# For any sets A and B, prove that(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A).

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]

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RS Aggarwal - Mathematics