Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

(i) A × (B ∩ C) = (A × B) ∩ (A × C). (ii) A × C is a subset of B × D.

(ii) A × C is a subset of B × D.

(i) Given: A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

To verify: A × (B ∩ C) = (A × B) ∩ (A × C)

As we see,

By definition if either of the two set P and Q is null set then P × Q will also be a null set. i.e. P × Q = ϕ .

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

And (A × C) = {(1,5), (1,6), (2,5), (2,6)}

From (1) and (2)

(ii) Given: A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}

To verify: A × C is a subset of B × D.

(A × C) = {(1,5), (1,6), (2,5), (2,6)}

(B × D) = {(1,5), (1,6), (1,7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8)}

As we see all the elements of set A × B are there in set B × D.

Hence, A × C is a subset of B × D.