Question

If $A$, $B$ and $C$ are three sets such that $A\cap B=A\cap CandA\cup B=A\cup C$, then

• A. $A=C$
• B. $B=C$
• C. $A\cap B=\varphi$
• D. $A=B$

Answer 1

The correct option is B

$B=C$

Given that $A\cup B=A\cup C$ and $A\cap B=A\cap C$
$\Rightarrow (A\cup B)\cap C=(A\cup C)\cap C$
$\Rightarrow (A\cap C)\cup (B\cap C)=C[\because (A\cup C)\cap C=C]$
$\Rightarrow (A\cap B)\cup (B\cap C)=C......\left(1\right)$

Again $A\cup B=A\cup C$
$(A\cup B)\cap B=(A\cup C)\cap B$
$B=(A\cap B)\cup (C\cap B)$
$=(A\cap B)\cup (B\cap C)......\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we get,
$B=C$