How can I prove that $(\mathbb Z,\leq_*)$ is isomorphic to $(\mathbb N,\leq )$

Question

We define the binary relation $\leq_*$ between integers by the following rule: For $n,m\in\mathbb Z$, $n\leq_∗m$ holds if and only if one of the following conditions is satisfied:

$|n|<|m|$, or

$|n|=|m|$ and $n<0\,$, or

$n=m$.

How can I prove that $(\mathbb Z,\leq_*)$is isomorphic to $(\mathbb N,\leq )$?

My thoughts:

I have to find a bijection $f : A_1 → A_2$. Then, I need to prove that $f$ is an isomorphism by showing that it is injective, surjective and that it preserves the order. However, I can't seem to find to find $f$. How can I do this?

Answer 1

Hint: $0 \leq_* -1 \leq_*1 \leq_* -2 \leq_* 2 \leq_*\dots$