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

Answer 1

The correct option is C $B=C$

$A$$\cup B=A\cup C$
$\Rightarrow n\left(A\cup B\right)=n\left(A\cup C\right)$
$\Rightarrow n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)=n\left(A\right)+n\left(C\right)-n\left(A\cap C\right)$
$\Rightarrow n\left(B\right)=n\left(C\right)$ since $A\cap C=A\cap B$
$\Rightarrow B=C$