Simplify (not $p$ and $q$) and (not $p$ or $q$)

Question

$(\neg P \wedge Q) \wedge (\neg P \vee Q)$

I have to simplify this using logic rules, and I can't seem to find the one rule to simplify this. Can you please help me? If yes, please tell me the properties used. Thank you

Answer 1

If you make a truth table, you'll see that:

• If $P,Q \longrightarrow$ FALSE
• If $P,\neg Q \longrightarrow$ FALSE
• If $\neg P,Q \longrightarrow$ TRUE
• If $\neg P,\neg Q \longrightarrow$ FALSE

So maybe the simplified result you're looking for is $$\boxed{\neg P\wedge Q}$$