Question

Let A = {x ∈ W | x < 2}, B = {x ∈ N | 1 < x < 4} and C = {3, 5}. Verify that (A ∪ B) × C = (A × C) ∪ (B × C)

Answer 1

A = {x ∈ W | x < 2} = {0, 1}

B = {x ∈ N | 1 < x < 4} = {2, 3, 4}

C = {3, 5}

(A ∪ B) × C = (A × C) ∪ (B × C)

L.H.S. = (A ∪ B) × C

A ∪ B = {0, 1, 2, 3, 4}

(A ∪ B) × C = {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} …………. (1)

R.H.S. = (A × C) ∪ (B × C)

(A × C) = {(0, 3), (0, 5), (1, 3), (1, 5)}

(B × C) = {(2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)}

(A × C) ∪ (B × C) = {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} ………… (2)

(1) = (2)

∴ L.H.S. = R.H.S.

Hence it is verified.