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

Question:

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

Solution:

Formula:

When two sets A and have m and elements respectively, then the number of onto functions from A to B is

$\left\{\begin{array}{l}\sum_{r=1}^{n}(-1)^{r} n C_{r} r^{m}, \text { if } m \geq n \\ o, \text { if } m

Here, number of elements in A = 4 = m
Number of elements in B = 2 = n

So, $m>\mathrm{n}$

Number of onto functions

$=\sum_{r=1}^{2}(-1)^{r} 2 C_{r} r^{4}$

$=(-1)^{1} 2 C_{1} 1^{4}+(-1)^{2} 2 C_{2} 2^{4}$

$=-2+16$

$=14$

 

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