Q20 of 58 Page 12

Using properties of sets prove the statements given

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

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


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