# Quotient set

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The distinct equivalence classes of a complete equivalence relation form its quotient set and partition the original carrier.

mko-quotient-set · definition · v0.10.0



## Mathematical statement

X/R is the set of all distinct equivalence classes, not a set of selected representatives. The classes cover X and are pairwise disjoint. The canonical projection sends each element to its class. Any quotient operation still needs a separate proof of independence from representative choice.





## Representative formula (exact TeX)

Q(X)





## Assumptions


- The finite carrier and certified equivalence relation are complete.
- Classes are normalized as sorted element arrays and deduplicated.





## Symbol scope


- Q — quotient set
- R — equivalence relation





## Intuition, definition, and misconceptions

List all distinct full classes.

The canonical projection value is a class, not a representative.

Coverage and pairwise disjointness give a partition; quotient operations separately require well-definedness.

A list of representatives is not the quotient set, and operations cannot be inherited without checking representative independence.





## Adopted derivation guide

[The quotient of a four-element carrier by parity](https://ocme.evemisslab.com/en/derivations/derivation-mko-quotient-set-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Equivalence class](https://ocme.evemisslab.com/en/mko/mko-equivalence-class/) — Quotient elements are distinct complete classes.
- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — The quotient itself is a set.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The canonical projection is a total map.
- [Equivalence relation](https://ocme.evemisslab.com/en/mko/mko-equivalence-relation/) — All classes come from the same relation.





## Verification boundary

Only finite quotient sets and their projections are constructed; no group or ring operations are automatically assigned.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add quotient rings of finite rings with representative-independent operations.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/quotient-set.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-quotient-set/index.md)