Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 3 / b / Solution

This is the pointwise order on a function space. Each of the three partial order axioms follows from the corresponding axiom in $B$.

For reflexivity, $f(x){⪯}_{B}f(x)$ for every $x$, so $f{⪯}_{\mathcal{F}}f$. For antisymmetry, if $f{⪯}_{\mathcal{F}}g$ and $g{⪯}_{\mathcal{F}}f$, then $f(x)=g(x)$ for every $x\in A$; equality of functions therefore gives $f=g$. For transitivity, if $f{⪯}_{\mathcal{F}}g{⪯}_{\mathcal{F}}h$, then $f(x){⪯}_{B}g(x){⪯}_{B}h(x)$ for every $x$, so $f{⪯}_{\mathcal{F}}h$.

Thus $(\mathcal{F},{⪯}_{\mathcal{F}})$ is a partially ordered set. The argument also covers an empty domain $A$, since then there is exactly one function $A\to B$.