1.

If A and B are sets, then prove that A – B, A ∩ B and B – A are pair wise disjoint.

Answer»

Let x ϵ A and y ϵ B 

A – B = The set of values of A that are not in B. 

A ∩ B = The set containing common values of A and B 

B – A = The set of values of B that are not in A. 

Two sets X and Y are called disjoint if, 

X ∩ Y = ϕ 

(A – B) ∩ (A ∩ B) = ((A – B) ∩ A) ∪ ((A – B) ∩B) 

(A – B) ∩ (A ∩ B) = ϕ ∪ ϕ 

(A – B) ∩ (A ∩ B) = ϕ 

Similarly, 

(B – A) ∩ (A ∩ B) = ((B – A) ∩ A) ∪ ((B – A) ∩B) 

(B – A) ∩ (A ∩ B) = ϕ 

Hence, 

the three sets are pair wise disjoint.



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