How can the graph of an equivalence relation be conceptualized? Consider a generic equivalence relation $R$ on a set $S$. By definition, if we partition $S$ using the relation $R$ into $\pi_S$, whose members are the congruence classes $c_1, c_2...$ then
$aRb \text{ iff a and b are members of the same congruence class in } \pi_S$.
But what is the domain and codomain of $R$? Is it $S \rightarrow S $ or $S^2 \rightarrow \text{{true,false}}$?
The reason I ask is to have an idea of the members of $graph(R)$. Does it only contain ordered pairs which are equivalent, i.e. $(a,b)$; or all elements of $S \times S$ followed by whether they are equivalent, i.e. $((a,b), true)$?
Moreover, what would the image of $s \in S$ under $R$ look like? If one suggests that $R(s)$ would return a set of all the equivalent members, then the former definition is fitting.
I suppose the source of confusion is that we rarely think of equivalence relations as 'mapping' from an input to an output, instead it tells us if two objects are similar in some way.

The domain is $S\times S$ and codomain is {true, false}.  Said another way, you can just think of any relation as a subset of $S\times S$.
