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

A preorder is a binary relation that is both reflexive and transitive: $a⪯a$ for every $a\in A$, and $a⪯b⪯c$ implies $a⪯c$.

A partial order is a preorder satisfying the antisymmetric relation axiom. In addition to reflexivity and transitivity, it satisfies $a⪯b$ and $b⪯a$ only when $a=b$.

A total order is a partial order in which every pair is comparable: for every $a,b\in A$, at least one of $a⪯b$ and $b⪯a$ holds. Both hold exactly when $a=b$.