Rational Numbers and Equivalence Classes

Question

Describe the rational numbers as the equivalence classes for an equivalence relation on certain pairs of integers.

Answer 1

Two rationals $a/b$ and $c/d$ are the same if and only if $ad=bc$. This is a hint to consider the equivalence relation $(a,b)\sim (c,d)$ if and only if $ad=bc$. There is a 1-1 correspondence between the set of equivalence classes of pairs of integers under this relation and the rational numbers. The bijection is given by $\overline{(a,b)}\mapsto \frac{a}{b}$. You have to verify that this map is well defined and that it is in fact a bijection.

EDIT: You have to consider only pairs $(a,b)$ with $b\neq 0$ because if you do not have this restriction the relation is not transitive (for example $(1,1)\sim (0,0)$ and $(0,0)\sim (1,2)$ but $(1,1)$ and $(1,2)$ are not related).