Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Discrete mathematics / 3 / c / iii / Solution

Every nonzero integer $a$ has $a^2>0$, proving reflexivity. Because $ab=ba$, the relation is symmetric. If $ab>0$ and $bc>0$, then $ac=(ab)(bc)/b^2>0$, proving transitivity. Thus this is an equivalence relation, whose two equivalence classes are the positive integers and the negative integers: