Let $A, B,$ and $C$ be sets such that $A \cup B = A \cup C$ and $A \cap B = A \cap C$. Show that $B = C$.

(N/A) Given: $A \cup B = A \cup C$ and $A \cap B = A \cap C$.
To prove: $B = C$.
Let $x \in B$.
Since $B \subseteq A \cup B$,we have $x \in A \cup B$.
Given $A \cup B = A \cup C$,it follows that $x \in A \cup C$.
This implies $x \in A$ or $x \in C$.
Case $I$: If $x \in A$,then since $x \in B$,we have $x \in A \cap B$.
Since $A \cap B = A \cap C$,it follows that $x \in A \cap C$,which implies $x \in C$.
Case $II$: If $x \in C$,then $x \in C$ is already satisfied.
In both cases,$x \in C$. Thus,$B \subseteq C$.
Similarly,by taking $x \in C$,we can show $C \subseteq B$.
Since $B \subseteq C$ and $C \subseteq B$,we conclude $B = C$.