Question

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

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

Answer 1

The correct option is B $B=C$

Let $x\in C$
Suppose $x\in A\Rightarrow x\in A\cap C$
$\Rightarrow x\in A\cap B(\because A\cap C=A\cap B)$
Thus $x\in B$
Again suppose $x\notin A\Rightarrow x\in C\cup A$
$\Rightarrow x\in B\cup A\Rightarrow x\in B$
Hence $C\subseteq B$ .......(1)
Similarly we can show that $B\subseteq C$ .....(2)
Commbining (1) and (2) we get $B=C$