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Q.
Let $A = \{1$, $2$, $3$, .... $n]$ and $B = \{a$, $b\}$. Then the number of surjections from $A$ into $B$ is
Relations and Functions - Part 2
Solution:
If $f$ : $A \to B$ is a function, the $f(1)$ can be chosen in two ways, $f(2)$ can be chosen in two ways, ..., $f(n)$ can be chosen in two ways.
Hence, $f$ can be chosen in $2 \times 2 \times ... \times 2 = 2^n$ ways
In total there are $2^n$ functions possible. Out of these two functions $f_1$ and $f_2$, defined as $f_1(i) = a\,\forall\,i = 1$, $2$, $ ... $, $n$ and $f_2(i) = b\, \forall \, i = 1$, $2$, $...$, $n$ are not surjective as range of $f_1$ is $\{a\} \ne B$ and $f_2$ is $\{b\} \ne B$.
Hence, the number of surjections from $A$ to $B$ is $2^n - 2$.