Example of a bijection from the set of real numbers to a subset of irrationals I need an example of a bijection from the set of real numbers to a subset of the irrationals.
I tried something like
$f(x)=x+\sqrt{2}$,
but where should I map $-\sqrt{2}$?

Enumerate $\Bbb Q$ as $q_n$ for $n\in\Bbb N$. Now define,
$$f(x)=\begin{cases}e^{2n+1} & x=q_n\\e^{2n} & x=e^n, n\in\Bbb N\\x&\text{otherwise}\end{cases}$$
Key point here is that $e$ is transcendental, so $e^n\notin\Bbb Q$ for any $n>0$. You can effectively replace $e$ by any other number with this property.  For example $\pi,\sqrt2^\sqrt2$, Liouville number, and so on and so forth.
