Quantification logic - Is my answer also correct? I have a problem:

Analaze the logic of the statements:
...
(d) Jane saw a police officer, and Roger saw one too.

The answer was:
$$∃x (P(x) ∧ S(j,x)) ∧ ∃y (P(y) ∧ S(r,y))$$
Where $P(x)$ is a police officer, $S(x,y)$ means "x saw y", $j$ is Jane, and $r$ is Roger.
My answer is different:
$$\forall x\exists p((x=j\vee x=r)\rightarrow S(x,p))$$
Where $p$ is a police officer.
Are both answers correct. If my answer is incorrect then why?
Edit: where does the downvote come from? It is a mathematics question even though it doesn't seem like!
 A: Your answer is almost correct.  However, you cannot use $p$ to refer to "a police officer".  When you write $\exists p$, that just means there exists $p$, with no further assumptions.  So $p$ could be any person (or thing!) at all, not necessarily a police officer.  To specify that the variable $p$ refers to a person who is a police officer, you need to use a predicate $P$ as in the first solution.  Note that it is further not correct to say "$P(x)$ is a police officer" as you did in describing the first solution.  Rather, $P(x)$ is the statement "$x$ is a police officer".
So let us modify your solution to: $$\forall x\exists p((x=j\vee x=r)\rightarrow (P(p)\wedge S(x,p))).$$
With this modification, your solution is correct.  For values of $x$ other than Jane or Roger, the statement inside $\forall x$ says nothing at all, since $x=j\vee x=r$ is false and so the implication is automatically true.  When $x$ is either Jane or Roger, the statement says there exists $p$ who is a police officer and who $x$ saw.  Since the quantifier $\exists p$ is inside $\forall x$, this $p$ can be different for different values of $x$, so it does not claim that Jane and Roger saw the same police officer.  Thus this has the same meaning as the given sentence.
