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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)
Solution
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.
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