If I consider the relation $R$ on the set $X={3,5,30,45}$ and I define it by this statement:
$\forall$ $m,n \in X$, $m$ $R$ $n$ if and only if there exists an integer $K$ such that $m=kn$.
How could I prove $R$ is a partial order on $X$? Would I have to draw a Hasse Diagram?
Thanks for your help.
It's not too difficult to show that $R$ (the divisibility relation) is a partial order on $\mathbb{N}$. Since $X \subseteq \mathbb{N}$, then $R$ is still a partial order on $X$ (since we don't lose any of the three properties (reflexivity, antisymmetry, and transitivity) of a partial order if take a subset of the elements we are ordering).
To show a partial order you just need to show that $R$ is transitive (if $m$ $R$ $n$ and $n$ $R$ $n^\prime$ then $m$ $R$ $n^\prime$), antisymmetric (i.e. there exists at most one relation between two distinct elements, so if $m$ $R$ $n$ and $n$ $R$ $m$ then $n = m$) and reflexive ($n$ $R$ $n$).