Answer : 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, y)    (A × B) and (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) and (a, b)    (A × C)

⇒ (a     A and b    B) and (a    A and b    C)

⇒ a     A and (b    B and 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]