Question

If A and B are sets, then prove that A - B, $A\cap B$ and B- A are pair wise disjoint.

Answer 1

We need to show that (A - B) $cap(A\cap B)=\varphi$, $(A\cap B)\cap (B-A)=\varphi$ and $(A-B)\cap (B-A)=\varphi$

The 3 sets A - B , $A\cap BandB-A$ may be represented by a venn diagram as follows.

It is clear from the diagram that the 3 sets are pairwise disjoint, but we shall giva a proof of it.

We first show that $(A-B)\cap (A\cap B)=\varphi$

Let $xϵ(A-B)$

$\Rightarrow xϵAandx\text{/}ϵB$

[by definition of A- B]

$\Rightarrow x\text{/}ϵA\cap B$. This is true for all $xϵ(A-B)$

Hence $(A-B)\cap (A\cap B)=\varphi$

Finally , we show that $(A-B)\cap (B-A)=\varphi$

Finally , we show that $(A-B)\cap (B-A)=\varphi$

We have,

A - B = {$xϵA:x\text{/}ϵB$}

and B- A ={$xϵB:x\text{/}ϵA$}

Hence, $(A-B)\cap (B-A)=\varphi$.