Download presentation

Presentation is loading. Please wait.

Published byTania Torell Modified about 1 year ago

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

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google