Show that the set of odd numbers is countable Problem: Show that the set of odd numbers is countable
Attempt: So for this problem, I just need to find a bijection from the natural numbers to the set of odd numbers. However, I find the claim "odd numbers" a bit ambiguous because it can be odd natural numbers or odd integers. However, I think that problem is about the odd integers. Would it be sufficient to show a bijection in the following manner?
1->1
2->-1
3->3
4->-3
5->5
6->-5      
and so on.

Your answer is correct but in case you want an explicit function then the following help . Define $f:\mathbb{N}\to \{\dots,-5,-3,-1,1,3,5,\dots\}$ by
$$
f(n)=
\begin{cases}
n&, \mbox{if $n$ is odd}\\
1-n&, \mbox{if $n$ is even}
\end{cases}
$$
for all $n\in\mathbb{N}$.
