Quantifiers and English: Existential and Universal difference Firstly, excuse my simplicity in describing the title; I couldn't find a proper title that could explain what I am confused about. The example looks at how we express the following statements in predicates and quantifiers,


*

*“Every student in this class has visited Mexico”

*“Some student in this class has visited Mexico”


The first statement is fairly easy. If we take $x$ as the domain of people, $S(x)=$ '$\text{x is a student}$', and $M(x)=$ '$\text{x visited Mexico}$'. We have $\forall x(S(x) \rightarrow M(x))$. Its clear that it wouldn't work if we had it like this $\forall x(S(x) \land  M(x))$. Because that would mean that all the domain of $x$ is $S(x)$ but that isn't true. 
Now, the confusion comes when we the second statement comes in. The answer is $\exists x(S(x) \land  M(x))$ but I don't see how  this representation $\exists x(S(x) \rightarrow M(x))$ can be wrong. 
 A: Your confusion probably lies in that you haven't clear what you're supposed to do. You want to express some facts:


*

*Every student in the class has visited Mexico

*At least one student in the class has visited Mexico


At this stage you're not asking if either is true or false. You just want a formal statement that expresses those facts.

Consider the truth table for $\to$:
$$
\begin{array}{cc|c}
A&B& A\to B \\
\hline
T & T & T \\
T & F & F \\
F & T & T \\
F & F & T
\end{array}
$$
Thus the statement $\exists x(S(x)\to M(x))$ is true if there's an individual in your domain who's not a student and either visited Mexico or not. Therefore, as soon as there's a non student, the statement is true.
You can also see it by substituting $\exists x$ with $\lnot\forall x\lnot$ and $S(x)\to M(x)$ by $\lnot S(x)\lor M(x)$:
\begin{gather}
\exists x(S(x)\to M(x))\\
\lnot\forall x\lnot(\lnot S(x)\lor M(x))\\
\lnot\forall x(\lnot\lnot S(x)\land \lnot M(x))\\
\lnot\forall x(S(x)\land \lnot M(x))
\end{gather}
Thus the initial statement is false if and only if
$$
\forall x(S(x)\land \lnot M(x))
$$
is true, that is, every individual is a student and no individual has visited Mexico.
Look at the truth table for $\land$, instead:
$$
\begin{array}{cc|c}
A&B& A\land B \\
\hline
T & T & T \\
T & F & F \\
F & T & F \\
F & F & F
\end{array}
$$
Now all should be clear: the statement is true if and only if you have $S(x)$ and $M(x)$ true for the same individual.
A: Maybe look at the negation of the second sentence, which would be:

None of the students has visited Mexico

or:

Every student in this class has not visited Mexico

So you would translate it as
$$ \forall x(S(x)\to\neg M(x))$$
or equivalently
$$ \forall x(\neg(S(x)\land M(x))$$
Something that's wrong for all must be true for some, i.e. $\forall x\neg\Phi$ is the same as $\exists x \Phi$. So we indeed arrive at
$$ \exists x(S(x)\land M(x)).$$
