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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
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B
Y is a proper subset of X
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C
X=Y
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D
X and Y are disjoint sets
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E
X/A=ϕ
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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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