MATH.2:4.2 - Make the proposed equality into an equivalence relation

Write a~b for the proposed identification. An equivalence relation is reflexive, symmetric and transitive. Its class [a] contains all objects identified with a; these classes partition A.

If the candidate is a resemblance, test the missing law before forming classes. For instance, on integers a~b defined by |a-b|≤1 is reflexive and symmetric but fails transitivity: 0 is related to 1 and 1 to 2, while 0 is not related to 2.

You can refine a proposed relation by retaining an additional property. Requiring both a~b and h(a)=h(b) remains an equivalence relation when ~ was one. Choose h from the failure that matters to the operation or query. The repaired relation still needs the operation test.