1. The problem asks for the negation of the statement $$\forall x (P(x) \to Q(x))$$. 2. Recall that the negation of a universal quantifier $$\forall x$$ is an existential quantifier $$\exists x$$, and the negation of an implication $$P(x) \to Q(x)$$ is $$P(x) \wedge \neg Q(x)$$. 3. Therefore, the negation is: $$\neg \forall x (P(x) \to Q(x)) = \exists x \neg (P(x) \to Q(x)) = \exists x (P(x) \wedge \neg Q(x))$$. 4. This means there exists at least one $$x$$ for which $$P(x)$$ is true and $$Q(x)$$ is false. 5. Among the options given, the correct negation is: $$\exists x (P(x) \wedge \neg Q(x))$$. Final answer: $$\exists x (P(x) \wedge \neg Q(x))$$.