If $\text{"}x^2>0\text{"}$ and $\text{"}x>0 \text{"}$, write a translation for $\text{"}$ $x^2>0$ whenever $x>0$ $\text{"}$?

Question

I'm reading Rosser's: Logic for Mathematicians.

In the first chapter, he asks to translate some sentences to It's logical form. There is this exercise:

If $\text{"P"}$ and $\text{"Q"}$ are translations for $\text{"}x^2>0\text{"}$ and $\text{"}x>0 \text{"}$, write a translation for $\text{"}$$x^2>0$ whenever $x>0$$\text{"}$.

I'm a bit confused. If I try to build a truth table, I'll have:

$$\begin{matrix} {x>0}&{x^2>0}&{}&{f(P,Q)}\\ {0}&{0}&{}&{?}\\ {0}&{1}&{}&{?}\\ {1}&{0}&{}&{0}\\ {1}&{1}&{}&{1} \end{matrix}$$

And I guess that I should fill the last line with $1$ (if $x>0$ then $x^2>0$) and the previous line with $0$ because it would contradict the one I just wrote (if $x>0$ then $x^2\leq 0$).

My confusion comes with the filling of the remaining lines: The question asks me to translate $\text{"}$$x^2>0$ whenever $x>0$$\text{"}$. So, it seems I should care only with the cases where $x>0$ and the rest of the cases are irrelevant but I guess I should fill them anyway. Can I fill them with anything or should I fill them with the behavior one should expect in the premise? If It's the second case, I guess that I should write:

$$\begin{matrix} {x>0}&{x^2>0}&{}&{f(P,Q)}\\ {0}&{0}&{}&{0}\\ {0}&{1}&{}&{0}\\ {1}&{0}&{}&{0}\\ {1}&{1}&{}&{1} \end{matrix}$$

I guess the zero in $f(P,Q)$ in the second line should be so because if $x\leq 0$ then there is no guarantee that $x^2>0$, it'll be so only when $x<0$. And the zero in the first line of $f(P,Q)$ should be so because if $x\leq0$ then $x^2\leq 0$ only when $x=0$.If my reasoning is correct, I believe the translation is: $P\wedge Q$. Is it correct?

Answer 1

(I'm unable to comment due to a lack of reputation, so feel free not to accept this answer since I'm just going to refer to older questions/answers)

"$x^2 > 0$ whenever $x>0$" means "if $x>0$ then $x^2 > 0$ like you said. This tends to be written as the implication "$x>0 \implies x^2 > 0$".

As for the other two rows, you may want to read the answers to these questions: In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True? and In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?.