MATH.2:2 - Problem

An equivalence relation makes classes, but by itself it does not determine usable operations on those classes. An operation defined by selecting a representative is ambiguous if a different representative changes the returned class.

There is a second loss to check: even a mathematically sound quotient may discard the quantity or distinction the receiver wants. The task is to construct an identification that preserves both the selected operations and the intended class-level question, or to expose why the proposed identification cannot do so.