I am tasked with changing the $\forall$ quantifier with the $\exists$ quantifier. However, I am not sure, whether I understood it correctly, and would like to ask for help.
As I understood, the overarching rule is: $\forall x\in\mathbb{Z}:p(x)$ is equal to $\neg(\exists x \in\mathbb{Z}):\neg(p(x))$
Applying this rule to two examples:
$\exists x\in\mathbb{N}$ , $\forall y\in\mathbb{Z}$ : x-y²x=0. When replacing the $\forall$ quantifier: $\exists x\in\mathbb{N}$ , $\neg(\exists y\in\mathbb{Z})$ : x-y²x≠0. Further, this statement is True for x = 0
$\forall x\in\mathbb{N}$, $\forall y\in\mathbb{Z}$ : $yx\in\mathbb{N}$. When replacing the $\forall$ quantifier two times: $\neg(\exists x\in\mathbb{N}$),$\neg(\exists y\in\mathbb{Z}$): $\neg(yx\in\mathbb{N}$). Further, this statement is false for all y < 0.
Especially, in the second example I am unsure whether two replacements will result in $\neg(yx\in\mathbb{N}$) or $(yx\in\mathbb{N}$)
Thanks!
The outer “$\forall x \in N$” quantifies all that follows. So all that follows must be negated, and then that negation can ripple inwards. In your case, replacing the outer for all in $\forall x\in\mathbb{N}$, $\forall y\in\mathbb{Z}$ : $yx\in\mathbb{N}$ with a there exists yields $$\neg\exists x\in\mathbb{N}: \neg(\forall y\in\mathbb{Z} : yx\in\mathbb{N}),$$ which, we can see by now applying that inner negation, amounts to $$\neg\exists x\in\mathbb{N}: (\exists y\in\mathbb{Z} : yx\not\in\mathbb{N}).$$
When mathematicians write in the real world, we often slightly abuse the terminology of quantification, saying things like “For all $x \in \bf N$ and $y \in \bf Z$…”. But the same rules of logic apply when negating, so in the negation of a statement quantified that way, one can replace the contracted for all with a contracted (and negated) there exists. In concept, this contraction is essentially the same thing as if we replaced the formal expression of your answer (above) by $$\neg\exists (x,y)\in\mathbb{N}\times\mathbb{Z} : yx\not\in\mathbb{N}.$$