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

Let A and B be two sets such that $A\cap X=B\cap X=\varphi$ and $A\cup X=B\cup X$ for some set X.

To show: $A=B$
It can be seen that
$A=A\cap (A\cup X)$
$=A\cap (B\cup X)$ $(A\cup X=B\cup X)$
$=(A\cap B)\cup (A\cap X)$ (Distributive law)
$=(A\cap B)\cup \varphi$ ($\because A\cap X=\varphi$)
$=A\cap B$ ......(1)

Now, $B=B\cap (B\cup X)$
$=B\cap (A\cup X)$ ($\because A\cup X=B\cup X$)
$=(B\cap A)\cup (B\cap X)$ (Distributive law)
$=(B\cap A)\cup \varphi$ ($\because B\cap X=\varphi$)
$=B\cap A$
$=A\cap B$ ......(2)
Hence, from (1) and (2), we get
$A=B$.