First and foremost, to easily parse these, it may be best to express these in quantifiers (even if you haven't done them yet: if nothing else, you can just use the words instead of the symbols):
- $\forall$ means "for all"
- $\exists$ means "there exists" or "there is some"
- $P(x)$ can be taken to mean "a proposition $P$ for which $x$ holds true"
- $\neg P(x)$ likewise means "$P$ is false for $x$"
Then the negation of a statement has you interchange every $\forall$ and $\exists$ for the other, and replace $P(x)$ with $\neg P(x)$.
For example: "all numbers in $S$ are positive" can be translated into
$$(\forall x \in S)(x > 0)$$
which has negation
$$(\exists x \in S)(x \le 0)$$
(where $P(x)$ is the statement $x>0$): the negation is "there is some number in $S$ which is nonpositive".
1 - Problem: All the integers are positive
My Answer: There exists at least one integer that is not positive
This is fine.
2 - Problem: The boiling temperature of water is 98 degrees Celsius
My Answer: The boiling temperature of water is NOT 98 degrees Celsius
I believe I can simplify this to: The boiling temperature of water is greater than 98 degrees
Your answer is fine; your simplification is not. The negation would just say that the boiling temperature is not 98 degrees.
You could make, perhaps, an argument that if the boiling temperature of water is 98 degrees, it is also certainly any higher number (up to the conditions in which you boil water), so this in negation would translate to a lower number.
However, boiling temperature is typically defined (under fixed conditions) as a single quantity. "The boiling point is 100 degrees" means "this is boiling at any temperature of at least 100 degrees" but not "higher temperatures are boiling points."
But this is mostly semantics.
The most correct statement from the viewpoint of the mathematics is that the negation has boiling temperature not equal to 98 degrees. After all, we're simply saying we know that it's not 98, but we don't know about it being 97 or 99.
3 - Problem: 100 is a multiple of 5
My Answer: 100 is NOT a multiple of 5
I believe I can simplify this to: 5 does not divide 100
Your answer and "simplification" are equivalent statements, but I wouldn't say the latter is simpler, because then you have to define what you mean by "divides," which is a bit of a nontrivial thing even if it seems obvious.
4 - Problem: All students do not have 3 legs
My Answer: It is not the case that all students do not have 3 legs
I believe I can simplify this to: There exists at least one student with 3 legs
These are equivalent statements and equally correct. It basically amounts to whether you say "$(\forall x)(P(x))$" has negation $(\neg \forall x)(P(x))$ or $(\exists x)(\neg P(x))$. These are equivalent, but we usually default to the latter since it's easier to handle and prove. (Similar ideas mentioned in a past question I asked here.)
5 - Problem: There are 13 items in a baker's dozen
My Answer: There are not 13 items in a baker's dozen
Correct.
