Understanding Crisp and Fuzzy Relations in Mathematical Set Theory

1. Fuzzy Relations Review Fuzzy Relations

2. Crisp Relation • Definition (Product set): Let A and B be two nonempty sets, the product set or Cartesian product AB is defined as follows, AB {(a, b) | aA, bB } • Extension to n sets A1A2...An = {(a1,...,an) | a1A1, a2A2, ... , anAn }

3. Crisp Relation Example:A {a1, a2, a3}, B {b1, b2} AB {(a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2)} Product set A  B

4. Crisp Relation A  A {(a1, a1), (a1, a2), (a1, a3), (a2, a1), (a2, a2), (a2, a3), (a3, a1), (a3, a2), (a3, a3)} Cartesian product A  A

5. Crisp Relation • Definition • Binary Relation R = { (x,y) | xA, yB }  A x B • n-ary Relation (x1, x2, x3, … , xn) R , RA1A2A3…An

6. A B A B f y x 1 1 y x R 2 2 ) ) ( R ran( R dom y x 3 3 Crisp Relation • Domain and Range dom(R) = { x | xA, (x, y) R for some yB } ran(R) = { y | yB, (x, y) R for some xA } dom(R) , ran(R) Mapping yf(x)

7. A B x f 1 y x 2 A B y1 x y2 Crisp Relation • Characteristics of relation (1) One-to-many  x  A, y1, y2  B (x, y1)  R, (x, y2)  R (2) Surjection (many-to-one) • f(A) B or ran(R) B. yB, x A, yf(x) • Thus, even if x1x2, f(x1) f(x2) can hold. One-to-many relation (not a function) Surjection

8. A B A B f y x 1 1 f y x 1 1 y x 2 2 y y x x 3 3 2 2 y 4 y x 3 3 y x 4 4 Crisp Relation (3) Injection (into, one-to-one) • for all x1, x2  A, x1  x2 , f(x1)  f(x2). • if R is an injection, (x1, y)  R and (x2, y)  R then x1  x2. (4) Bijection (one-to-one correspondence) • both a surjection and an injection. Injection Bijection

9. y x Crisp Relation • Representation of Relations (1)Bipartigraph representing the relation by drawing arcs or edges (2)Coordinate diagram plotting members of A on x axis and that of B on y axis Relation of x2 + y2 4 Binary relation from A to B

10. Crisp Relation (3) Matrix (4) Digraph MR (mij) i 1, 2, 3, …, m j  1, 2, 3, …, n Matrix Directed graph

11. Crisp Relation • Operations on relations R, SAB (1) Union T  R  S If (x, y) R or (x, y) S, then (x, y) T (2) Intersection T  R  S If (x, y) R and (x, y) S, then (x, y) T. (3) Complement If (x, y) R, then (x, y)  RC (4) Inverse R-1 {(y, x) BA | (x, y) R, xA, yB} (5) Composition T RAB, SBC , TSRAC T {(x, z) | xA, yB, zC, (x, y) R, (y, z) S}

12. Types of Relation on a set • Reflexive relation xA (x, x) R or R(x, x) = 1, xA • irreflexive if it is not satisfied for some xA • antireflexive if it is not satisfied for all xA • Symmetric relation (x, y) R (y, x) R or R(x, y) = R(y, x), x, yA • asymmetric or nonsymmetric when for some x, yA, (x, y) R and (y, x) R. • antisymmetric if for all x, yA, (x, y) R and (y, x) R

13. 2 1 3 2 4 Types of Relation on a Set • Transitive relation For all x, y, zA (x, y) R, (y, z) R (x, z) R 3 1 4 (b) R (a) R Transitive Closure

14. Types of Relation on a Set • Equivalence relation (1) Reflexive xA (x, x) R (2) Symmetric (x, y) R (y, x) R (3) Transitive relation (x, y) R, (y, z) R (x, z) R

15. Types of Relation on a Set • Equivalence classes a partition of A into n disjoint subsets A1, A2, ... , An (a) Expression by set (b) Expression by undirected graph Partition by equivalence relation (A/R)  {A1, A2}  {{a, b, c}, {d, e}}

16. Types of Relation on a Set • Compatibility relation (tolerance relation) (1) Reflexive relation (2) Symmetric relation xA (x, x) R (x, y) R (y, x) R (b) Expression by undirected graph (a) Expression by set Partition by compatibility relation

17. Pre-order relation (1) Reflexive relation xA (x, x) R (2) Transitive relation (x, y) R, (y, z) R (x, z) R A e e b a f a d b, d f, h g c c h g Types of Relation on a Set (b) Pre-order (a) Pre-order relation Pre-order relation

18. Types of Relation on a Set • Order relation (1) Reflexive relation xA (x, x) R (2) Antisymmetric relation (x, y)  R (y, x) R (3) Transitive relation (x, y) R, (y, z) R (x, z) R • strict order relation (1’) Antireflexive relation xA (x, x) R • total order or linear order relation (4) x, yA, (x, y) R or (y, x) R

19. Types of Relations on a Set • Comparison of relations

20. Fuzzy Relation • Definition of fuzzy relation • Crisp relation • membership function R(x, y) R : AB {0, 1} • Fuzzy relation • R : AB [0, 1] • R = {((x, y), R(x, y))| R(x, y)  0 , xA, yB}

21. R 1 0.5 ( a ( a ( ... , b ) ) , a b b ) , 1 1 1 2 1 2 Fuzzy Relation Fuzzy relation as a fuzzy set

22. a c b d Fuzzy Relation • Example Crisp relation R R(a, c)  1, R(b, a)  1, R(c, b)  1 and R(c, d)  1. Fuzzy relation R R(a, c)  0.8, R(b, a)  1.0, R(c, b)  0.9, R(c, d)  1.0 a 0.8 c 1.0 1.0 0.9 b d (a) Crisp relation (b) Fuzzy relation corresponding matrix crisp and fuzzy relations

