Question

Let A = The set of all natural number less than 8, B = The set of all prime numbers less than 8, C = The set of even prime number. Verify that (A ∩ B) × C = (A × C) ∩ (B × C)

Answer 1

A = {1, 2, 3, 4, 5, 6, 7}

B = {2, 3, 5, 7}

C = {2}

(A ∩ B) × C = (A × C) ∩ (B × C)

A ∩ B = {1, 2, 3, 4, 5, 6, 7} ∩ {2, 3, 5, 7}

= {2, 3, 5, 7}

(A ∩ B) × C = {2, 3, 5, 7} × {2}

= {(2, 2) (3, 2) (5, 2) (7, 2)} ….(1)

A × C = {1, 2, 3, 4, 5, 6, 7} × {2}

= {(1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (7, 2)}

B × C = {2, 3, 5, 7} × {2}

= {(2, 2) (3, 2) (5, 2) (7, 2)}

(A × C) ∩ (B × C) = {(2, 2) (3, 2) (5, 2) (7, 2)} ….(2)

From (1) and (2) we get

(A ∩ B) × C = (A × C) ∩ (B × C)