8.1 Relations on Sets

Definition: Relation

Let A and B be sets. A relation R from A to B is a subset of A ⨯ B. Given an ordered pair (x,y) in A ⨯ B, x is related to y by R, written x R y, if and only if, (x, y) is in R. The set A is called the domain of R and the set B is called its co-domain.

• x R y means that (x, y) ∈ R

Examples

• Less-than Relation (defined from R to R): x L y ↔ x < y
• Congruence Modulo 2 Relation (defined from Z to Z): m E n ↔ m - n is even

Definition: Inverse Relation

Let R be a relation from A to B. Define the inverse relation R^-1 from B to A as follows:

• R^-1 = {(y, x) ∈ B ⨯ A | (x, y) ∈ R}

Definition: Relation on a Set

A relation on a set A is a relation from A to A.

In Class Exercises

• 3. The congruence modulo 3 relation, T, is defined from Z to Z as follows: For all integers m and n, m T n ↔ 3 | (m - n)
  • Is 10 T 1? Is 1 T 10? Is (2, 2) ∈ T? Is (8, 1) ∈ T?
  • List 3 integers n such that n T 0.
• 11. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" relation. That is, for every ordered pair (x, y) ∈ A ⨯ B, x S y ↔ x | y. State explicitly which ordered pairs are in S and S^-1.
• 15. Let A = {2, 3, 4, 5, 6, 7, 8} and define a relation R on A as follows: For every x, y ∈ A, x R y ↔ x | y. Draw the directed graph of the relation defined by R.
• 19. Let A = {2, 4} and B = {6, 8, 10} and define relations R and S from A to B as follows: For every (x, y) ∈ A ⨯ B,
  • x R y ↔ x | y and
  • x S y ↔ y - 4 = x
  State explicitly which ordered pairs are in A ⨯ B, R, S, R ∪ S and R ∩ S.