Question:
For all sets A and B, A – (A – B) = A ∩ B
Solution:
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