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If $A, B, C$ are three sets such that $A \cup B=A \cup C$ and $A \cap B=A \cap C$, then which one of the following is correct?
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The correct answer is:
$B=C$ only
Let $A, B, C$ be three sets such that $A \cup B=A \cup C$ and $A \cap B=A \cap C$
Let $x \in A \cup B \Rightarrow x \in A$ or $x \in B$
(1) Since, $A \cup B=A \cup C$ there fore
$x \in A$ or $x \in C$
Also, Given $A \cap B=A \cap C$ $x \in A \cap B \Rightarrow x \in A$ and $x \in B$
and $x \in A$ and $x \in C(\because A \cap B=A \cap C)$
Thus, from $(1),(2),(3)$, and $(4)$, we have $B=C$ only
Let $x \in A \cup B \Rightarrow x \in A$ or $x \in B$
(1) Since, $A \cup B=A \cup C$ there fore
$x \in A$ or $x \in C$
Also, Given $A \cap B=A \cap C$ $x \in A \cap B \Rightarrow x \in A$ and $x \in B$
and $x \in A$ and $x \in C(\because A \cap B=A \cap C)$
Thus, from $(1),(2),(3)$, and $(4)$, we have $B=C$ only
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