A relation $\mathrm{R}$ is defined on the set of all positive integers by:
$x\mathrm{R}y$ if and only if $y = 3^k\cdot x$ for some non-negative integer $k$.
Prove that $\mathrm{R}$ is a partial order.
A relation $\mathrm{R}$ is defined on the set of all positive integers by:
$x\mathrm{R}y$ if and only if $y = 3^k\cdot x$ for some non-negative integer $k$.
Prove that $\mathrm{R}$ is a partial order.
As you mentioned in a comment, you need to show that the relation is reflexive, antisymmetric, and transitive. And you know that the reflexive step uses the fact that $3^0=1$. From there, you ask how you carefully convey the fact that the relation is reflexive.
You want: For all positive integers $x$, $x\mathrm{R}x$. This means that for all positive integers $x$, there exists a nonnegative integer $k$ such that $x=3^k\cdot x$.
To show this, let $x$ be given, and take $k=0$, which is a nonnegative integer. Then $x=3^0\cdot x$ shows that $x\mathrm{R}x$.
Hint for antisymmetry: $3^k\cdot 3^j=1\Rightarrow k=j=0$.
Hint for transitivity: A product of powers of $3$ is a power of $3$.