1. CLASSICAL RELATIONS AND FUZZY RELATIONS

2. 報告流程 卡氏積 (Cartesian Product) 明確關係 (Crisp Relations)
Cardinality
Operations
Properties
合成 (Composition)
模糊關係 (Fuzzy Relations)
Fuzzy Cartesian Product and Compositon
Noninteractive Fuzzy Sets
Crisp Tolerance and Equivalence Relations
Fuzzy Tolerance and Equivalence Relations
Value Assignments
Cosine Amplitude
Max-min Method
Other Similarity Methods

3. Cartesian Product Producing ordered relationships among sets
X × Y = {(x,y)│x∈X, y∈Y}
All the Ar = A
A1 × A2 × ……. × Ar = Ar

4. Cartesian Product Example 3.1 Set A = { 0,1 } Set B = { a, b, c }
A × B = {(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)}
B × A = {(a,0),(a,1),(b,0),(b,1),(c,0),(c,1)}
A × A = A2 = {(0,0),(0,1),(1,0),(1,1)}
B × B = B2 ={(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)}

5. Crisp Relations Measure by characteristic function：χ
X × Y = {(x,y)│x∈X, y∈Y}
Binary relation
χX×Y(x,y)= 1, (x,y) ∈ X × Y
0, (x,y) X × Y
χR(x,y)= 1, (x,y) ∈ X × Y
0, (x,y) X × Y

6. Crisp Relations EX： X={1,2,3} Y={a,b,c} a b c 1 R = 2 3
Sagittal diagram
Relation Matrix
a b c 1
R = 2 3

7. Crisp Relations Example 3.2 (一) (二) X={1,2} Y={a,b} 1 a
Locations of zero b
R={(1,a),(2,b)} R X × Y
(二)
A={0,1,2}
UA：universal relation IA：identity relation
以 A2 為例
UA = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)}
IA = {(0,0),(1,1),(2,2)}

8. Crisp Relations Example 3.3 Continous universes
R={(x,y) | y ≥ 2x, x∈X, y∈Y}
χR(x,y)= 1, y ≥ 2x
0, Y < 2x

9. Cardinality of Crisp Relations
X：n elements Y：m elements
n X ：the cardinality of X
n Y ：the cardinality of Y
Cardinality of the relation
n X × Y = nX * nY
power set
The cardinality ：P(X × Y)
n P(X × Y) = 2(nXnY)

10. Operations on Crisp Relations
Union
Intersection
Complement
Containment
Identity
(Ø → O and X → E)

11. Properties of Crisp Relations
交換律(Commutative law)
結合律(Associative law)
分配律(Distributive law)
乘方(Involution)
冪等律(Idempotence)
狄摩根定律(De Morgan’s law)
排中律(Low of Excluded Middle)

12. Composition R={(X1,Y1),(X1,Y3),(X2,Y4)} S={(Y1,Z2),(Y3,Z2)}
Composition oeration
Max-min composition
T=R。S
Max-product comositon

13. Composition Example 3.4 Max-min composition y1 y2 y3 y4 z1 z2
R= x S= y1
x y2
x y3 y4
z1 z2
T= x1 x2 x3

14. Fuzzy Relations Membership function Cardinality of Fuzzy Relations
Interval [0,1]
Cartesian space X × Y =>
Cardinality of Fuzzy Relations
Universe is infinity

15. Operations on Fuzzy Relations
Union
Intersection
Complement
Containment

16. Properties of Fuzy Relations
排中律(Low of Excluded Middle)在Fuzzy 集合中並不成立 !

17. Fuzzy Cartesian Product
Cartesian product space
Fuzzy relation has membership function
Example 3.5

18. Fuzzy Composition Fuzzy max-min composition
Fuzzy max-product composition
不論 crisp 或 fuzzy 的composition

19. Fuzzy Composition Example 3.6 X={x1,x2} Y={y1,y2} Z={z1,z2,z3}
Max-min composition
Max-product compositon

20. Noninteractive Fuzzy Sets
Fuzzy set on the Cartesian space X =X1 × X2
noninteractive
interactive

21. Noninteractive Fuzzy Sets
Example 3.7

22. Noninteractive Fuzzy Sets
Example 3.7(續)
Cartesian product

23. Noninteractive Fuzzy Sets
Example 3.7(續)
Max-min composition
Example 3.8

24. Noninteractive Fuzzy Sets
Example 3.9

25. Tolerance and Equivalence relations
自返性(reflexivity)
對稱性(symmetry)
傳遞性(transitivity)

26. Crisp Eqivalence Relation
自返性(reflexivity)
(xi,xi) R or
對稱性(symmetry)
(xi,xj) R (xj,xi) R or
傳遞性(transitivity)
(xi,xj) R and (xj,xk) R (xi,xk) R

27. Crisp Tolerance Relation
Also called proximity relation
Only the reflexivity and symmetry
Can be reformed into an equivalence relation
By at most (n-1) compositions with itself

28. Crisp Tolerance Relation
Example 3.10
X={x1,x2,x3,x4,x5}={Omaha, Chicago, Rome, London, Detroit}
R1 does not properties of transitivity
e.g. (x1,x2) R1 (x2,x5) R but (x1,x5) R1

29. Crisp Tolerance Relation
Example 3.10(續)
R1 can become an equivalence relation through two compositions

30. Fuzzy tolerance and equivalence relations
自返性(reflexivity)
對稱性(symmetry)
傳遞性(transitivity)

31. Fuzzy tolerance and equivalence relations
Fuzzy tolerance relation Can be reformed into an equivalence relation
By at most (n-1) compositions with itself

32. Fuzzy tolerance and equivalence relations
Example 3.11
It is not transitive
One composition
Reflexive and symmetric

33. Fuzzy tolerance and equivalence relations
Example 3.11(續)

34. Value assignments Cartesian product Closed-from expression
Simple observation of a physical process
No variation
model the process crisp relation
Y= f(X)
Lookup table
Variability exist
Membership values on the interval [0,1]
Develop a fuzzy relation
Linguistic rules of knowledge
If-then rules
Classification
Similarity methods in data manipulation

35. Cosine Amplitude
X={x1,x2,….,xn}
xi={ }

36. Cosine Amplitude Example 3.12 r12=0.836 Regions x1 x2 x3 x4 x5
Xi1—Ratio with no damage
0.3
0.2
0.1
0.7
0.4
Xi2—Ratio with medium damage
0.6
Xi3—Ratio with serious damage

37. Cosine Amplitude Example 3.12(續) Tolerance relation
Equivalence relation

38. Max-min Method rij= where i, j =1,2,…n Example 3.13
Reconsider Example 3.12
Tolerance relation

39. Summary
Q & A