Chap. 7 Relations: The Second Time Around

1. Chap. 7 Relations: The Second Time Around

2. Binary Relation For sets A, B, any subset of A╳Bis called a (binary) relation from A to B. Any subset of A╳A is called a (binary) relation on A.

3. Reflexive Relation • e.g. Given a finite set A with |A|=n. Then, • The number of relations on A is . • 2. The number of reflexive relations on A is .

4. Symmetric Relation e.g. Given a finite set A with |A|=n. Then, 1. The number of symmetic relations on A is . 2. The number of reflexive and symmetic relations on A is .

5. Transitive Relation Let A={1, 2, 3, 4}. Which of the following relation is transitive? a) R1={(1,1), (2,3), (3,4), (2,4)}. b) R2={(1,3), (3,2)}. O X because (1,3), (3,2)∈R2 but (1,3)∉R2 .

6. Antisymmetric Relation Let A={1, 2, 3}. Which of the following relation is antisymmetric? a) R1={={(1,1), (2,2)}. b) R2={(1,2), (2,1), (2,3)}. O X because (1,2), (2,1)∈R2 but 1≠2.

7. Partial Ordering Relation Which of the following relation is a partial order? a) The relation R on the set Z is defined by aRb, or (a, b)∈R, if a≤b. b) Let n∈Z+, For x,y ∈Z, the modulo relation R is defined by xRy if x-y is a multiple of n. O X because it is not antisymmetric.

8. Example 7.15 Let A={1, 2, 4, 8, 16}, the set of positive integer divisors of 16. Define the relation R on the set A by xRy if x divides y. Then, the order pairs from A╳A that comprise R: R= {(1,1), (1,2), (1,4), (1,8), (1,16), (2,2), (2,4), (2,8), (2,16), (4,4), (4,8), (4,16), (8,8), (8,16), (16,16)}.

9. Example 7.15 (2) 1. (c,d)∈R⇔ and , Where m, p∊N with 0≤m≤p≤4. 2. Each possibility for m, p is simply a selection of size 2 from a set of size 5, the set {0,1,2,3,4}, where the repetitions are allowed. 3. Thus, the number of ways to choose m, p is 5. Therefore, the number of order pairs in R is 15.

10. Example 7.15 (3) Let A={1, 2, 3, 4, 6, 12}, the set of positive integer divisors of 12. Define the relation R on the set A by xRy if x divides y. Then, the order pairs from A╳A that comprise R:

11. Example 7.15 (4) 1. (c,d)∈R⇔ where 3. Thus, the number of ways to choose m, p is 4. Similarly, the number of ways to choose n, q is 5. Therefore, the number of order pairs in R is

12. Equivalence Relation Let A={1, 2, 3}. Which of the following is a equivalence relation? O O O O

13. Composite Relation

14. Theorem 7.1 1. 2. It suffices to show ⇔ 3. (Proof of only if part.) ⇒ ⇒ because ⇒ because ⇒ because 4. The proof of if part is similar.

15. Power of Relation

16. (0,1)-Matrix

17. Relation Matrix Let R be a binary relation from A to B. Then, the relation matrix for R, denoted by M(R), is the |A|╳|B| (0,1)-matrix whose entries mi,j, for 1≤i≤|A| and 1≤j≤|B|, are given by Let A={1,2,3,4}, B={w,x,y,z}, and C={5,6,7}. If then

18. Multiplication of Relation Matrices

19. Multiplication of Relation Matrices

20. Multiplication of Relation Matrices 1. It suffices to show [ ]i,j=1 ⇔ [ ]i,k=1 and [ ]k,j=1 for some k. 2. [ ]i,j=1 ⇔ (ai,cj)∊ ⇔ (ai,bk)∊ and (bk,cj)∊ for some bk ⇔ [ ]i,k=1 and [ ]k,j=1 for some k

21. Some Results about Relation Matrices

22. Comparison of (0,1)-Matrices

23. Identity of (0,1)-Matrix

24. Transpose of (0,1)-Matrix

25. Theorem 7.2

26. Directed Graph

27. Relation and Directed Graph

28. Poset Let

29. Hasse Diagram

30. Hasse Diagram (2) e.g.

31. Total Order Which of the following relation is a total order?

32. Total Order (2) O . . . . . .

33. Total Order (3) X

34. Total Order (4) O

35. Topological Sorting

36. Topological Sorting

38. Maximal and Minimal Elements

39. Maximal and Minimal Elements (2)

40. Theorem 7.3 • 1. • 2. • 3. • 4.

41. Least and Greatest Elements Which of the following partial orders has a least element and a greatest element ?

42. Which of the following partial orders has a least element and a greatest element ? O O X X

43. Theorem 7.4 1. 2. It suffices to show 3. 4. 5. 6. x=y

44. Partition Let . Which of the following determines a partition of A ? O O O

45. Equivalence Class

46. Theorem 7.6 • 1. It suffices to show • 2. This is clearly true because • b) (⇒) 1. • 2. Then, • 3. • 4. • 5. • 6. • 7.

47. Theorem 7.6 (2) b) (⇐) 1. 2. c) 1. 2. 3. 4. 5. 6. 7.

48. Theorem 7.7 1. 2. 3. (x,x)∊R ⇒ 4. (x,y)∊R ⇒ 5. (x,y)∊R and (y,z)∊R ⇒ R is reflexive. x and y are in the cell of the partition ⇒ (y,x)∊R ⇒ R is symmetric. x, y, and z are in the cell of the partition ⇒ (x,z)∊R ⇒ R is transitive.

49. Example 7.59 1. 2. 3. 4.

50. Example 7.59 (2) 1. 2.