Question

Given A = {1, 2, 3}, B = {2, 3, 5}, C = {3, 4} and D = {1, 3, 5}, check if (A ∩ C) × (B ∩ D) = (A × B) ∩ (C × D) is true?

Answer 1

A = {1, 2, 3}, B = {2, 3, 5}, C = {3, 4} D = {1, 3, 5}

A ∩ C = {1, 2, 3} ∩ {3, 4}

= {3}

B ∩ D = {2, 3, 5} ∩ {1, 3, 5}

= {3, 5}

(A ∩ C) × (B ∩ D) = {3} × {3, 5}

= {(3, 3) (3, 5)} ….(1)

A × B = {1, 2, 3} × {2, 3, 5}

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

C × D = {3, 4} × {1, 3, 5}

= {(3, 1) (3, 3) (3, 5) (4, 1) (4, 3) (4, 5)}

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

From (1) and (2) we get

(A ∩ C) × (B ∩ D) = (A × B) ∩ (C × D)

This is true.