Which of the following equality is not true.

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A ∩ (B ~ C) = A ∩ B ~ (A ∩ C)

A ~ (A ∩ B) = A ~ B

A ~ (B ~ C) = (A ~ B) ∪ (A ∩ C)

A ~ (B ∆ C ) = (A ~ B) ∆ (A ~ C)

answer is D.

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Detailed Solution

For equality (a),A ∩ (B ~ C) = A ∩ (B ∩ C')= ϕ ∪ (A ∩ B ∩ C')= (A ∩ B∩ A') ∪ (A ∩ B ∩ C')= A ∩ B ∩ (A' ∪ C')= A∩ B ~ (A ∩ C)For equality (b),A ~ (A ∩ B) = A ∩ (A' ∪ B')= (A ∩ A') ∪ (A ∩ B')= ∅ ∪ (A ∩ B') = A ∩ B' = A ~ BFor equality (c)A ~ (B ~ C) = A ~ (B ∩ C') = A ∩ (B' ∪ C)= (A ∩ B') ∪ (A ∩ C)= (A ~ B) » (A « C)For (d) LetA={1, 2, 3, 4, 5}, B={3, 4, 5},C={1, 2, 3}So, B∆C={4, 5} ∪ {1, 2}={1, 2, 4, 5}Thus A (B∆C)={3}A~ B={1, 2}; A~ C={4, 5}Therefore (A ~B) ∆ (A~ C)=({1, 2}~ {4, 5})∪({4, 5}~ {1, 2})= {1, 2} ∪ {4, 5} = {1, 2, 4, 5).Hence A ~ (B ∆ C) ≠ (A ~ B) ∆ (A ~ C)

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