construct a bijection from $\Bbb R\setminus \Bbb Z$ to $\Bbb R\setminus \Bbb N$

Question

How do I construct a bijection from $\Bbb R\setminus \Bbb Z$ to $\Bbb R\setminus \Bbb N$?

I understand that I have to send some elements from $\Bbb R\setminus \Bbb Z$ to $\{0,-1,-2,\ldots\}$ in $\Bbb R\setminus \Bbb N$, but I'm unable to do that. Thanks in advance.

Are you sure you need to construct a bijection explicitly, and not just prove that one exists? The Cantor-Bernstein theorem will do the latter for you easily. — hmakholm left over Monica, Aug 07 '17 at 14:46

score 1 · Answer 1 · answered Aug 07 '17 at 14:53

Hint: Consider separately each of the open intervals $(a,a+1) \subset \mathbb{R} \setminus \mathbb{Z}$ where $a \in \mathbb{Z}$. Label each such interval by its left endpoint $a$. Now can you construct a bijection from this set to $\mathbb{N}$? Convert them back into intervals using the same definition - you should argue that the result is still a bijection but it shouldn't be too difficult.

score 1 · Answer 2 · answered Aug 07 '17 at 14:55

One idea would be: Define a bijection $(0,1)\to(0,1]$ and apply that to each of the unit intervals to the left of $0$ while keeping everything to the right of $0$ unchanged.

Another idea would be: Leave everything except the negative half-integers unchanged and map $-n-\frac12$ to $-\frac12n$.

The possibilities are endless.

construct a bijection from $\Bbb R\setminus \Bbb Z$ to $\Bbb R\setminus \Bbb N$

2 Answers2