Does $m*n \iff (m, n)=n$ define a partial order / an equivalence relation on $\{1,\ldots, 9\}$?

Question

$Note: (m, n) = n$ means that the greatest common divisor between $m$ and $n$ is $n$.

The relation $*$ is reflexive because, being $x$ any element of $A$,

$x * x$ since $(x, x) = x$

The $*$ relation isn't symmetric, for example,

$2*1$ but $1\require{cancel} \cancel{*}2$

Is it correct and is it well demonstrated what I did? Also, I can not see if the relation is transitive and / or antisymmetric.

Answer 1

Note that $$(m, n) = n \iff n|m$$

Thus the relation on $A$ is reflexive, transitive, antisymmetric and of course not symmetric.

The antisymmetric part is because all elements of A are positive so if $ n|m $ and $ m|n$, then $m=n$.