I'm having some trouble with a proof, and I'm sure I'm overthinking something simple.
If we let $\mathbb{N}$ be the set of natural numbers, and define a relation $R$ on $\mathbb{N} \times \mathbb{N}$ (to produce $x,y$ pairs) such that $xRy$ if $y$ is a multiple of $x$, then does the multiple itself have to be an element of $\mathbb{N}$? i.e., does the multiple itself have to be a natural number?
Obviously, it does work for any natural number multiple, but that would ensure that $y>x$ always. What about is $x>y$ and the multiple is a fraction? I'm wondering how to interpret the definition of a the relation $R$: apply it to all parts of the relation, or just to $"x"$ and $"y"$.
The answer to your first quest (“does the multiple itself have to be an element of $\mathbb N$?”) is “yes”. Your universe is the set $\mathbb N$ and if $x,y\in\mathbb N$, then $xRy$ means that the natural number $y$ is a multiple of the natural number $n$.