Using properties of sets prove the statements given
For all sets A and B, A – (A – B) = A ∩ B

Question

Using properties of sets prove the statements given For all sets A and B, A – (A – B) = A ∩ B

Philoid · Accepted Answer

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

Using properties of sets prove the statements givenFor all sets A and B, A – (A – B) = A ∩ B