Order of statements in implication The question is from Exercise 13 of part 1.4 in Rosen's "Discrete Mathematics and Its Applications" (5th edition): "let $M(x,y)$ be "$x$ has sent y an e-mail message", where the universe of discourse consists of all students in your class".
The sentence to be translated to predicate language is:
"g) there is a student in your class who has sent everyone else in your class an e-mail message"
My solution is:
$$\exists x \ \forall y \ \ (M(x,y)\implies x \neq y) $$  
The solution in the instructor's manual of Rosen's book has the statements of the implication in reverse:
$$\exists x \ \ \forall y \ \ (x \neq y \implies M(x,y))$$  
Which one is correct? To me it seems that the solutions manual is incorrect. The solution from the manual is true if the antecedent is F and the consequent T, which would mean that if the student is himself/herself, then he/she has sent himself/herself an e-mail message, so it seems to be wrong in my mind because the original idea was to exclude sending messages to oneself. The first solution allows x to send y an e-mail message only if it is not sent to oneself and if the e-mail is not sent, then it doesn't matter to whom it is sent.
 A: Statement $1$ is $\exists x\forall y (M(x,y) \implies x\ne y)$
This can be interpreted as:

There exists a student such that for all students, if student $x$ has sent student $y$ an email, student $x$ is not student $y$.

This doesn't implies that the student has sent everyone but themselves an email, or even any student an email, just that they haven't sent an email to themselves.
Statement $2$ is $\exists x\forall y (x\ne y \implies M(x,y))$
This is correct. It can be interpreted as:

There exists a student $x$ such that for all students, if student $x$ is not student $y$, then they have sent student $y$ an email.

And statement $2$ also means that student $x$ has not sent an email to themselves.
A: An other approach by the writing the negation of your statement and the book's statement :


*

*The negation of $\exists x \ \forall y \ \ (M(x,y)\implies x \neq y) $ is :
For all $x$ there exist $y$ such that $x$ sent $y$ an email and ($x$ and $y$ is the same person).


In other word : 

For all $x$,  $x$ sent an email to himself.



*

*The negation of $\exists x \ \ \forall y \ \ (x \neq y \implies M(x,y))$ is :


For all $x$ there exist $y$ such that $x$ is different from $y$ and 
$x$ didn't  send an email to $y$
In other word :

For all $x$ there exist $y$ different from $x$ such that $x$ didn't  send an
  email to $y$.

You can see that this second statement  is the exact negation of the one wanted.
Note that for the statement of the negation I used the fact that $(P \Longrightarrow  Q)$ is the same as $\lnot P \vee Q$
