Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 1 / b / ii / Solution

For the pointwise order on real polynomials, $p(x)\le p(x)$ proves reflexivity. Two opposite inequalities imply $p(x)=q(x)$ for every real $x$; the polynomials are therefore equal, proving the antisymmetric relation axiom. Chaining the inequalities at every $x$ proves transitivity.

To disprove total order, take $p(x)=x$ and $q(x)=0$. The inequality $p⪯q$ fails at $x=1$, whereas $q⪯p$ fails at $x=-1$. The order is partial but non-total.