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Question
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify the following:
A ∩ (B ∆ C) = (A ∩ B) ∆ (A ∩ C)
Solution
A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8},
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B – C = {3}
C – B = {7, 8}
B ∆ C = (B – C) ∪ (C – B)
= {3, 7, 8}
∴ A ∩ (B ∆ C) = {3} .........(i)
A ∩ B = {3, 4}
A ∩ C = {4}
∴ (A ∩ B) ∆ (A ∩ C)
= [(A ∩ B) – (A ∩ C)] ∪ [(A ∩ C) – (A ∩ B)]
= {3} .........(ii)
From (i) and (ii), we get
A ∩ (B ∆ C) = (A ∩ B) ∆ (A ∩ C)
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