# Simplify $(p\land q)\lor(p\land \neg q)$

So I was asked to simplify this statement $S$:

$$(p \land q) \lor (p \land \neg q)$$

My understanding is that it needs to have a similar truth table, though I'm not sure if that's exactly right. I tried simplifying it, but I can't think of any way to make it simpler and logically equivalent.

Any help is appreciated.

• Distributivity. – Git Gud Feb 21 '15 at 21:26
• @GitGud has pointed the way to an algebraic simplification. As an informal alternative, does the truth value of $q$ have any effect at all on the truth value of $S$? – Brian M. Scott Feb 21 '15 at 21:30

Using the distributive rule on the first line, \begin{align} (p\land q) \lor (p \land \lnot q) &\equiv p \land\underbrace{(q \lor \lnot q)}_{\large\text{true}}\\ & \equiv p\end{align}