How should I begin when trying to determine a proposition whose truth value is unknown? I've been doing many exercises that ask me something like

With whatever sets X, Y, A, and B, determine the truth value of
$(X \subseteq (A \triangle B) \land Y \subseteq A) \implies X \cap Y = \emptyset$

Since obviously I don't know whether the end value will be true or false, I have a pretty hard time when determining it, because it seems like I require some "intuition", assume that it is either true or false, and attempt to prove that. If my assumption is wrong, then my entire proof leads to nowhere, and I waste my time.
I usually end up determining their value because I realize that my attempt to prove them true is leading nowhere, so then I attempt to prove them false. Time wasted, I guess.
Is there a proper way to start with this kind of problem?
 A: With this sort of problem you can often make a good start by drawing a Venn diagram. If the circles in the diagram below represent the sets $A$ and $B$, the red region represents $A\mathbin{\triangle}B$. If $X$ is a subset of the red region, and $Y$ is a subset of the lefthand circle, must $X\cap Y=\varnothing$? Clearly not: $X$ and $Y$ might both be the lefthand half of the red region, i.e., $A\setminus B$. Now just build a concrete counterexample using that idea: you might, for instance, let $A=\{0,1\}$, $B=\{1,2\}$, and $X=Y=\{0\}$.

With a bit more experience you won’t need the diagram. You might, for instance, see right away that if $A\cap B=\varnothing$, then $A\mathbin{\triangle}B=A\cup B$, so as long as $A$ is non-empty, you can set $X=Y=A$ and get a counterexample. Or you might reason it out like this. Points of $A\mathbin{\triangle}B$ are points that are in exactly one of $A$ and $B$; surely we can find $A$ and $B$ so that some of these are in $A$, and we can then let $Y$ be the set of those points.
A: Don't worry - you'll gain some sort of feel for how to do these questions until they become easy, or obvious.  
In fact, you can make this very question obvious now.  For a problem in elementary set theory, such as this one, you can easily visualize the question by drawing an Euler diagram: draw the set $A,B$ as intersecting circles, and fill in the region corresponding to $A\triangle B$ (if you haven't seen this before, I'll just tell you - it's everything except the region in the middle).  If a set $X$ is contained in this region and a set $Y$ is contained in $A$, must it be the case that $X$ and $Y$ don't intersect?  If it must, then try and use your diagram to find a proof.  If you can find a counterexample in the Euler diagram, use it to construct an explicit counterexample: draw $X$ and $Y$ on the diagram in the specified regions in such a way that they do intersect, and then fill in every separate region on the diagram with a distinct number.  Define $A,B,X,Y$ to be the sets of the numbers that those circles enclose on the diagram, and show mathematically that this choice of $A,B,X,Y$ doesn't satisfy the conjecture.
