Predicate Logic Negation Question

Question

What is the equivalent of $\neg (\forall x) (P(x) \vee Q(x))$? Will $P(x) \vee Q(x)$ be negated too? Or is just $\forall x$ negated?

Answer 1

$$\neg (\forall x)(Px \lor Qx)$$

By quantifier negation (QN) rules, $$(\exists x)\neg(Px \lor Qx)$$

By DeMorgan's Law

$$(\exists x)(\neg Px \land \neg Qx)$$

Answer 2

the Quantifier Negation Law says that:

$$\neg (\forall x ) \varphi(x) \Leftrightarrow (\exists x) \neg \varphi(x)$$

for any formula $\varphi(x)$

Hence:

$$\neg (\forall x)(P(x) \lor Q(x)) \Leftrightarrow (\exists x)\neg (P(x) \lor Q(x))$$

Now, you can either leave the statement this way, or you can push the negation further in to get:

$$(\exists x)(\neg P(x) \land \neg Q(x))$$

Which one is it? Your choice!