Question

Let A and B be sets. If $A\cap X=B\cap X=\Phi$ and $A\cup X=B\cup X$ for some set X. Show that $A=B.$

Answer 1

Here $A\cup X=B\cup X$ for some X

$\Rightarrow A\cap (A\cup X)=A\cap (B\cup X)$

$\Rightarrow A=(A\cap B)\cup (A\cap X)$ $[\because A\cap (A\cup X)=A]$

$\Rightarrow A=(A\cap B)\cup \Phi$ $[\because A\cap X=\Phi ]$

$\Rightarrow A=A\cap B$

$\Rightarrow A\subset B...\left(i\right)$

Also $A\cup X=B\cup X$

$\Rightarrow B\cap (A\cup X)=B\cap (B\cup X)$

$\Rightarrow (B\cap A)\cup (B\cap X)=B$ $[\because B\cap (B\cup X)=B]$

$\Rightarrow (B\cap A)\cup \Phi =B$ $[\because B\cap X=\Phi ]$

$\Rightarrow B\cap A=B$

$\Rightarrow B\subset A...\left(ii\right)$

From (i) and (ii), we have $A=B$