How to define an injective and surjective function from $\mathbb{Z}$ to $\mathbb{N}$?

Question

I want to show that $|\mathbb{Z}|=|\mathbb{N}|$. FWIW, I think again that I must define a injective and surjective function from $\mathbb{Z}$ to $\mathbb{N}$. But how? Is there any proof as to how could one define such functions and based on what information?

Answer 1

Since you are to show that $\mathbb{N}$ and $\mathbb{Z}$ have the same cardinality, you're correct: you need to find a bijection (hence both injective and surjective) between $\mathbb Z $ and $\mathbb N$

One bijection between $\mathbb{Z}$ and $\mathbb{N}$ is the function $f: \mathbb{Z} \to \mathbb{N}$, defined by:

$$f(k) = \begin{cases} \\ \\ 2k & \quad k \in \mathbb Z,\; k>0 \\ \\ -2k + 1 & \quad k\in \mathbb Z, \; k \leq 0 \\ \\ \end{cases} $$

In words, you are simply mapping positive integers to positive even integers, and non-positive integers to positive odd integers.

Answer 2

You can do something like this, $f\colon\mathbb{Z} \to \mathbb{N}$. By $f(0) = 0,\; f(1) = 1, \;f(-1) = 2,\; f(2) = 3,\; f(-2) = 4,\;$ etc. This gives a bijection from $\mathbb{Z}$ to $\mathbb{N}$.

I leave it as an exercise for the reader to give an explicit formula for the function $f$.