Quantified statement with contrapositive

Question

I have this statement

$$\forall x: P(x) \implies Q(x)$$

If I want to take the contra positive with the $\forall$ change to an $\exists$.

Answer 1

The only way that "taking the contrapositive" makes sense is in order to write the contrapositive of the quantified implication in parentheses. For this, there is no need to change the quantifier:

$$\forall x\,(P(x)\rightarrow Q(x)) \equiv \forall x\,(\lnot Q(x) \rightarrow \lnot P(x))$$

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Now, if you want to negate your posted expression, we work with moving the negation from the outside to the inside: $$\begin{align}\lnot \forall x\,(P(x)\rightarrow Q(x)) &\iff \exists x\Big(\lnot(P(x)\rightarrow Q(x))\Big) \\ \\ & \iff \exists x\Big(\lnot(\lnot P(x) \lor Q(x))\Big)\\ \\\ &\iff \exists x\,(P(x)\land \lnot Q(x)) \end{align}$$