Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 1 / e

Each of the $m$ elements of $A$ independently receives one of $n$ images in $B$. The multiplication principle therefore gives $n^m$ functions. In particular an empty set domain has exactly one function to any $B$, so here $0^0$ is interpreted as $1$; a nonempty domain has no function to an empty set codomain.

A bijection between finite sets requires equal cardinality. When $m=n$, list the elements of $A$ and choose their distinct images in $n,n-1,\dots ,1$ ways. ThusThe factorial convention $0!=1$ includes the unique bijection from the empty set to itself.

For a surjection, the nonempty fibres $f^{-1}(b)$ form a partition of a set $A$ into exactly $n$ blocks. Choose these unlabelled blocks in $S(m,n)$ ways, then assign the $n$ distinct elements of $B$ to them in $n!$ ways. ConsequentlyIt is zero when $m<n$. With $S(0,0)=1$ and $S(m,0)=0$ for $m>0$, the same formula also handles an empty set codomain.

Choose the image of a function as a $j$-element subset of $B$, then choose a surjection onto that selected image as in (iii). ThereforeThe answer is zero when $j>n$. If $j=0$, it is one for $m=0$ and zero for $m>0$, as the Stirling numbers of the second kind conventions require.