Discrete Mathematics - Partial Order Equivalence Relations

Question

Let S be a set of positive integers and define a relation ρ on S as follows: ∀a, b ∈ S, aρb if and only if a ≤ b and a and b have the same number of positive divisors.

Let S = Z+. Prove that (S, ρ) is a partial order.

Answer 1

If I understood correctly your relation $\rho$ it's easy to show it's order relation for th partial order you can take $a=3$ anb $b=9$, and therefore a is b are not ordered by $\rho$

Answer 2

let $x,y \quad and \quad z \in \mathbb{Z}$

1) x and x have he same number of positive divisors and $x\leq x$ this implies $x \rho x$

2) $x \rho y$ and $y\rho x$ this implies ($x\leq y$ and $y\leq x $) this implies $x=y$ because$(\mathbb{Z},\leq)$ is ordered

3) $x \rho y$ and $y\rho z\quad $ this equivalent to say
($x\leq y$ and $y\leq z$, and x ,y and z have the same number of positive divisors ) this imples $x\leq z$ and x and z have the same number of positive divisors so $x\rho z$

$\rho$ is not total order total it's just partial order (look the answer above )