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 × B) – (A × C)

Answer 1

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

B = {2, 3, 5, 7}

C = {2}

A × (B – C) = (A × B) – (A × C)

B – C = {2, 3, 5, 7} – {2}

= {3, 5, 7}

A × (B – C) = {1, 2, 3, 4, 5, 6, 7} × {3, 5, 7}

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

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

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

A × C = {1, 2, 3, 4, 5, 6, 7} × {2}

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

(A × B) – (A × C) = {(1, 3) (1, 5) (1, 7) (2, 3) (2, 5) (2, 7) (3, 3) (3, 5) (3, 7) (4, 3) (4, 5) (4, 7) (5, 3) (5, 5) (5, 7) (6, 3) (6, 5) (6, 7) (7, 3) (7, 5) (7, 7)} ….(2)

From (1) and (2) we get

A × (B – C) = (A × B) – (A × C)