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.