy matrices Sum: Example

35. 35 Operations on fuzzy matrices Max product: C = A ・ B=AB= Example

36. 36 Max product Example

37. 37 Max product Example

38. 38 Max product Example

39. 39 Operations on fuzzy matrices Scalar product: Example

40. 40 Operations on fuzzy relations Union relation For n relations

41. 41 Union relation Example

42. 42 Operations on fuzzy relations Intersection relation For n relations

43. 43 Intersection relation Example

44. 44 Operations on fuzzy relations Complement relation: Example

45. 45 Composition of fuzzy relations Max-min composition Example

46. 46 Composition of fuzzy relations

47. 47 Composition of fuzzy relations Example

48. 48 Composition of fuzzy relations Example

49. 49 Composition of fuzzy relations