How do I prove that $R=\{(x,y) \in S \times S : x\text{ divides }y\}$ is antisymmetric?

Question

$S=\{1, 2, 3,\ldots, 1000\}$

$R=\{(x,y) \in S \times S: x \mid y\}$

My attempt:

Assume $xRy$ and $yRx$. Then $x=ym$ and $y=xn$ for $m$, $n$ in Natural numbers.

-So $x=xxn..$ that gets me nowhere.

-So $\left(\frac yn\right)=ym$, $\left(\frac1n\right)=m$. Then $x=\left(\frac yn\right)$ and $y=xn$. Then I just get $y=xn$ twice.

I think that I need to prove that $m = n$, but I get $m=\left(\frac1n\right)$. Any suggestions?

Answer 1

Since $x \sim y$ and $y \sim x$, we have (as you said) $x = ym$ and $y = xn$ for $n,m \in \Bbb N$.

Therefore we have, $$xy = xynm \implies 1 = nm \iff n = m = 1.$$

The result now follows.