Question

Find the number of all onto functions from the set $\left\{1,2,3,....,n\right\}$ to itself.

Answer 1

Step 1: Compute the total number of one-one functions in the set $\left\{1,2,3\right\}$.

As $f$ is onto, every element of $\{1,2,3\}$ will have a unique pre-image.

Element	Number of possible pairings
$1$	$3$
$2$	$2$
$3$	$1$

Total number of one-one function

$$\begin{gathered} =3\times 2\times 1 \\ =6 \end{gathered}$$

Step 2: Compute the total number of onto functions in the given set.

As $f$ is onto, every element of $\{1,2,3....n\}$ will have a unique pre-image.

Element	Number of possible pairings
$1$	$n$
$2$	$n-1$
$3$	$n-2$
.	.
.	.
$n-1$	$2$
$n$	$1$

Total number of one-one function

$$\begin{gathered} =n\times (n–1)\times (n–2)\times \dots \dots .\times 2\times 1 \\ =n! \end{gathered}$$

Hence, the number of all onto functions from the set $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}\mathrm{,}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{,}\mathrm{n}\mathrm{\}}$ to itself is $n!$.