Question

Show that $A\cap B=A\cap C$ need not imply $B=C$

Answer 1

Let $A=\{0,1\}$
$B=\{0,2,3\}$
$C=\{0,4,5\}$
So, $A\cap B=\left\{0\right\}$
and $A\cap C=\left\{0\right\}$
Here, $A\cap B=A\cap C=\left\{0\right\}$
However, $B\ne C$ as $2\in B$ and $2\notin C$