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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)
Sum
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Solution
Given:
A = {0, 1} (since A = {x ∈ W ∣ x < 2}
B = {2, 3} (since B = {x ∈ N ∣ 1 < x < 4}
C = {3, 5}
B ∪ C ={2, 3, 5}
Compute A × (B ∪ C)
A × (B ∪ C) = {(0, 2), (0, 3), (0, 5), (1, 2), (1, 3), (1, 5)}
Compute A × B and A × C
A × B = {(0, 2), (0, 3), (1, 2), (1, 3)}
A × C = {(0, 3), (0, 5), (1, 3), (1, 5)}
Compute (A × B) ∪ (A × C)
(A × B) ∪ (A × C) = {(0, 2), (0, 3), (0, 5), (1, 2), (1, 3), (1, 5)}
Compare both sides
A × (B ∪ C) = (A × B) ∪ (A × C)
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