To find the negation of the statement $P$, we first need to restate it clearly:

Statement $P$: There is a rational number $x$ in the set $S$ such that $x > 0$.

This can be formally written as: $P: \exists x \in S \text{ such that } x \text{ is rational and } x > 0.$

Now, to negate this statement, we'll apply the negation to the entire statement.

The negation of "there exists" ($\exists$) is "for all" ($\forall$). Therefore, the negation of $P$ is:

Negation of $P$: For all rational numbers $x$ in $S$, $x \leq 0$.

This can be formally written as: $\neg P: \forall x \in S, \text{ if } x \text{ is rational, then } x \leq 0.$

So, the correct negation of statement $P$ is:

For every rational number $x$ in $S$, $x \leq 0$.

Would you like to go over the reasoning again or explore related concepts? Here are some related questions:

1. How do you negate a statement involving "for all" ($\forall$)?
2. What are some examples of rational numbers?
3. How does the negation of an "if-then" statement work?
4. What is the difference between rational and irrational numbers?
5. How do you express a logical statement in symbolic form?

Tip: When negating a statement, it's helpful to break it down into its basic logical components.