Question

Let $X$ and $Y$ be two non-empty sets such that $X\cap A=Y\cap A=\varphi$ and $X\cup A=Y\cup A$ for some non-empty set $A$. Then which of the following is true?

• A. $X$ is a proper subset of $Y$
• B. $Y$ is a proper subset of $X$
• C. $X=Y$
• D. $X$ and $Y$ are disjoint sets
• E. $X/A=\varphi$

Answer 1

The correct option is C $X=Y$

We have, $X\cup A=Y\cup A$
$\Rightarrow X\cap (X\cup A)=X\cap (Y\cup A)$
$\Rightarrow X=(X\cap Y)\cup (X\cap A)$ $[\because X\cap (X\cup A)=X]$
$\Rightarrow X=(X\cap Y)\cup \varphi$ $[\because X\cap A=\varphi ]$
$\Rightarrow X=X\cap Y$ .....(i)
Again, $X\cup A=Y\cup A$
$\Rightarrow Y\cap (X\cup A)=Y\cap (Y\cup A)$
$\Rightarrow (Y\cap X)\cup (Y\cap A)=Y$
$\Rightarrow (Y\cap X)\cup \varphi =Y$
$\Rightarrow Y\cap X=Y$
$\Rightarrow X\cap Y=Y$ ....(ii)
From equations (i) and (ii), we get
$X=Y$