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 • Membership Function • Universal Set • user specify (crisp 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]

5. Crisp set and Fuzzy set • Example • Universal set • Crisp Set • Fuzzy Set

6. Expression of fuzzy set

7. Example of fuzzy set

8. Support • Support of A • example

9. Height • The maximum value of the membership degree

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

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

12. -cut

13. Level set • Level set • Example

14. Convex fuzzy set

15. Non-convex fuzzy set

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

17. Subset of fuzzy set

18. Standard operation of fuzzy set • Complement • Union • Intersection

19. Complement • Example

20. Union • Example

21. Intersection • Example

22. Other operations • Disjunctive sum (exclusive OR)

23. Other operations

24. Other operations

25. Product set

26. Product set

27. Product set • A={a1,a2} B={b1,b2} C={c1,c2} • AxBxC = {(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)}

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. Fuzzy relation • Afuzzy relation is a fuzzy setdefined on the Cartesian product of crisp sets A1, A2, ..., Anwhere tuples (x1, x2, ..., xn)may have varying degrees of membership within the relation. • The membership gradeindicates the strength of the relation present between the elements of the tuple.

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

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

32. α-cut of fuzzy relation • Example

33. α-cut of fuzzy relation

34. Operations on fuzzy matrices Sum: Example

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

36. Max product Example

37. Max product Example

38. Max product Example

39. Operations on fuzzy matrices Scalar product: Example

40. Operations on fuzzy relations Union relation For n relations

41. Union relation Example

42. Operations on fuzzy relations Intersection relation For n relations

43. Intersection relation Example

44. Operations on fuzzy relations Complement relation: Example

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

46. Composition of fuzzy relations

47. Composition of fuzzy relations • Example

48. Composition of fuzzy relations • Example

49. Composition of fuzzy relations