How do I derive logical statements simply?

Question

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 $\neg x \lor y$?

Answer 1

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}