1. Chapter 3 FUZZY RELATION AND COMPOSITION
Chi-Yuan Yeh

2. Outline Product set Crisp / fuzzy relations
Composition / decomposition
Projection / cylindrical extension
Extension of fuzzy set / fuzzy relation
Fuzzy distance between fuzzy sets

3. Product set

4. Product set

5. 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)}

6. 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 :

7. Fuzzy relation
A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets A1, A2, ..., An where tuples (x1, x2, ..., xn) 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.

8. Representation methods
Bipartigraph
(Crisp)
(Fuzzy)

9. Representation methods
Matrix
(Crisp)
(Fuzzy)

10. Representation methods
Digraph
(Crisp)
(Fuzzy)

11. Domain and range of fuzzy relation

12. Domain and range of fuzzy relation
Fuzzy matrix

13. Operations on fuzzy matrices
Sum:
Example

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

15. Max product
Example

16. Max product
Example

17. Max product
Example

18. Operations on fuzzy matrices
Scalar product:
Example

19. Operations on fuzzy relations
Union relation
For n relations

20. Union relation
Example

21. Operations on fuzzy relations
Intersection relation
For n relations

22. Intersection relation
Example

23. Operations on fuzzy relations
Complement relation:
Example

24. Composition of fuzzy relations
Max-min composition
Example

25. Composition of fuzzy relations

26. Composition of fuzzy relations
Example

27. Composition of fuzzy relations
Example

28. Composition of fuzzy relations

29. α-cut of fuzzy relation
Example

30. α-cut of fuzzy relation

31. Decomposition of relation

32. Decomposition of relation

33. Decomposition of relation

34. Projection / cylindrical extension

35. Projection / cylindrical extension

36. Projection in n dimension

37. Projection

38. Projection

39. Projection

40. Projection

41. Projection / cylindrical extension

42. Cylindrical extension

43. Cylindrical extension

44. Cylindrical extension
x1 = 0 : x，x1 = 1 : y
x2 = 0 : a， x2 = 1 : b
x3 = 0 : α， x3 = 1 : β

45. Cylindrical extension
Join(R123’，R123’’) = C(R123’)∩C(R123’’)
= Min(R123’，R123’’)
= R123’’’

46. Extension of fuzzy set
A crisp function Let then

47. Extension of fuzzy set
There are two universal sets And We can obtain B by A and R, use

48. Extension of fuzzy set By

49. Extension of fuzzy set
If A is a fuzzy set and R is We can also get B by A an R, use

50. Extension of fuzzy set
By use

51. Extension of fuzzy set
If A is a fuzzy set and R is a fuzzy relation We can get B by using

52. Extension of fuzzy set By

53. Extension of fuzzy set
Extension of a crisp relation

54. Extension of fuzzy set

55. Extension by fuzzy relation

56. Extension by fuzzy relation

57. Extension by fuzzy relation

58. Extension by fuzzy relation

59. Extension by fuzzy relation

60. Extension by fuzzy relation

61. Fuzzy distance between fuzzy sets
nonnegative

62. Fuzzy distance between fuzzy sets

63. Fuzzy distance between fuzzy sets

64. Fuzzy distance between fuzzy sets

65. Fuzzy distance between fuzzy sets

66. Thanks for your attention!