1. Chap. 7 Relations: The Second Time Around

2. Binary Relation
For sets A, B, any subset of A╳B is 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.
c) The relation R on the set A={1,2,3,4} is defined by aRb if a|b. O
Total Order X
because it is not antisymmetric. O

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 O

13. Equivalence Relation 2.
Equivalence Class

14. Directed Graph V: vertex set E: edge set V: set of vertices
E: subset of V ╳ V

15. Relation and Directed Graph

16. Poset
Let

17. Hasse Diagram

18. Hasse Diagram (2)
e.g.

19. Total Order Which of the following relation is a total order? O X O
. . .

20. Maximal and Minimal Elements

21. Theorem 7.3 1. 2. 3. 4.

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

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

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

25. Equivalence Class

26. Theorem 7.6 1. It suffices to show . 2. This is clearly true because .
b) (⇒) 1. It suffices to show
2. To show , we need to show for all , .
3. Clearly,
4. Thus,

27. Theorem 7.6 (2) 5. To show , we need to show for all . 6. 7. b) (⇐) 1.2.

28. Theorem 7.6 (3)
c) 1. 2. 3. 4. 5. 6. 7.

29. 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.

30. Example 7.59 1. 2. 3. 4.

31. Example 7.59 (2) 1. 2.