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.

  • 1
    $\begingroup$ Distributivity. $\endgroup$ – Git Gud Feb 21 '15 at 21:26
  • 2
    $\begingroup$ @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 $S$? $\endgroup$ – Brian M. Scott Feb 21 '15 at 21:30

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

  • $\begingroup$ Thanks! Didn't think it was as simple as using distribution $\endgroup$ – TheAtomicPeter Feb 21 '15 at 21:50
  • $\begingroup$ You're welcome! $\endgroup$ – Namaste Feb 21 '15 at 21:52

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