Propositional Logic Discrete Maths

Question

There is a student in the class who has sent an e-mail to everyone else in the class,Consider domain as all students in a class.

Assume M(x,y) means x has send an email to y

 

Answer given is :-  ∃x∀y[(x≠y)→M(x,y)]

 it's gonna result in True if the x=y  and the student has sent himself an email

Shall we use ^ consider instead of -> ?Please correct

Comments

No, if ^ is used it says only x exist in the class.

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^ specifies the properties of selected things from the sql table(if there exists one), right Arjun?

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@Arjun Sir, it will not say only x exist,it will say at least one x exist for all y.Please correct