This is, I am sure, an easy question but something I did not see in my undergrad years and wikipedia does not define it. For a function or a relation or a property to be invariant under an equivalence relation, I have the definition. For a set, I do not. There is no function implied from the set to itself, so the notion of an invariant set under a function does not make sense to me in this context.
Both ideas that I have for what this could mean are contradictory. The first is that if x~y and x is in the set, so is y; that is, the set treats objects that are equivalent in the same way. The second is that the set remains the same object in the category of sets after imposing the equivalence relation. That is, if x~y, then exactly one of x or y can be in the set.
I am reading from Marker's "Introduction to Model Theory," defn of interpretable structure: "For each symbol of L_0 we can find definable ~-invariant sets on X"