Okay, I asked a question earlier today, Congruence Class, pertaining to finding equivalence classes. I already know how to solve such problems, now my question is, what is the importance of having/knowing what an equivalence class is? Does it reveal any sort of information about the relation?
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Yes. In nice algebraic structures (as groups, rings, modules, Boolean algebras, etc), one congruence class determines the whole congruence relation.
Considering the quotient by an equivalence (or congruence) relation is the counterpart of division of integer numbers.