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If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then show that $\mathrm{B}=\mathrm{C}$.
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Verified Answer
Let $x \in A$ and $x \in B \Leftrightarrow x \in A \cup B$
$$
\begin{aligned}
&\Leftrightarrow x \in A \cup C \quad(\because A \cup B=A \cup C) \\
&\Leftrightarrow x \in C \\
&\therefore \quad B=C . \\
&\text { Let } x \in A \text { and } x \in B \Leftrightarrow x \in A \cap B \\
&\Leftrightarrow x \in A \cap C \quad(\because A \cap B=A \cap C) \\
&\Leftrightarrow x \in C \\
&\therefore B=C
\end{aligned}
$$
$$
\begin{aligned}
&\Leftrightarrow x \in A \cup C \quad(\because A \cup B=A \cup C) \\
&\Leftrightarrow x \in C \\
&\therefore \quad B=C . \\
&\text { Let } x \in A \text { and } x \in B \Leftrightarrow x \in A \cap B \\
&\Leftrightarrow x \in A \cap C \quad(\because A \cap B=A \cap C) \\
&\Leftrightarrow x \in C \\
&\therefore B=C
\end{aligned}
$$
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