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
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
(i) To
verify: A × (B ∩
C) = (A × B) ∩
(A × C)
We
have B ∩
C = {1, 2, 3, 4} ∩
{5, 6} = Φ
∴L.H.S.
= A × (B ∩
C) = A × Φ
= Φ
A × B = {(1,
1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
A × C = {(1,
5), (1, 6), (2, 5), (2, 6)}
∴
R.H.S.
= (A × B) ∩
(A × C) = Φ
∴L.H.S.
= R.H.S
Hence,
A × (B ∩
C) = (A × B) ∩
(A × C)
(ii) 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)}
We
can observe that all the elements of set A × C are the elements
of set B × D.
Therefore,
A
× C is a subset of B × D.
(i) To verify: A × (B ∩ C) = (A × B) ∩ (A × C)
We have B ∩ C = {1, 2, 3, 4} ∩ {5, 6} = Φ
∴L.H.S. = A × (B ∩ C) = A × Φ = Φ
A × B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}
∴ R.H.S. = (A × B) ∩ (A × C) = Φ
∴L.H.S. = R.H.S
Hence, A × (B ∩ C) = (A × B) ∩ (A × C)
(ii) 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)}
We can observe that all the elements of set A × C are the elements of set B × D.
Therefore, A × C is a subset of B × D.
