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 :
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 :
$$X = Y$$
Solution :
$$\eqalign{
& {\text{Suppose }}a\, \in \,X{\text{ and }}a\, \in \,A \cr
& \Rightarrow a\, \in \,X \cup A \cr
& \Rightarrow a\, \in \,Y{\text{ and }}a\, \in \,A\,\,\left( {\because \,X \cup A = Y \cup A} \right) \cr
& \Rightarrow a\, \in \,Y \cap A \cr
& \Rightarrow Y \cap A{\text{ is non - empty}} \cr
& {\text{This contradicts that }}Y \cap A = \phi \cr
& {\text{So, }}X = Y \cr} $$