CLASSICAL RELATIONS AND FUZZY RELATIONS

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) • Cardinality • Operations • Properties • 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. a b c 1 R = 2 3 Crisp Relations • EX： X={1,2,3} Y={a,b,c} • Relation Matrix • Sagittal diagram

7. Crisp Relations • Example 3.2 • (一) • X={1,2} Y={a,b} 1 a • Locations of zero 2b • 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 • T=R。S

13. Composition • Example 3.4 • Max-min composition y1 y2 y3 y4 z1 z2 R= x1 S= y1 x2 y2 x3y3 y4 z1 z2 T= x1 x2 x3

14. Fuzzy Relations • Membership function • 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 • Max-min composition

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 or

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) R1 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 • 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

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