Question

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

L.H.S. = A × (B ∩ C)

(B ∩ C) = {3}

A × (B ∩ C) = {(0, 3), (1, 3)} …(1)

R.H.S. = (A × B) ∩ (A × C)

(A × B) = {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)}

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

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

(1) = (2) ⇒ L.H.S. = R.H.S.

Hence it is verified.