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

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

To show: $B = C$

Let $x \in B$

$\Rightarrow x \in A \cup B \quad[B \subset A \cup B]$

$\Rightarrow x \in A \cup C \quad[A \cup B=A \cup C]$

$\Rightarrow x \in A$ or $x \in C$

Case $I$

Also, $x \in B$

$\therefore x \in A \cap B$

$\Rightarrow x \in A \cap C \quad[\because A \cap B=A \cap C]$

$\therefore x \in A$ and $x \in C$

$\therefore x \in C$

$\therefore B \subset C$

Similarly, we can show that $C \subset B$

$\therefore B=C$