For any three sets A, B and C, (A – B) ∩ (C – B) is equal to ____________.

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Solution

For A, B and C
(A – B) ∩ (C – B) = (A ∩ B^C) ∩ (C ∩ B^C)
= A ∩ B^C ∩ B^C ∩ C (∵ intersection is associative)
= A ∩ (B^C ∩ B^C) ∩ C
= A ∩ B^C ∩ C (∵ B^C∩ B^C = B^C)
= A ∩ C ∩ B^C
Hence, (A – B) ∩ (C – B) = (A ∩ C) – B

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