How can I put this sentence into prenex form

Question

I have this sentence, and want to put it in Prenex form:

$$ \lnot(( \exists y)( \forall z) Q(y,z)) $$

Is it this?:

$$ ( \forall y)( \exists z) \lnot Q(y,z) $$

Answer 1

Yes ... basically the relevant rule is:

When you push/pull a negation sign past an adjacent quantifier, the quantifier flips into its dual.

So pushing inwards, $\neg\forall x\varphi$ is equivalent to $\exists x\neg\varphi$. Pulling outwards $\forall x\neg\varphi$ is equivalent to $\neg\exists x\varphi$.

That's because $\forall$ and $\neg\exists\neg$ come to the same: so $\neg\forall x\varphi$ is equivalent to $\neg\neg\exists x\neg\varphi$ is equivalent to $\exists x\neg\varphi$. Similarly for $\exists$ and $\neg\forall\neg$, and so on (all classically, of course).

So $\lnot\exists y\forall z Qyz$ is equivalent to $\forall y\neg\forall z Qyz$ is equivalent to $\forall y\exists z\neg Qyz$

[Minor point. Your notation is unnecessarily over-bracketed according to most modern versions of FOL syntax.]