Proving set equality Z = {2a-3b, a,b $\in$ N}

Question

I'm trying to prove that the above sets are equal. The integers can be mapped using the natural numbers and 2a-3b. My thought process is to use induction to prove that you can get positive numbers, all negative numbers, and then 0, but I'm having trouble with the inductive step.

Answer 1

If $m<0$, set $a=b=-m$. If $m>0$, set $a=2m,b=m$. For $m=0$, set $a=3,b=2$.