Some Trouble Understanding set theory I'm currently in a discrete mathematics class and we've recently been discussing set theory. I feel like I have basic understanding of how to actually prove set relations when a question asks to do so. However I am having a lot of trouble when initially presented with questions, where I am asked to determine if a statement is true or false. The approach we were taught was to set up ven diagrams in order to help us. However I find that I get very lost in certain types of questions, especially ones where one set contains another. Here is an example of a problem I struggled with:
One of these is true and one is false, provide a proof for both:
(1) For all sets A, B and C, if A-B is a subset of A-C then C is a subset of B
(2) For all sets A, B and C, if C is a subset of B then A-B is a subset of A-C.
I understand that the first one is false and the second one is true. However when first presented with the problems the only way I was able to solve it was by plugging in sets, until one was false. I was hoping that someone would be able to provide me with a better approach to making sense of these type of problems, and possibly how I could represent these questions with a diagram. 
Thanks!

Part (1):

The light gray area represents $A - B$. If you combine the dark and light gray areas it is $A - C$. Evidently, $A - B \subset A - C$ but $C \not\subset B$. How did I come up with this diagram? I purposely drew $B$ and $C$ so that $C \not\subset B$, and then I drew $A$ so that $B$ overlapped more with it than $C$ did. To prove that the statement is false, we have provided a counterexample, so we are done. (Remember that it is not true that $A - B \subset A - C \implies C \not\subset B$! A counterexample can be constructed for that too.
Part (2):
Since you were asking about how to intuitively approach these questions, let us not use Venn diagrams here. Instead, let us logically proceed from the givens. We know $C \subset B$: if $x \in C$ then $x \in B$. Equivalently, we have the contrapositive: $x \not\in B \implies x \not\in C$. Now, take any $y \in A - B$, which means $y \in A$ and $y \not\in B$. We just said that $y \not\in B \implies y \not\in C$, so we have $y \in A$ and $y \not\in C$. But this means $y \in A - C$! Thus $y \in A - B \implies y \in A - C$ which is the same as $A - B \subset A - C$.
Now how did I get this proof? I just followed logical deductions and definitions and tried to coherently put them together. For you, this may be easier than thinking graphically to come up with a diagram like the one above, or it may be more difficult, but as @IttayWeiss said, both will be greatly helped by practice to develop intuition.
