Question

Let $A,B$ and $C$ be the sets such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$. Show that $B=C.$

Answer 1

Given: $A\cup B=A\cup C$ and $A\cap B=A\cap C$
$\Rightarrow (A\cup B)\cap C=(A\cup C)\cap C$
$\Rightarrow (A\cap C)\cup (B\cap C)=C[\because (A\cup C)\cap C=C]$
$\Rightarrow (A\cap B)\cup (B\cap C)=C\cdots \left(i\right)$

Again, $A\cup B=A\cup C$
$\Rightarrow (A\cup B)\cap B=(A\cup C)\cap B$
$\Rightarrow B=(A\cap B)\cup (C\cap B)$
$\Rightarrow B=(A\cap B)\cup (B\cap C)\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right),$ we get
$B=C.$ Hence proved.