Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 2 / b / Solution

For positive integer pairs, reflexivity is $ab=ba$. If $ad=bc$, reversing the equality gives $cb=da$, establishing symmetry. For transitivity, suppose $(a,b)R(c,d)$ and $(c,d)R(e,f)$. Then $ad=bc$ and $cf=de$, whenceSince $d>0$, cancellation gives $af=be$. Thus $(a,b)R(e,f)$, and $R$ is an equivalence relation.

Membership in the equivalence class of $(4,6)$ means $4b=6a$, or $2b=3a$. Since $2$ and $3$ are coprime, $a=2r$ and $b=3r$ for a positive integer $r$. Conversely every such pair satisfies the equality. ThereforeThese pairs are precisely the positive numerator-denominator representations of the rational number $2/3$.