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

You use ^ whe you need to test the clauses independently.

When x=y, M(x,y)=false . So, the result is true and this precisely what we want, right?

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But question says everyone else. So how this is true for a person sending to himself?

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F -> F is true ,right?

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What i am getting is that a student who has sent mail to everyone else in class means everyone else except him and thats why i am getting confused whether he has sent mail to himself or not?

If he has sent then F->F will be true else we will use ^.Please clear this point

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Consider the that Rahul sends mail to everyone except himself. Now, let the sql table describing the information be:

student in class             Has been sent mail

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Sushant                          Yes

Arjun                               Yes

Rahul                               No

 

Now, what functional dependency would you derive to describe the students that have been sent mail by Rahul?

Note that in ^ operator, you need to test the operands of ^ independently while for ->, you need to check only the premise.

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I am asking that whether we should consider that a person has sent a mail to himself or not

Two possible answers :

1.   ∃x∀y[(x≠y)→M(x,y)] :- In this ,it will be true even when x=x,as implication becomes true,So there exits a student who has sent a mail to everone(including himself)

2.  ∃x∀y[(x≠y)^M(x,y)]:-This means there is a student who has sent a mail to everyone except himself.

Nowthe only difference is 1. answer allowing x=x,where as second is not.

But what actually is implied by the question i am not getting

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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