I was solving below question from Kenneth Rosen and I had a doubt in it. Let $M(x,y)$ be "$x$ has sent $y$ an e-mail message" and $T(x,y)$ be "$x$ has telephoned $y$", where the domain consists of all students in the class. Using quantifiers, express the below statement:
"There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class."
This seems to me like there are two different students $x$ and $y$ (say) who both have messaged each other or for all students $z$ ($z$ is not any one of $x$ and $y$), these both persons $x$ and $y$ have telephoned $z$. So, I wrote the expression as
$\exists x \, \exists y ( ( (x \neq y) \land M(x,y) \land M(y,x) ) \lor (\forall z ((z \neq x) \land (z \neq y)) \Rightarrow (T(x,z) \land T(y,z))))$
But Rosen's key says:
$\exists x \exists y (x \neq y \land \forall z((z \neq x \land z \neq y) \Rightarrow (M(x,z) \lor M(y,z) \lor T(x,z) \lor T(y,z))))$
And this seems to me like there are two different persons $x$ and $y$ such that for all other persons other than $x$ and $y$, even if $x$ has sent message to all such $z$ then this expression will be true. But this is not something which is asked by the statement. Please help me to know if my analysis was correct. And if it is wrong, then what is the implied meaning of Rosen's Key.
You shouldn't have $M(x,y)$. "Between them" is referring to the fact that everyone else in the class was messaged (or telephoned) by one of those two; not that the two messaged each other. (The wording is somewhat confusing.)
Notice it says "between them have sent an e-mail message to [...] everyone else in the class." Your interpretation would be correct if it said "between them have sent an e-mail message, or telephoned everyone else in the class."