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]