Bijection from $\mathbb R$ to $\mathbb R - \mathbb N$ Could someone provide an example of a bijection from $\mathbb R$ to $\mathbb R - \mathbb N$? 
 A: Define $f: \mathbb R \rightarrow \mathbb R- \mathbb N$ as follows:
For any non-integer $x$, $f(x) = x$.
For any negative integer $x$, $f(x) = 2x$.
For any non-negative integer $x$, $f(x) = -2x-1$.
Loosely speaking, this will thrust the negative integers into the even negative integers, the non-negative integers into the odd negative integers, and the rest of $\mathbb R$ into itself.
A: Hint
Consider $\mathbb{R} = \left( \mathbb{R}-\mathbb{Z} \right) \cup \color{black}{\mathbb{Z}}$ and $\mathbb{R}-\mathbb{N} = \left( \color{black}{\mathbb{R}-\mathbb{Z}} \right) \cup \color{black}{\mathbb{Z^-}}$, so
$$\mathbb{R} \to \mathbb{R}-\mathbb{N}$$
becomes
$$\left( \color{green}{\mathbb{R}-\mathbb{Z}} \right) \cup \color{blue}{\mathbb{Z}} \to \left( \color{green}{\mathbb{R}-\mathbb{Z}} \right) \cup \color{red}{\mathbb{Z^-}}$$
Now map all non-integers to itself and send the elements of $\color{blue}{\mathbb{Z}}$ to $\color{red}{\mathbb{Z^-}}$.
Can you find a bijection for this last part? If not, hoover over:

 For example: send the positive integers to the even negative integers and the negative integers to the odd negative integers.

