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

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

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