Why the given statement can't be expressed using predicates and quantifiers in the way described in details? Suppose there is a statement -


  Some students in this class has visited Mexico.


Solution given is: considering the universe of discourse for the variable $x$ consists of all the people. Our solution becomes $\exists{x}[S(x)\land{M(X)}]$ where the predicate $S(x)$ tells $x$ is a student in the class and $M(x)$ tells $x$ has visited Mexico. 

The solution also tells $\exists{x}[S(x)\implies{M(X)}]$ is wrong because this expression is true when there is not someone in the class. So the statement can't be described in this way.

My questions is that only - What's wrong in giving true when there is not someone in the class (or true even when the someone is not student) ? Why exactly can't the statement be expressed using implication?
Rephrasing that second version back into English:


  There exists a person $x$, such that if $x$ is a student, then $x$ has gone to Mexico.


If there exists anyone who's not a student, this ends up being true. "If false then [anything]" is always true, so when "$x$ is a student" is false, "if $x$ is a student, then $x$ has gone to Mexico" is true—whether or not $x$ has gone to Mexico.

What you want is:


  There exists a person $x$, such that $x$ is a student, and $x$ has gone to Mexico.


That is indeed what the first given solution means.