Question

The number of surjective functions from a set $A=\{1,2,3,4\}$ onto a set $B=\{1,2\}$ is

• A. $12$
• B. $14$
• C. $16$
• D. $8$

Answer 1

The correct option is B $14$

Total number of functions from $A$ to $B=2^4$
The constant functions $f\left(x\right)=1$ and $f\left(x\right)=2$ are the only functions which are not surjective.
$\therefore$ Number of surjective functions $=2^4-2=14$

Alternatively,
If $X$ has $m$ elements and $Y$ has $n$ elements, then the number of surjective functions $f:X\to Y$is given by
$\sum _{k=0}^n(-1{)}^k{}^n{C}_k(n-k{)}^m$
Here, $m=4,n=2$
$\therefore$ Number of surjective functions
$=16-2+0=14$