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Question
For all sets A and B, A – (A – B) = A ∩ B
Solution
Given: There are two sets A and B
To prove: A – (A – B) = A ∩ B
Take L.H.S
A – (A – B)
= A – (A ∩ B’) ......[∵ A – B = A ∩ B’]
= A ∩ (A ∩ B’)’
= A ∩ [A’ ∪ (B’)’] ......[∵ (A ∩ B)’ = A’ ∪ B’]
= A ∩ (A’ ∪ B) ......[∵ (B’)’ = B]
= (A ∩ A’) ∪ (A ∩ B)
∵ Distributive property of set:
(A ∩ B) ∪ (A ∩ C) = A ∩ (B ∪ C)}
= Φ ∪ (A ∩ B) ......[∵ A ∩ A’ = Φ]
= A ∩ B
= R.H.S
Hence Proved
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