Putting negations before predicates using quantifiers

Question

I'm trying to transform this logical statement $\neg\forall x\exists y(P(x, y)\lor Q(x, y))$ to have
negation before predicates and I'm asking for a little explanation. Are these steps
I performed earlier correct or am I wrong, and how to do that the proper way?

$\neg\forall x\exists y(P(x, y)\lor Q(x, y))$

My steps:

1. $\exists x\neg(\exists y(P(x, y)\lor Q(x, y)))$

2. $\exists x\forall y\neg(P(x, y)\lor Q(x, y))$

3. $\exists x\forall y(\neg P(x, y)\land\neg Q(x, y))$

So this is the final statement I get :

$$\exists x\forall y(\neg P(x, y)\land\neg Q(x, y))$$

Is this correct? You guys are the best 🙂

Answer 1

Yes, you seem to have a clear understanding of bringing the negation into what is being quantified.

For you're first step, you could have started with $$\exists x \lnot\exists y\Big((P(x, y) \lor Q(x, y)\Big).$$

But either way, the steps you demonstrate are correct.