Question

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

A × (B ∩ C) = (A × B) ∩ (A × C)

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}

To prove: A × (B ∩ C) = (A × B) ∩ (A × C)

B ∩ C = {8}; A = {1, 2, 5, 7}

So A × (B ∩ C) = {1, 2, 5, 7} × {8}

= {(1, 8), (2. 8), (5, 8), (7, 8)}

Now A x 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)}    ......(1)

A × C = {(1, 1), (1, 5),(1, 8), (1, 10), (2, 1), (2, 5), (2, 8), (2, 10), (5, 1), (5, 5), (5, 8), (5, 10), (7, 1), (7, 5), (7, 8), (7, 10)}

(A × B) ∩ (A × C) = {(1, 8), (2, 8), (5, 8), (7, 8)}   ......(2)

(1) = (2)

⇒ A × (B ∩ C) = (A × B) ∩ (A × C)