How do I derive logical statements simply? For example if I want to show the equation for $x \rightarrow y$, using truth tables it is the same as:
$$(\neg x \wedge \neg y) \lor (\neg x \wedge y) \lor (x \wedge y)$$
Is there a methodical way to reduce something like this to its simple form $\neg x \lor y$?

You can prove this way:
\begin{align}
((\neg x \wedge \neg y) \lor (\neg x \wedge y) \lor (x \wedge y))&\iff ((\neg x\land(\neg y\lor y))\lor (x \wedge y))
\\
&\iff((\neg x\land 1)\lor (x \wedge y))
\\
&\iff(\neg x\lor (x \wedge y))
\\
&\iff((\neg x\lor x)\land (\neg x\lor y))
\\
&\iff(1\land (\neg x\lor y))
\\
&\iff(\neg x\lor y)
\end{align}
