Q. 83.5( 53 Votes )

Show that for any

Answer :

To Prove: A = (A B) (A – B)


Proof: Let X ϵ A


Now, we need to show that X ϵ (A B) (A – B)


In Case I,


X ϵ (A B)


X ϵ (A B) (A B) (A – B)


In Case II,


X A B


X B or X A


X B (X A)


X A – B (A B) (A – B)


A (A B) (A – B) (i)


It can be concluded that, A B A and (A – B) A


Thus, (A B) (A – B) A (ii)


Equating (i) and (ii),


A = (A B) (A – B)


Now, we need to show, A (B – A) A B


Let us assume that,


X ϵ A (B – A)


X ϵ A or X ϵ (B – A)


X ϵ A or (X ϵ B and X A)


(X ϵ A or X ϵ B) and (X ϵ A and X A)


X ϵ (B A)


A (B – A) (A B) (iii)


Now, to prove: (A B) A (B – A)


Let y ϵ AB


yϵ A or y ϵ B


(y ϵ A or y ϵ B) and (X ϵ A and X A)


y ϵ A or (y ϵ B and y A)


y ϵ A (B – A)


Thus, A B A (B – A) (iv)


Using (iii) and (iv), we get:


A (B – A) = A B


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