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

• 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

Answer 1

Answer: (C)

Suppose $a\in X$ and $a\in A$
$\Rightarrow a\in X\cup A\Rightarrow a\in Y\cup A$
$\Rightarrow a\in Y$ and $a\in A\left(\therefore X\cup A=Y\cup A\right)$
$\Rightarrow a\in Y\cap A\Rightarrow Y\cap A$ is no-empty
This contradicts that $Y\cap A=f$
So, $X=Y$