I am familiar of the equivalence class on an element in a set (given some equivalence relation), but is there a such thing as equivalence class of a set?
I ran into this reading paper Encoding Data Structures by Rajeev Raman: "Given a set of objects S, and a set of queries Q, consider the equivalence class C on S induced by Q, where two objects from S are equivalent if they provide the same answer to all queries in Q."
This doesn't seem to be any different from how we normally talk about equivalence classes: the author defines a binary relation over a set $S$, where any two objects from the set are related 'if they provide the same answer to all queries'. This relation is obviously reflexive, symmetric, and transitive, and hence is an equivalence relation. And the equivalence class $C$ is still defined relative to any object $o$ in the set, in that it is the set of all objects from that set standing in that equivalence relation to the object $o$. So this is all still relative to elements of the set.