Negating a Symbolic Expression/Logic

Question

Negate the following and simplify as much as you can...

$$\exists x ~\forall y~(p(y) \to \forall z~q(z))$$

How would I negate this expression. Not sure how to start it.

Answer 1

Begin with placing a negation sign.

$\qquad\neg \exists x ~\forall y~(p(y) \to \forall z~q(z))$

Then use deMorgan's Rules for quantifier duality, and the negation of a conditional.

$\qquad\begin{align}\neg \exists x~\phi ~&\equiv~\forall x~\neg \phi\\[1ex]\neg \forall y~\psi~&\equiv~\exists y~\neg\psi\\[1ex]\neg(\chi\to \xi)~&\equiv~\chi\wedge\neg\xi\end{align}$