Negation of a statement? Confused

Question

My book wants me to give the negation of the defining statement below

“A set of Real Numbers S has the Archimedian property iff

∀ a, b ∈ S, ∃ n ∈ N such that na > b”

Now, I know logically in general the negation of x>y would be x ≤ y

However, it seems to me that here the negation should be

“∃ a, b ∈ S such that ∀ n ∈ N, na = b” as opposed to “∃ a, b ∈ S such that ∀ n ∈ N, na ≤ b”

Because, if na < b then that implies bn > a for n = 1, which would mean it does the fit the original defition.

SIDE NOTE: If I’m understanding this property correctly, does any set S of real numbers have the archimedian property iff S = ø and 0 ∉ S.

Thanks so much!

Answer 1

No. What you do here may work for $n=1$, but note that the statement is about any $n$.

Also, as a piece of advice for future work, when you are asked to provide a logical negation, you should just work with that. Yes, given some intended interpretation, you can sometimes write it as a different statement (e.g if the domain is natural numbers, we can use $x=0$ instead of $x <1$, but logically those two statements are not equivalent.

Answer 2

You want the negation of $na>b$.

This cannot be $na=b$, because there are cases where both $na>b$ and $na=b$ are false, for example if $n=a=b=2$.

Even in the context with the quantifiers, your proposed negation does not mean the opposite of the original statement. For example, your proposed negation is not satisfied by $S=\{-1,1\}$, even though that set also fails to satisfy the original definition that it's supposed to be a negation of.

In general, when you negate a statement, you should not be thinking about implications -- just about getting the opposite truth value in every situation.