# Equivalence class

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

For a fixed complete equivalence relation, the class of an element a consists of every carrier element related to a.

mko-equivalence-class · definition · v0.10.0



## Mathematical statement

Given a complete finite carrier X and a certified equivalence relation R, define [a]={x in X:aRx}. If aRb then [a]=[b]; any two equivalence classes are equal or disjoint. Sampled relations or relations that fail the equivalence axioms must not be used to form canonical classes.





## Representative formula (exact TeX)

K(R)





## Assumptions


- The carrier 0 through n−1 and the complete relation-pair set are explicit and first pass full reflexivity, symmetry and transitivity checks.
- The class uses that same relation and carrier, not a visual guess at clustering.





## Symbol scope


- K — equivalence class
- R — complete equivalence relation





## Intuition, definition, and misconceptions

[a] lists exactly every element x for which aRx.

Related representatives produce the same class.

Classes are equal or disjoint and cover the carrier; quotient sets are a separate node.

An uncertified relation does not justify class formation, and two unequal equivalence classes cannot retain a nonempty overlap.





## Adopted derivation guide

[Equivalence class: certify a complete finite relation first, then form classes, with a non-equivalence as contrast](https://ocme.evemisslab.com/en/derivations/derivation-mko-equivalence-class-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Equivalence relation](https://ocme.evemisslab.com/en/mko/mko-equivalence-relation/) — Classes use the same previously certified relation.
- [Set membership](https://ocme.evemisslab.com/en/mko/mko-set-membership/) — Class membership is equivalent to aRx.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — Each class is a subset of the carrier.





## Verification boundary

Only the specified complete finite equivalence relation is handled; quotient operations and their independence from representative choice are not constructed.

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 sets and quotient rings of finite rings.





## 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/equivalence-class.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-equivalence-class/index.md)