State the following sequent as a theorem

Question

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

Not really sure how to do this, if someone could help me, it would be much appreciated.

Answer 1

What system are you using? I will do using natural deduction:

You eliminate the negation and get $\neg p \lor \neg q$ as hypothesis and need to prove $\bot$. Now you use disjunction elimination for $\neg p \lor \neg q$ and using $p \land q$ you get $\bot$.

Answer 2

To prove $\rm P\wedge Q\vdash \neg(\neg P\vee Q)$ by contradiction, we shall need to show: $\rm \{P\wedge Q, \neg P\vee \neg Q \}\vdash \bot$.

If we assume $\rm P\wedge Q$, then ....

If we assume $\rm \neg P\vee \neg Q$, then ....

Therefore the two assumptions are a contradition.

Thus if we assume one we will conclude the other is false.