Negations of Statements (Regular Sentences)

Question

I'm having trouble trying to understand the negation of certain sentences.

Negate the following statements:

1. At least two of my library books are overdue
2. One of my two friends misplaced his homework assignment
3. No one expected that to happen
4. It's surprising that two students received the same exam score

Now the answer is this (according to my professor):

1. At most one of my library book is overdue
2. Both of my friends did not misplaced their homework
3. Someone expected that to happen
4. It's not surprising that two students received the same exam score

Now my question is, when do I treat the sentence as two cases? For example number 2. It's treating "friends" and "misplaced" as two cases where number 1 only treated the number of books as the case but it did not treat "overdue" as a case. I apologize if I'm using the term "case" incorrectly.

For example I thought number 1 would be:

At most one of my library book is not overdue

And I thought number two would be:

It's not surprising that two students didn't receive the same exam score

Maybe my logic is a bit fuzzy (pun intended) right now. Could someone help me clear up on when I should treat a sentence as one or two cases?

Answer 1

The negation of a sentence in English is almost (but not quite) always expressible by prefixing the whole sentence with "It's not the case that $\ldots$".

So the negation of

It's surprising that two students received the same exam score

can be expressed as

It isn't the case that it's surprising that two students received the same exam score.

Of course, that isn't the most natural English. So the optional task now is to rephrase it a bit more naturally (though this is a matter of wanting elegance than a logical requirement). No problem!

It isn't the case that it's surprising that $p$

is plainly just long-winded for

It isn't surprising that $p$!

So that's the general technique illustrated. To express the negation of a proposition expressed in English, (i) prefix with "It's not the case that". And then, if you want or you are explicitly asked for the most natural English rendering, (ii) rephrase.

Thus, step (i) the negation of

At least two of my library books are overdue

can be expressed by

It's not the case that at least two of my library books are overdue

Now, step (ii), can we think of a more natural equivalent? Well,

It's not the case that at least two blah

is equivalent to

At most one blah

(think about it!), so a more natural expression of the negation of the library proposition is

At most one of my library books are overdue.

And so it goes.

Answer 2

You're problem is that you're negating EVERY part of the proposition, so you're just back to the original (with some horrid syntax). Remember when you were in elementary school playing tag or something and you'd play the smart-aleck saying stuff like, "I'm not not $it$", meaning you really are? The same goes here. You only need to negate the main part of the proposition. Think of it this way:

P=At least two of my library books are overdue.

Phrase it mathematically; let n be the number of library books that are overdue. Then (1) is just asserting $n\geq2$. If $n$ is not $\geq2$, then $n<2$, so ~P = Less than two of my library books are overdue, or At most one of my library books is overdue. You can also phrase it using "not overdue": Assume you have $m$ library books. Then ~P is equivalent to "At least $m-1$ of my library books are NOT overdue".

But this latter phrasing requires that you know the cardinality of the set of your library books and that this cardinality is finite and that's too many assumptions. We mathematicians like less assumptions :)