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Using properties of sets prove the given statement.For all sets A and B, A – (A – B) = A ∩ B |
Answer» According to the question, There are two sets A and B To prove: A – (A – B) = A ∩ B L.H.S = A – (A – B) Since, A – B = A ∩ B’, we get, = A – (A ∩ B’) = A ∩ (A ∩ B’)’ Since, (A ∩ B)’ = A’ ∪ B’, we get, = A ∩ [A’ ∪ (B’)’] Since, (B’)’ = B, we get, = A ∩ (A’ ∪ B) Since, distributive property of set ⇒ (A ∩ B) ∪ (A ∩ C) = A ∩ (B ∪ C), we get, = (A ∩ A’) ∪ (A ∩ B) Since, A ∩ A’ = Φ, we get, = Φ ∪ (A ∩ B) = A ∩ B = R.H.S Hence Proved |
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