# Show that for any

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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