Number of bijections between infinite cardinal I'm trying to prove that if ${\kappa}$ is an infinite cardinal, then there are $2^{\kappa}$ bijective functions from ${\kappa}$ to ${\kappa}$. I would greatly appreciate any tips. Thank you.   
 A: I would do it differently.
The number of bijections is clearly not more than $\kappa^\kappa=2^\kappa$.
Let $S$ be the set of subsets of $\kappa$ that leave more than $1$ element in the complement. Clearly $S$ has the same cardinality as $2^\kappa$.
To show that there are at least $2^\kappa$ bijections we give an injection from $S$ to  the set of bijections of $\kappa$. To do this we send each $s\in S$ to a bijection $\sigma$ of $\kappa$ such that $s$ is the set of fixed points of $\sigma$.
A: We know that there are at most $\kappa^\kappa\le(2^\kappa)^\kappa=2^{\kappa^2}=2^\kappa$ bijections from $\kappa$ to $\kappa;$ we have to show that there are at least $2^\kappa$ bijections. Since $|\{0,1\}\times\kappa|=2\kappa=\kappa,$ it will suffice to exhibit $2^\kappa$ bijections from $\{0,1\}\times\kappa$ to $\{0,1\}\times\kappa,$ which we do as follows.
For each set $S\subseteq\kappa$ define
$$f(\langle i,\ \alpha\rangle)=\begin{cases}
\langle1-i,\ \alpha\rangle\ \text{ if }\ \alpha\in S,\\
\ \ \ \ \ \ \ \langle i,\ \alpha\rangle\ \text{ if }\ \alpha\notin S.\\
\end{cases}$$
