Question

By taking suitable sets A, B, C, verify the following results:

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)

Answer 1

To prove the following results let us take U = {1, 2, 5, 7, 8, 9, 10}

A = {1, 2, 5, 7}

B = {2, 7, 8, 9}

C = {1, 5, 8, 7}

A × B = {(1, 2), (1, 7), (1, 8), (1, 9) (2, 2), (2, 7), (2, 8), (2, 9) (5, 2), (5, 7), (5, 8), (5, 9) (7, 2), (7, 7), (7, 8), (7, 9)}

B × A = {(2, 1), (2, 2), (2, 5), (2, 7) (7, 1), (7, 2), (7, 5), (7, 7) (8, 1), (8, 2), (8, 5), (8, 7) (9,1), (9, 2), (9, 5), (9, 7)}

L.H.S. (A × B) ∩ (B × A) = {(2, 2), (2, 7), (7, 2), (7, 7)}  ......(1)

R.H.S. A ∩ B = {2, 7}

B ∩ A = {2, 7}

(A ∩ B) × (B ∩ A) = {2, 7} × {2, 7}

= {(2, 2), (2, 7), (7, 2), (7, 7)}   ......(2)

(1) = (2)

⇒ L.H.S = R.H.S