group of functions $\Bbb N \to \Bbb N$

Question

$F$ is the group of all functions fron $\Bbb N$ to $\Bbb N$, $S$ is a relation on $F$: for $f,g\in F: (f,g)∈K \iff$ for all $n\in\Bbb N, f(n)≤g(n)$.

So what is $g$ or $f$ ? Are they the outputs of $n$ ? Give me an example of element of $S$

I wrote that $S$ is

1. partially order

2. not a total order (because that $f(n)$ and $f(n+1)$ are not comparable)

3. And that order don't have maximal element or greatest element

4. And has infinite minimal elements and one least Element

Thanks. edit F，g can Be n,n+1

Answer 1

I'm assuming these functions from $\Bbb N \to \Bbb N$ are just functions, and not bijections. It's not very clear what this question wants, but looking at what you have:

1. I agree.

2. I disagree that this is sufficient. If $f(n) = n$, then the functions $f(n)$ and $g(n) \overset{\text{def}}{=} f(n + 1)$ are comparable.

3. I agree, there are no maximal or greatest elements in $S$. Should you maybe justify this; stating why, given any function $f: \Bbb N \to \Bbb N$, you can always find some $g: \Bbb N \to \Bbb N$ with $(f, g) \in S$?

4. I agree there's a least element in $S$, you should probably say what it is. I'm not convinced there are infinitely many minimal elements. Can you justify that, using the definition of minimal elements, and demonstrating what minimal elements of $S$ must look like?