If A, B, C are three sets such that $A \cap B = A \cap C and A\cup B =A\cup C$, then 

Let $x\in B \Rightarrow x \in A \cup B$

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

If $x\in A , then x\in A\cap B=A\cap C\Rightarrow x\in C$

So, $x\in B\Rightarrow x\in C$

Similarly        $x\in B\Rightarrow x\in C ,$ Hence B = C .