# Order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$

• How can one prove the existence of an order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$?

• Can you give an example of such a bijection?

• What is $\mathbb{Q}^*$? Nonzero rationals? – Batominovski Feb 16 '16 at 21:38
• Yes. Is $\mathbb{Q}\backslash\lbrace{0}\rbrace$ better? – Albert Feb 16 '16 at 21:43
• @Eliott: Yes, much. – hmakholm left over Monica Feb 16 '16 at 21:43
• The quickest way to an existence proof is of course to know that all countable dense linear orders without first or last elements are isomorphic... – hmakholm left over Monica Feb 16 '16 at 21:46
• You really ought to say something about what you've tried, but have a look at en.wikipedia.org/wiki/Dense_order. – Rob Arthan Feb 16 '16 at 21:46

See the proof of theorem 3.7 here: http://www.math.wustl.edu/~freiwald/ch8.pdf

Choose an irrational number $\alpha$.

Let $x_1, x_2, \ldots$ be a strictly increasing sequence of rational numbers that converge towards $\alpha$.

Let $y_1, y_2, \ldots$ be a strictly decreasing sequence of rational numbers that converge towards $\alpha$.

Then define $f:\mathbb Q\to\mathbb Q\setminus\{0\}$ as:

• $f$ maps $(-\infty,x_1]$ to $(-\infty,-1]$ by subtracting $x_1+1$ from everything.
• For every $n$, $f$ maps $[x_n,x_{n+1}]$ to $[-\frac1n,-\frac1{n+1}]$, by linear interpolation between the endpoints.
• For every $n$, $f$ maps $[y_{n+1},y_n]$ to $[\frac1{n+1},\frac1n]$, by linear interpolation between the endpoints.
• $f$ maps $[y_1,\infty)$ to $[1,\infty)$ by subtracting $y_1-1$ from everything.