Question -
Answer -
Given:
A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}
(i) A x C ⊂ B x D
Let us consider LHS A x C
A × C = {1, 2} × {5, 6}
= {(1, 5), (1, 6), (2, 5), (2, 6)}
Now, RHS
B × D = {1, 2, 3, 4} × {5, 6, 7, 8}
= {(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)}
Since, all elements of A × C is in B × D.
∴We can say A × C ⊂ B × D
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
Let us consider LHS A × (B ∩ C)
(B ∩ C) = ∅
A × (B ∩ C) = {1, 2} × ∅
= ∅
Now, RHS
(A × B) = {1, 2} × {1, 2, 3, 4}
= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
(A × C) = {1, 2} × {5, 6}
= {(1, 5), (1, 6), (2, 5), (2, 6)}
Since, there is no common element between A × B and A × C
(A × B) ∩ (A × C) = ∅
∴ A × (B ∩ C) = (A × B) ∩ (A × C)