Question

$A\cap X=B\cap X=\varphi \&A\cup X=B\cup X$ prove that $A=B$

Answer 1

Consider given that,

$A$ and $B$ be any two sets.

If $A\cap X=B\cap X=\varphi andA\cup X=B\cup X$ for some set X.
Then, show that $A=B$.

Suppose that $a\in A$ then if $A\cap X=\varphi$, $a\notin$$X$

If $A\cup X=B\cup X$ $i.e.$ $A\cup X=B\bigcup X-X=B$ , $a\in B$

Then,

$\text{a}\in A\to \text{a}\in B$ is equivalent to say that $A\subseteq B......\left(1\right)$

Let now, $a\in B$

If $B\cap X=\varphi$ then $p\notin X$ and if $B\cup X=A\cup X$ but $a\notin X$, $i.e.$

$B\cup X=A\cup X-X=A$
$\text{a}\in B\to \text{a}\in A$

That means $B\subseteq A.......\left(2\right)$

By equation (1) and (2), we get

$A=B$

Hence proved.