For any sets A, B

Given: A, B and C three sets are given.

Need to prove: A × (B ∪ C) = (A × B) ∪ (A × C)

Let us consider, (x, y) A × (B ∪ C)

⇒ x A and y (B ∪ C)

⇒ x A and (y B or y C)

⇒ (x A and y B) or (x A and y C)

⇒ (x, y) (A × B) or (x, y) (A × C)

⇒ (x, y) (A × B) ∪ (A × C)

From this we can conclude that,

⇒ A × (B ∪ C) ⊆ (A × B) ∪ (A × C) ---- (1)

Let us consider again, (a, b) (A × B) ∪ (A × C)

⇒ (a, b) (A × B) or (a, b) (A × C)

⇒ (a A and b B) or (a A and b C)

⇒ a A and (b B or b C)

⇒ a A and b (B ∪ C)

⇒ (a, b) A × (B ∪ C)

From this, we can conclude that,

⇒ (A × B) ∪ (A × C) ⊆ A × (B ∪ C) ---- (2)

Now by the definition of the set we can say that, from (1) and (2),

A × (B ∪ C) = (A × B) ∪ (A × C) [Proved]