Question

# Let $$X$$ and $$Y$$ be two non-empty sets such that $$X\cap A=Y\cap A=\phi$$ 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=ϕ

Solution

## The correct option is C $$X=Y$$We have, $$X\cup A=Y\cup A$$$$\Rightarrow X\cap \left( X\cup A \right) =X\cap \left( Y\cup A \right)$$$$\Rightarrow X=\left( X\cap Y \right) \cup \left( X\cap A \right)$$         $$\left[ \because X\cap \left( X\cup A \right) =X \right]$$$$\Rightarrow X=\left( X\cap Y \right) \cup \phi$$         $$\left[\because X\cap A=\phi \right]$$$$\Rightarrow X=X\cap Y$$              .....(i)Again, $$X\cup A=Y\cup A$$$$\Rightarrow Y\cap \left( X\cup A \right) =Y\cap \left( Y\cup A \right)$$$$\Rightarrow \left( Y\cap X \right) \cup \left( Y\cap A \right) =Y$$$$\Rightarrow \left( Y\cap X \right) \cup \phi =Y$$$$\Rightarrow Y\cap X=Y$$$$\Rightarrow X\cap Y=Y$$                  ....(ii)From equations (i) and (ii), we get$$X=Y$$Mathematics

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