Given A = {1,2,3}, B = {2,3,5}, C = {3,4} and D = {1,3,5}, check if (A

1.	Given A = {1,2,3}, B = {2,3,5}, C = {3,4} and D = {1,3,5}, check if (A ∩ C) x (B ∩ D) = (A x B) ∩ (C x D) is true?
Answer» 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) x (B ∩ D) = {3} x {3,5} = {(3, 3)(3, 5)} … (1) A x 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 x D = {3,4} × {1,3,5} = {(3,1) (3, 3) (3, 5) (4,1) (4, 3) (4, 5)} (A x B) ∩ (C x D) = {(3, 3) (3, 5)} … (2) From (1) and (2) we get (A ∩ C) x (B ∩ D) = (A x B) ∩ (C x D) This is true.

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

Answer»

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) x (B ∩ D) = {3} x {3,5}

= {(3, 3)(3, 5)} … (1)

A x 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 x D = {3,4} × {1,3,5}

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

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

From (1) and (2) we get

(A ∩ C) x (B ∩ D) = (A x B) ∩ (C x D)

This is true.