Question

Write the formula to find the number of onto functions from set $A$ to set $B$.

Answer 1

Formula to find the number of onto functions from set $A$ to set $B$.

Onto function

Let the number of elements in set $A$ be $n$.

Let the number of elements in set $B$ be $m$.

Total number of functions from $A$ to $B$ are,

$m^n$

The formula to determine the number of functions that is onto is given by:

$m^n-\left({}_1^m\right){\left(m-1\right)}^n+\left({}_2^m\right){\left(m-2\right)}^n-\left({}_3^m\right){\left(m-3\right)}^n+.....-\left({}_{m-1}^m\right){\left(1\right)}^n$

When, $n<m$, the number of onto functions is zero.

When, $n=m$, the number of onto functions is $m!$.

Therefore, the number of onto function from $A$ to $B$ can be find with the formula,

$m^n-\left({}_1^m\right){\left(m-1\right)}^n+\left({}_2^m\right){\left(m-2\right)}^n-\left({}_3^m\right){\left(m-3\right)}^n+.....-\left({}_{m-1}^m\right){\left(1\right)}^n$