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There is a student in your class who has sent everyone else in your class an e- mail message.

M(x,y)=x has sent an e-mail message to y Domain-All students of your class

My solution: (there exists x)(for all y)((x!=y)<->M(x,y))

The solution in Solution Manual (there exists x)(for all y)((x!=y)->M(x,y))

The only difference is the biconditional and the implies.

How does the meaning change with biconditionals and implies?

Diya
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    With $\leftrightarrow$, the formula includes the information that this student didn't send himself an e-mail message. With $\to$, it doesn't say whether or not there was such a self-message. – Andreas Blass Jun 30 '19 at 11:58
  • @AndreasBlass which one of the answer is correct?mine or that of the solution manual ?As I have not considered that the student didn't send himself an e-mail message – Diya Jun 30 '19 at 12:38
  • The book's answer is correct, because the English sentence that you want to translate doesn't say whether this student sent himself an e-mail , so an accurate translation shouldn't say that either. – Andreas Blass Jun 30 '19 at 18:29

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Remember that $A\leftrightarrow B$ is a shorthand for $(A\to B)\land (B\to A)$.

Your sentence says:

If student $y$ is not the sender, then the sender sent an e-mail to $y$, AND, if the sender sent an e-mail to $y$, then $y$ is not the sender.

Or in other words:

The sender sent everyone else a message AND did not send a message to himself.

The sentence from the solution manual says:

If student $y$ is not the sender, then the sender sent an e-mail to $y$.

This is exactly the meaning from the exercise statement, without saying anything about the sender sending himself messages. This is as it should, since nothing of the latter sort is mentioned in the exercise statement.

So it seems the solution manual is correct.

Probably your confusion comes from the interpretation of the sentence in natural language (which is ambiguous) as contrasted with a logical interpretation.

In daily speech, when we say "Bob sent everybody else a message", we usually implicitly mean that Bob didn't send himself a message, for one because sending yourself messages is kind of strange, and additionally because we would have otherwise probably said the shorter sentence "Bob sent everybody a message". From a linguistic perspective these two sentences have a different meaning that is most likely mutually exclusive.

However, from a logical viewpoint, if Bob sends everybody a message, then Bob also sends everybody who is not Bob a message. So the latter sentence is logically a strictly weaker statement than the first one: it does not make any claims about mails Bob sends himself.

It's comparable to the logical "joke" of answering the question "Do you want coffee or tea?" with "Yes" or of answering "Could you pass the butter?" with "Yes, I could" and not undertaking any action. In logic, you take statements literally.

Vsotvep
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