Question

Let A, B, and C be any sets. Show that A × (B – C) = (A × B) – (A × C).

Can you help justify the proof

Answer 1

If × denotes the Cartesian product, and if a ∈ A and b ∈ B, then (a, b) ∈ A × B.

First prove that A × (B - C) ⊆ (A × B) - (A × C):

Let a ∈ A and b ∈ B - C.

By definition of the product, (a, b) ∈ A × B.

By definition of complement, b ∈ B and b ∉ C.

So (a, b) ∈ A × B and (a, b) ∉ A × C.

By definition of complement, (a, b) ∈ (A × B) - (A × C).

Therefore, A × (B - C) ⊆ (A × B) - (A × C).

Prove the other direction, that (A × B) - (A × C) ⊆ A × (B - C):

Let (a, b) ∈ (A × B) - (A × C).

By definition complement, (a, b) ∈ (A × B) and (a, b) ∉ (A × C).

By definition of the product, a ∈ A, b ∈ B, and b ∉ C.

By definition of complement, b ∈ B - C.

By definition of the product, (a, b) ∈ A × (B - C).

Therefore, (A × B) - (A × C) ⊆ A × (B - C).

Hence the two sets are equal. QED

Answer 2

AB-AC=AB-AC

1. Factor the first problem into AB-AC
2. Multiply and add the second problem and you will get AB-AC
3. AB-AC is equal to AB-AC