Given: A and B two sets are given.

Need to prove: (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)

Let us consider, (x, y) (A × B) ∩ (B × A)

⇒ (x, y) (A × B) and (x, y) (B × A)

⇒ (x A and y B) and (x B and y A)

⇒ (x A and x B) and (y B and y A)

⇒ x (A × B) and y (B × A)

⇒ (x, y) (A × B) ∩ (B × A)

From this, we can conclude that,

⇒ (A × B) ∩ (B × A) ⊆ (A ∩ B) × (B ∩ A) ---- (1)

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

⇒ a (A ∩ B) and b (B ∩ A)

⇒ (a A and a B) and (b B and b A)

⇒ (a A and b B) and (a B and b A)

⇒ (a, b) (A × B) and (a, b) (B × A)

⇒ (a, b) (A × B) ∩ (B × A)

From this, we can conclude that,

⇒ (A ∩ B) × (B ∩ A) ⊆ (A × B) ∩ (B × A) ---- (2)

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

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A) [Proved]