Chap. 7 Relations: The Second Time Around

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.

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 .

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 .

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 .

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.

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

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)}.

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.

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:

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

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

Equivalence Relation 2. Equivalence Class

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

Relation and Directed Graph

Poset Let

Hasse Diagram

Hasse Diagram (2) e.g.

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

Maximal and Minimal Elements

Theorem 7.3 1. 2. 3. 4.

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

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

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

Equivalence Class

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, .

Theorem 7.6 (2) 5. To show , we need to show for all . 6. 7. b) (⇐) 1. 2.

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

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.

Example 7.59 1. 2. 3. 4.

Example 7.59 (2) 1. 2.