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Question
If A = {x : x ∈ Z, −2 < x ≤ 4}, B = {x : x ∈ W, x ≤ 5}, C = {− 4, −1, 0, 2, 3, 4} verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Solution
A = {x : x ∈ Z, −2 < x ≤ 4} = {−1, 0, 1, 2, 3, 4}
B = {x : x ∈ W, x ≤ 5} = {0, 1, 2, 3, 4, 5}
C = {−4, −1, 0, 2, 3, 4}
A ∪ (B ∩ C)
B ∩ C = {0, 1, 2, 3, 4, 5} ∩ {−4, −1, 0, 2, 3, 4}
= {0, 2, 3, 4}
A ∪ (B ∩ C) = {−1, 0, 1, 2, 3, 4} ∪ (0, 2, 3, 4}
= {−1, 0, 1, 2, 3, 4} ...(1)
(A ∪ B) ∩ (A ∪ C)
A ∩ B = {0, 1, 2, 3, 4}
A ∩ C = {−1, 0, 2, 3, 4}
(A ∩ B) ∪ (A ∩ C) = {0, 1, 2, 3, 4} ∪ {−1, 0, 2, 3, 4}
= {−1, 0, 1, 2, 3, 4} ...(2)
From (1) and (2), it is verified that
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
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