What happens when you apply a NOT to a universal quantifier

Question

If I have $\forall x: P(x)$, what is the equivalent function if I attach a not to the whole function:

$\neg (\forall x: P(x))$

Answer 1

Read the statement out: not (for all x, p(x) holds). So if not for all x, there must be some x such that p(x) doesn't hold. So:

Then you have $\exists x : \lnot P(x)$.

Answer 2

you'll have $$\exists x: \neg P(x)$$