How to formally define a bijection $f\colon (A/\sim)\to R$, where $R\subseteq A$ is a set of representatives?

Question

Given a set $A$ and an equivalence relation $\sim$ on $A$, how can I formally define a bijection $f\colon (A/\sim)\to R$, where $R\subseteq A$ is a set of representatives? Clearly, $[a]_\sim\mapsto a$ is not well-defined, because $a\ne b\in [a]_\sim\Longrightarrow [b]_\sim=[a]_\sim$, but $f([b]_\sim)=b\ne a$.

Answer 1

It's easier to define a bijection $g$ from $R$ to $A/{\sim}$ given by $g(a)=[a]_\sim$; then the bijection from $A/{\sim}$ to $R$ is simply $f=g^{-1}$.

If we want an explicit formula for $f$, perhaps $f([a])=[a]\cap R$ works.