Prove that if $A\subseteq B$ then $A\setminus C\subseteq B\setminus C$ How can I prove that if $$A \subseteq B$$ then $$A\setminus C \subseteq B \setminus C$$? 
I'm a bit lost. I have an idea that follows like so:
Let $x \in A, x \notin C$, then $\{x\} \in P(A\setminus C); \{x\} \in P(B\setminus C)$, while saying that nullsets are subsets of nullsets if $x \in C$.
However, I believe that there is a more straightforward solution. How should I prove the statement above?

I'm not sure why you involve the power set there. You also seemed to have switched from $\in$ to $\subseteq$ somehow when you are talking about null subsets.
Basically, this is a very straightforward application of the assumption, $A\subseteq B$, and the information that $x\notin C$. I will leave it to you to rethink it.
