Question

Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.

Answer 1

Formula:

When two sets A and B have m and n elements respectively, then the number of onto functions from A to B is
$\left\{\begin{array}{l}{\sum }_{r=1}^n{\left(-1\right)}^{r}n{C}_r{r}^m,\text{if}m\ge n\\ o,\text{if}m<n\end{array}\right.$

Here, number of elements in A = 4 = m
Number of elements in B = 2 = n
So, m > n
Number of onto functions
$$\begin{gathered} ={\sum }_{r=1}^2{\left(-1\right)}^{r}2C_r{r}^4 \\ ={\left(-1\right)}^{1}2C_1{1}^4+{\left(-1\right)}^{2}2C_2{2}^4 \\ =-2+16 \\ =14 \end{gathered}$$