23. Fuzzy Relation • Operation of Fuzzy Relation 1) Union relation  (x, y) AB RS (x, y)  Max [R (x, y), S (x, y)] R (x, y) S (x, y) 2) Intersection relation R  S(x) = Min [R(x, y), S(x, y)] = R(x, y)S(x, y) 3) Complement relation  (x, y) AB R (x, y)  1 - R (x, y) 4) Inverse relation For all (x, y) AB, R-1 (y, x) R (x, y)

24. Fuzzy Relation • Examples

25. Fuzzy Relation • (Standard) Composition • For (x, y) AB, (y, z) BC, RS (x, z) = Max [Min (R (x, y), S (y, z))] y =  [R (x, y) S (y, z)] y MR  SMRMS • Example =>

26. Fuzzy Relation => Composition of fuzzy relation Note: Matrix Multiplication

27. Fuzzy Relation • -cut of fuzzy relation R = {(x, y) | R(x, y) , xA, yB} : a crisp relation. Example

28. Fuzzy Relation • Decomposition of Fuzzy Relation • Example

29. Fuzzy Relation • Projection • Example

30. Fuzzy Relation • Projection in n dimension • Cylindrical extension C(R) (a, b, c) R (a, b) aA, bB, cC • Example

31. Types of Fuzzy Relations • Reflexive • Irreflexive • Antireflexive • Epsilon Reflexive • Symmetric • Asymmetric • Antisymmetric

32. Types of Fuzzy Relations • Transitive (max-min transitive) • Non-transitive: For some (x,z), the above do not satisfy. • Antitransitive: • Example: X = Set of cities, R=“very far” Reflexive, symmetric, non-transitive

33. Types of Fuzzy Relations • Transitive Closure • Crisp: Transitive relation that contains R(X,X) with fewest possible members • Fuzzy: Transitive relation that contains R(X,X) with smallest possible membership • Algorithm:

34. Types of Fuzzy Relations • Fuzzy Equivalence or Similarity Relation • Reflexive, symmetric, and transitive • Decomposition: • Partition Tree

35. Types of Fuzzy Relations • Fuzzy Compatibility or Tolerance Relation • Reflexive and symmetric • Maximal compatibility class and complete cover • Compatibility class • Maximal compatibility class: largest compatibility class • Complete cover: Set of maximal compatibility classes • Maximal alpha-compatibility class • Complete alpha-covers • Note: Relation from distance metrics forms tolerance relation in clustering.

36. Fuzzy Morphism • Homomorphism • Preserve relations by a function • Example: Log function preserves the order of real data.

37. Other Compositions • Sup-I composition • INF-omega i composition • Degree of Implication • i=min: a < b then 1, otherwise b. • INF-omega i composition

38. Extension of fuzzy set • Extension by relation • Extension of fuzzy set xA, yB yf(x) or x f -1(y) for yB if f -1(y) Example: A {(a1, 0.4), (a2, 0.5), (a3, 0.9), (a4, 0.6)}, B {b1, b2, b3} f -1(b1)  {(a1, 0.4), (a3, 0.9)}, Max [0.4, 0.9]  0.9 B' (b1)  0.9 f -1(b2)  {(a2, 0.5), (a4, 0.6)}, Max [0.5, 0.6]  0.6 B' (b2)  0.6 f -1(b3)  {(a4, 0.6)} B' (b3)  0.6 B'  {(b1, 0.9), (b2, 0.6), (b3, 0.6)}

39. Extension of Fuzzy Set • Extension principle • Extension principle A1 A2  ... Ar ( x1x2 ... xr )  Min [ A1 (x1), ... , Ar(xr) ] f(x1, x2, ... , xr) : XY

40. Extension of Fuzzy Set • Example:

41. Extension of fuzzy set • Extension by fuzzy relation For xA, yB, and B B B' (y)  Max [Min (A (x), R (x, y))] xf -1(y) • Example For b1Min [A (a1), R (a1, b1)]  Min [0.4, 0.8]  0.4 Min [A (a3), R (a3, b1)]  Min [0.9, 0.3]  0.3 Max [0.4, 0.3]  0.4 B' (b1)  0.4 For b2, Min [A (a2), R (a2, b2)]  Min [0.5, 0.2]  0.2 Min [A (a4), R (a4, b2)]  Min [0.6, 0.7]  0.6 Max [0.2, 0.6]  0.6 B' (b2)  0.6 For b3, Max Min [A (a4), R (a4, b3)]  Max Min [0.6, 0.4]  0.4  B' (b3)  0.4 B'  {(b1, 0.4), (b2, 0.6), (b3, 0.4)}

42. Extension of Fuzzy Set • Example A {(a1, 0.8), (a2, 0.3)} B {b1, b2, b3} C {c1, c2, c3} B'  {(b1, 0.3), (b2, 0.8), (b3, 0)} C'  {(c1, 0.3), (c2, 0.3), (c3, 0.8)}

43. Extension of fuzzy set • Fuzzy distance between fuzzy sets • Pseudo-metric distance (1) d(x, x)  0, x  X (2) d(x1, x2) d(x2, x1), x1, x2 X (3) d(x1, x3) d(x1, x2) d(x2, x3), x1, x2, x3 X + (4) if d(x1, x2)=0, then x1 = x2 metric distance • Distance between fuzzy sets • , d(A, B)() Max [Min (A(a), B(b))] d(a, b)

44. Extension of Fuzzy Set • Example A {(1, 0.5), (2, 1), (3, 0.3)} B {(2, 0.4), (3, 0.4), (4, 1)}