Question

Let $A$be a set with $n$ elements. The number of onto functions from $A$to $A$ is?

• A. $n^n$
• B. $n^n-n!$
• C. $\frac{n^n}{n!}$
• D. $n!$

Answer 1

The correct option is D

$n!$

Explanation for correct answer:

Given the set $A$has $n$ elements.

$A=\left\{a_1,a_2,...,a_n\right\}$.

Functions from $A$ to$A$, $f\left(a_m\right)=a_n$, where $m$ and $n$ranges from $1$ to $n$

For $m=1$ it has $n$possibilities. likewise $m$ ranges from $1$ to $n$.

Therefore, the total number of distinct functions from $A$ to $A$, is $n^n$

To find the number of onto functions:

If $A$ and $B$ are the two sets, if for every element of $B$, there is at least one or more element matching with set $A$ , it is called the onto function.

In the case of the onto function, each element of the image has a pre-image on the first set. So for $a$ we have $n$ possible functions, for $b$ we have $(n-1)$ functions and so on.

Therefore, the number of onto functions from $A$ to $A$ is $n\times n-1\times ...\times 1=n!$

Hence, option (D) is the correct answer