Show that the set of odd integers has the same cardinality as $\{2^n\mid n\in\mathbb N\}$ How do you show that the set of odd integers $(2k + 1)$ has the same cardinality as the set of positive powers of $2$ $(2^n)?$

This is a bijection from the integers $\mathbb{Z}$ to the odd integers:
$$
f(k) = 2k + 1.
$$
This is a bijection from the nonnegative integers, $\mathbb{N}$, to the positive powers of two:
$$
g(n) = 2^n.
$$
So, you have to show these two functions, $f$ and $g$, are bijections.
Finally, you may already know that $\mathbb{Z}$ and $\mathbb{N}$ have the same cardinality, so that means all four sets have the same cardinality.
If you don't already know $\mathbb{Z}$ and $\mathbb{N}$ have the same cardinality, you could try to make a bijection between those, too.
