Discrete Mathematical Structures by Pearson

2-2 Discrete Mathematical Structures

Thus, if R ⊆ A × B and (a, b) ∈ R, then we say that a is related to b by relation R and

write a R b.

Symbolically, R = {(a, b): a ∈ A, b ∈ B and a R b}

If A = B, then R is said to be a binary relation on A and R = {(a, b): a, b ∈ A and a R b}

If an element a is not related to b by relation R, we write a

R

b. If A is the set of all real

numbers, then we can define different relations on A.

For example,

1. x R y if x < y

2. x R y if x ≤ y

3. x R y if x > y

4. x R y if x ≥ y

Example 1 Let A = {1, 2, 3, 4}. Define R by x R y if x < y

Then 1R2, 1R3, 1R4, 2R3, 2R4, 3R4.

Thus, the relation R is completely ...