# Question 14270: An Equivalence Class Contains Every Related Carrier Element [Return to catalogue page 72](https://ocme.evemisslab.com/en/questions/page/72/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261029-r07-equivalence-class-foundations/#q-20261014-advance-20261029-r07-equivalence-class-0020) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0027-14270 · q-20261014-advance-20261029-r07-equivalence-class-0020 ## Original question and givens Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1. Give (element,class). - G001 — Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1. Give (element,class). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-equivalence-class](https://ocme.evemisslab.com/en/mko/mko-equivalence-class/) ## Step-by-step derivation S001 · impliesBefore complete carrier Fin(6);raw relation=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]] After sorted distinct ordered relation=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]];missing reflexive=[];missing reverses=[];missing transitive triples=[] Conditions: none Qualify reflexivity,symmetry and every composable pair's transitivity before forming classes.Do not silently deduplicate invalid inputs or infer equivalence from samples or connected clustering. Step identifiers and dependenciesRule: equivalence_relation_complete_parent_qualification · Uses: G001 S002 · impliesBefore first class:representative=1;carrier candidate=0 After ordered relation pair present=No;class membership=No Conditions: none By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself. Step identifiers and dependenciesRule: equivalence_class_every_carrier_element · Uses: G001, S001 S003 · impliesBefore first class:representative=1;carrier candidate=1 After ordered relation pair present=Yes;class membership=Yes Conditions: none By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself. Step identifiers and dependenciesRule: equivalence_class_every_carrier_element · Uses: G001, S002 S004 · impliesBefore first class:representative=1;carrier candidate=2 After ordered relation pair present=No;class membership=No Conditions: none By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself. Step identifiers and dependenciesRule: equivalence_class_every_carrier_element · Uses: G001, S003 S005 · impliesBefore first class:representative=1;carrier candidate=3 After ordered relation pair present=Yes;class membership=Yes Conditions: none By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself. Step identifiers and dependenciesRule: equivalence_class_every_carrier_element · Uses: G001, S004 S006 · impliesBefore first class:representative=1;carrier candidate=4 After ordered relation pair present=No;class membership=No Conditions: none By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself. Step identifiers and dependenciesRule: equivalence_class_every_carrier_element · Uses: G001, S005 S007 · impliesBefore first class:representative=1;carrier candidate=5 After ordered relation pair present=No;class membership=No Conditions: none By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself. Step identifiers and dependenciesRule: equivalence_class_every_carrier_element · Uses: G001, S006 S008 · impliesBefore first class:all carrier labels tested with representative=1 After complete class in increasing order=[1, 3] Conditions: none The entire carrier has been inspected;this list is the representative's full equivalence class,not a local sample. Step identifiers and dependenciesRule: class_complete_inventory_not_sample · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (1, [1, 3]) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 9810d8583677225544d3b9841e61a91c3ef3930146e013b5ae8f3093f4da6f9a - candidate record SHA-256: 36077ef28bc2a301d8d2c47bce2ea49da4e6067b83304fb2c4204e2bf971fbaa - semantic review receipt SHA-256: e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe - question-first witness SHA-256: a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r07-equivalence-class-0020.json)