Question

If $n\ge 2$ then the number of surjections that can be defined from $\{1,2,3,....n\}$ onto $\{1,2\}$ is

• A. $n^2-n$
• B. $n^2$
• C. $2^n$
• D. $2^n-2$

Answer 1

The correct option is D $2^n-2$

$1$ can give output in $2$ ways either $1$ or $2$
$2$ can give output in $2$ ways either $1$ or $2$, and so on till n
hence we can say that $1,2,\mathrm{3...},n$ can give output in $2^n$ ways
but in this $2$ functions will be such that all input $1,2,3,...,n$ will give output only $1$ or only $2.$
Therefore, number of surjective functions is $2^n-2$
Hence, option 'D' is correct.