Simplifying a logical compound statement

Question

I need to simplify $(p \vee r) \wedge (\neg p \vee \neg r)$ (if possible and using the laws of logic)

I tried to substitue $s: (\neg p \vee r)$ but that made it even worse.

Can anyone suggest an alternate way to approach this problem? Not looking for a solution, but a hint on how I can make some progress.

Answer 1

the answer is $r$. In fact using distribution law we have, $(p{\wedge{\sim}p)\vee}r=r{\vee}F=r$

Answer 2

Note that by simple testing, you can tell what's going on.

If $p$ is true, the result is $r$.

If $p$ is false, the result is $r$.

So, the expression is equivalent to just plain old $r$.

This is a simple tactic that may help you simplify some expressions: just fix one variable and see what falls out, then do the same with the opposite value for the variable. You may be able to make an observation based on this experiment. If not, little is lost.

Answer 3

For the modified question: If $p$ is true, $r$ must be false.

If $p$ is false, $r$ must be true.

So the formula is equivalent to "$p\iff\neg r$".