# English Statement to Logic Expression Using quantifiers

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 for x and y consists of all students in your class.

What would be the logic expression for 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."

Kenneth Rosen has below given answer for it.

$$\exists x \exists y(x \neq y \land \forall z((z \neq x \land z \neq y)\rightarrow (M(x,z) \lor M(y,z) \lor T(x,z) \lor T(y,z))))$$

But I think it should be

$$\exists x \exists y(x \neq y \land \forall z((z \neq x \land z \neq y)\rightarrow ((M(x,z) \land M(y,z)) \lor (T(x,z) \land T(y,z)))))$$

Which means that two different students have either Sent message to everyone else in the class OR telephoned everyone else in the class.Means Either these two students x and y have both sent message to everyone else in the class OR both telephoned everyone else in the class and even if both the cases are true then also it is fine but one of them(Either message or Telephone) MUST be true.

Is my claim correct?