Question

Let $A$ and $B$ be sets. If $A\cap X=B\cap X=\varphi$ and $A\cup X=B\cup X$ for some set $X,$ show that $A=B$

Answer 1

Given: $A\cap X=B\cap X=\varphi$ and $A\cup X=B\cup X$
Let $A=A\cap (A\cup X)$
$\Rightarrow A=A\cap (B\cup X)$ $\{\because A\cup X=B\cup X\}$
$\Rightarrow A=(A\cap B)\cup (A\cap X)\left\{\text{Using distributive law}\right\}$
$\Rightarrow A=(A\cap B)\cup \varphi \{\because A\cap X=\varphi \}$
$\Rightarrow A=A\cap B\cdots \left(i\right)$
Let $B=B\cap (B\cup X)$
$\Rightarrow B=B\cap (A\cup X)\{\because A\cup X=B\cup X\}$
$\Rightarrow B=(B\cap A)\cup (B\cap X)\left\{\text{Using distributive law}\right\}$
$\Rightarrow B=(B\cap A)\cup \varphi \{\because A\cap X=\varphi \}$
$\Rightarrow B=B\cap A$
$\Rightarrow B=A\cap B\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right),$
$A=B$ Hence proved.