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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 × B) ∪ (A × C)
Solution
A = {x ∈ W | x < 2} = {0, 1}
B = {x ∈ N | 1 < x < 4} = {2, 3, 4}
C = {3, 5}
L.H.S. = A × (B ∪ C)
B ∪ C = {2, 3, 4} ∪ {3, 5}
= {2, 3, 4, 5}
A × (B ∪ C) = {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5)} ……….(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, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (0, 5), (1, 5)} ….(2)
(1) = (2), L.H.S. = R.H.S.
Hence it is proved.
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