Question

Let $A=\{1,2,3,\cdots ,n\}$ and $B=\{a,b\}$. Then, the number of surjections from $A$ into $B$ is

• A. ${}^n{P}_2$
• B. $2^n-2$
• C. $2^n-1$
• D. None of the above

Answer 1

Answer: (B)

Here, $A=\{1,2,3,\cdots ,n\}$ and $B=\{a,b\}$
Since, every element of domain $A$ has two choices i.e., a or $b$.
$\therefore$ Number of functions will be $2^n$.
Also, there will be a case when all the elements of $A$ will map to $a$ only then, since $b$ is left which do not have pre image in $A$.
$\therefore$ In such a case, the function from $A$ to $B$ is not onto.
Similarly, when all elements of $A$ will map to bonly, then also the function from $A$ to $B$ is not onto.
$\therefore$ Total number of onto functions is $2^n-2$.