# Equivalence relation

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

A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity.

mko-equivalence-relation · definition · v0.10.0



## Mathematical statement

A relation R on X is an equivalence relation when every x satisfies xRx, xRy implies yRx, and xRy together with yRz implies xRz. This slice checks each obligation on a complete finite carrier and complete set of relation pairs.





## Representative formula (exact TeX)

E(R)





## Assumptions


- The carrier is labeled 0 through n−1 and the relation is fully listed as distinct ordered pairs.
- Equivalence classes are emitted only after all three checks pass; one counterexample suffices to refute its corresponding axiom.





## Symbol scope


- E — equivalence-relation record
- R — binary relation





## Intuition, definition, and misconceptions

Reflexivity checks every diagonal pair.

Symmetry checks the reverse of every relation pair.

Transitivity checks every composable pair of relation pairs.

A grouped-looking relation diagram, or reflexivity and symmetry alone, does not establish equivalence.





## Adopted derivation guide

[Equivalence relation: checking the three axioms on a complete finite relation and emitting the classes](https://ocme.evemisslab.com/en/derivations/derivation-mko-equivalence-relation-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Binary relations and set projections](https://ocme.evemisslab.com/en/mko/mko-relation/) — An equivalence relation is a relation on a fixed carrier.
- [Logical connectives](https://ocme.evemisslab.com/en/mko/mko-logical-connectives/) — The axioms use implication and conjunction.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — The three axioms quantify over all carrier elements.





## Verification boundary

Equivalence classes are derived only after full qualification.

not_formalized

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





## Next obligation

Add equivalence classes, quotient sets and cardinality comparisons.





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