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?


I'd say that the important point is 'between them': the English sentence is not saying that everyone in the class (apart from them) has received either two e-mails or two calls from them. It's saying that between them they have reached everyone in the class somehow, i.e. everyone in the class has been e-mailed or called atleast once by one of the two.

As an (hopefully) clarifying example, consider a class made up by Mark (M), Elizabeth (E), Susan (S) and Richard (R), and suppose this: Mark called Richard, Elizabeth e-mailed Susan. This situation satisfies the statement you are asked to formalize: there are two people (M and E) who between them e-mailed or called everyone else in the room.

Your formula is much stricter: in order to be satisfied, we would need Elizabeth to call Richard and Mark to e-mail Susan as well.

  • $\begingroup$ Got your point and my pitfall.Thanks :) $\endgroup$ – user3767495 Feb 20 at 3:36

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