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  4. Let X and Y be two non-empty sets such that X ∩ A = Y ∩ A = φ and X ∪ A = Y ∪ A for some non-empty set A. Then

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Q. 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

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Solution:

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, $X = Y$