TITLE: Is there a bijective function mapping Natural numbers to Natural numbers, other than $f(n) = n$?
QUESTION [5 upvotes]: Is there a function that can be bijective, with the set of natural numbers as domain and range, other than $f(n) = n$?

REPLY [9 votes]: There are uncountably many of such maps.
In fact, let $A$ be any subset of $\Bbb N=\{1,2,3,\ldots\}$ such that both $A$ and $\Bbb N\setminus A$ are infinite (for example, $A$ could be the set of primes or the set of perfect squares).
Then we can define $a(n):=$ $n$th smallest element of $A$, $b(n):=$ $n$th smallest element of $\Bbb N\setminus A$, and 
$$ f(n)=\begin{cases}a(\tfrac n2)&\text{if $n$ is even}\\b(\tfrac{n+1}2)&\text{if $n$ odd}\end{cases}$$
Different $A$ will give different $f$, hence there are at leadt as many $f$ as there are $A$ - and that's continuum-many.

REPLY [7 votes]: Sure. Consider the function $f: \Bbb N \to \Bbb N$ defined by
$$
f(n) = \cases{
n - 1 & if $n$ is even \\
n + 1 & if $n$ is odd.}
$$
This function is a non-identity bijection. As Hagen von Eitzen notes, depending on your definition of $\Bbb N$, swap $+$ and $-$ in the definition of $f$ if $\Bbb N = \{0, 1, 2, 3, \ldots\}$.