representing a sentence with quantified statements

Question

My approach to this question:
$$ \exists x(P(x)\to R(x)) $$

I cannot verify if my answer is correct, any help to verify my answer would be appreciated and if I did wrong any help to explain why would also be appreciated.

Answer 1

Your answer is on the right track, but it is saying “if $x$ is a student then $x$ is shy.”

We are looking for a person who is a student $\textit{and}$ is shy. Reason why, if someone is a student doesn’t necessarily make them shy, but there is a student that is shy.

So the correct translation would be, $\exists x (P(x) \land R(x))$. This translation is saying $x$ is shy $\textit{and}$ $x$ is a student.