Discrete Mathematical Structures by Pearson

1-14 Discrete Mathematical Structures

Example 2 If A, B, and C are three sets and A ⊆ B, then prove that A × C ⊆ B × C.

Solution: Let (a, c) be any element in A × C, then

(a, c) ∈ A × C ⇒ a ∈ A and c ∈ C

⇒ a ∈ B and c ∈ C (since A ⊆ B)

⇒ (a, c) ∈ B × C

Hence,

A × C ⊆ B × C

Example 3 Prove that A × (B - C ) = A × B - A × C

Proof: Let (x, y) be any element in A × (B - C ), then

(x, y) ∈ A × (B - C ) ⇔ x ∈ A and y ∈ (B - C )

⇔ x ∈ A and (y ∈ B and y ∉ C )

⇔ (x ∈ A and y ∈ B) and (x ∈ A and y ∉ C )

⇔ (x, y) ∈ A × B) and (x, y) ∉ (A × C )

⇔ (x, y) ∈ [(A × B) - (A × C )]

Hence,

A × (B - C ) = (A × B) - (A × C )

Example 4 Prove that A × (B ∪ C ) = (A × B) ∪ (A × C )

Proof: Let (x, y) be any element in A × (B ∪ C ), then

(x, y) ∈ A × (B ∪ C ) ⇔ x ∈ A and y ∈ B