How to prove that $\mathbb{Z} \sim E^*$ where $E^*$ is the set of positive even integers

Question

Prove that $\mathbb{Z} \sim E^*$ where $E^*$ is the set of positive even integers. Intuitevely I know this to be true, but cannot construct a proof.

Could someone show a proof?

Answer 1

In interpret your $\sim$ as "there is a bijection between them".

Instead of giving a bijection directly, as both set are countable, I think it is much more intuitive to consider the following two bijections (I take $0\in\mathbb N$ to be given):

$$f:\mathbb N\to E^*, f(n)=2(n+1)$$

and

$$g:\mathbb N\to\mathbb Z, f(n)=\begin{cases}(n-1)/2&n\text{ odd}\\-n/2&n\text{ even}\end{cases}$$

You can verify for both cases easily (by induction) that they are bijections. We then just take $g^{-1}\circ f$ as the desired bijection between $\mathbb Z$ and $E^*$.