From: Vilnis Detlovs and Karlis Podnieks, Introduction to Mathematical Logic, in the the usual predicate logic context, with $Person$, $Department$ and $working$ as predicates, they advise:
For “each person is working in some department”, the correct option is: $${ \forall x (Person(x) \to \exists y (Department(y) \land working(x,y))) }$$ While the following is incorrect: $${ \forall x \exists y (Person(x) \land Department(y) \to working(x,y)) }$$
I'm looking for an explanation as to why the incorrect version is incorrect, e.g. what sentence it's equivalent to, and in which cases it's different from the correct one.
Reference http://podnieks.id.lv/
The problem with the second statement is that it is true when $y$ is not a department and $x$ is a person. This is because when $\text{Department}(y)$ is false, the hypothesis is false and the implication is true. Thus it does not ensure that each person works in some department. We may have some person not working in any department at all but the second statement will still be true.