I would like to understand something regarding the nested quantifiers in discrete math. In the following question part (c):
Let $M(x,y)$ be "$x$ has sent $y$ an e-mail message", and $T(x,y)$ be "$x$ has telephoned $y$", where the domains consist of all students in your class. Use quantifiers to express each of the following statements. (Assume that all e-mail messages that were sent are received.)
(c) There is a student in your class who has sent everyone else in your class an e-mail message.
Is this answer correct:
$$\exists x\,\forall y\,[(x\ne y)\to M(x,y)]$$
And if yes, why to use the implication although it's gonna result in True if the $x = y$ and the student has sent himself an email (Which should be False not True).
Also when should we use and ($\land$) and not $\to$?
Thank you :D
Yes, $\exists x\,\forall y\,[(x\ne y)\to M(x,y)]$ correctly translates There is a student in your class who has sent everyone else in your class an e-mail message. The italicized statement does not say whether this student has sent himself an e-mail: provided that the student has sent everyone else in the class an e-mail, the statement is true both if he has sent himself an e-mail as well and if he has not done so.
And that’s exactly what the formal version says: $(x\ne y)\to M(x,y)$ is true if $x=y$ both when $M(x,y)$ is true — i.e., when $x$ has sent himself an e-mail — and when $M(x,y)$ is false — i.e., when $x$ has not sent himself an e-mail.
Your last question is too broad for me to be able to give you a useful answer.