Question

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

(i) (A ∩ B) x C = (A x c) ∩ (B x C) 

(ii) A x (B – C) = (A x B) – (A x C)

Answer 1

A = {1, 2, 3, 4, 5, 6, 7} 

B = {2, 3, 5, 7} 

C = {2}

(i) (A ∩ B) x C = (A x C) ∩ (B x C) 

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

= {2, 3, 5, 7} 

(A ∩ B) x C = {2, 3, 5, 7} x {2} 

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

A x C = {1,2, 3, 4, 5, 6, 7} x {2} = {(1,2) (2, 2) (3, 2) (4, 2) (5.2) (6, 2) (7, 2)} 

B x C = {2, 3, 5, 7} x {2} = {(2, 2) (3, 2) (5, 2) (7, 2)} 

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

From (1) and (2) we get 

(A ∩ B) x C = (A x C) ∩ (B x C)

(ii) A x (B – C) = (A x B) – (A x C) 

B – C = {2, 3, 5, 7} – {2}

= {3,5,7} 

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

= {(1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6.2) (7,2)} 

(A x B) – (A x 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 x (B – C) = (A x B) – (A x C)