Every non-empty subset of $\Bbb{N}$ (whether it’s finite or infinite) has a minimum. One can’t say the same about $\Bbb{Z}$. Find a total order relation $≺$ on $\Bbb{Z}$ such that every non-empty subset of $\Bbb{Z}$ has a minimum under the $≺$ relation.
What I did
I am thinking that the divisibility relation will work to define a total order relation on the integers such that every non-empty subset of $\Bbb{Z}$ has a minimum.
I would first define $a ≺ b$ if $∣a∣$ divides $∣b∣$, where $∣x∣$ denotes the absolute value of $x$.
$a≺b$ if and only if $∣a∣$ is a divisor of $∣b∣$. If $∣a∣=∣b∣$, then $a≺b$ if $a<b$.
This relation satisfies the following properties: Totality, Reflexivity, Transitivity.
With this relation, every non-empty subset of $\Bbb{Z}$ will have a minimum element under $≺$. Because, for any $a$ in the subset, there will always be an integer with the smallest absolute value that divides $∣a∣$, and that integer will be the minimum element.
Am I correct?
