Question

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

Answer 1

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’) ∪ (A ∩ B’)

∵ Distributive property of set:

(A ∩ B) ∪ (A ∩ C) = A ∩ (B ∪ C)}

= Φ ∪ (A ∩ B’)   ......[∵ A ∩ A’ = Φ]

= A ∩ B’

= A – B   ......[∵ A – B = A ∩ B’]

= R.H.S

Hence Proved