Set theory, operation with products, union and intersection I need to show the following using logical connectives:
$B\setminus (B \setminus A)=A \cap B $ 
$(A \setminus B)\cup(B\setminus A)=(A \cup B)\setminus(A \cap B)$
$(A\times B\setminus C )=(A\times C)\setminus(B \times C)$
My attempt at the first one was futile, as I ran into the following:
$x \in B\setminus (B \setminus A)\iff(x\in B \land (\neg (x \in B \land (\neg(x \in A)))))\iff(x\in B \land  (\neg x \in B \lor x \in A))$

At least for first statement, use distributive law after De Morgan's:
$$\begin{align}
x\in B\setminus(B\setminus A)&\iff x\in B\wedge \neg(x\in B\setminus A)\\
&\iff x\in B \wedge\neg(x\in B\wedge\neg x\in A)\\
&\iff x\in B\wedge(\neg x\in B\vee x\in A)\\
&\iff \overbrace{(x\in B\wedge \neg x\in B)}^\text{contradiction}\vee(x\in B\wedge x\in A)\\
&\iff x\in B\wedge x\in A\\
&\iff x\in A\cap B
\end{align}$$
