# Question 15482: Representative Invariance Does Not Define Quotient Operations [Return to catalogue page 78](https://ocme.evemisslab.com/en/questions/page/78/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261030-r04-quotient-set-foundations/#q-20261014-advance-20261030-r04-quotient-set-0082) 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. R0029-15482 · q-20261014-advance-20261030-r04-quotient-set-0082 ## Original question and givens Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim). - G001 — Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim). ## 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-quotient-set](https://ocme.evemisslab.com/en/mko/mko-quotient-set/) ## Step-by-step derivation S001 · impliesBefore complete carrier Fin(6);original relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]] After set of pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]];reflexive/symmetric/transitive=[Yes, Yes, Yes] Conditions: none Read pairs with set semantics;duplicate entries do not change the relation.Check reflexivity,symmetry and transitivity before forming the quotient.Connected components of a non-equivalence relation are not its quotient classes. Step identifiers and dependenciesRule: quotient_complete_equivalence_not_connectedness_only · Uses: G001 S002 · impliesBefore least uncovered carrier label=0;all labels related from it=[0, 2] After next canonical full class=[0, 2];covered labels=[0, 2] Conditions: none List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label. Step identifiers and dependenciesRule: quotient_full_classes_in_least_representative_order · Uses: G001, S001 S003 · impliesBefore least uncovered carrier label=1;all labels related from it=[1, 3, 4, 5] After next canonical full class=[1, 3, 4, 5];covered labels=[0, 1, 2, 3, 4, 5] Conditions: none List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label. Step identifiers and dependenciesRule: quotient_full_classes_in_least_representative_order · Uses: G001, S002 S004 · impliesBefore all complete canonical classes=[[0, 2], [1, 3, 4, 5]];original carrier=[0, 1, 2, 3, 4, 5] After cover whole carrier=Yes;pairwise disjoint=Yes;class count=2 Conditions: none Equivalence classes cover the original carrier and are pairwise disjoint,so each element lies in exactly one class.The empty family on an empty carrier also satisfies empty coverage and disjointness;class counts alone are not the partition test. Step identifiers and dependenciesRule: quotient_partition_cover_and_disjoint_not_count_only · Uses: G001, S003 S005 · impliesBefore original representatives a=0;b=1;related=No;full classes=[[0, 2], [1, 3, 4, 5]] After related implies same class=Yes;no_quotient_operation_claim Conditions: none If a~b then their classes agree.When unrelated,the original implication is true without asserting class equality.This is representative independence of the class,not well-definedness of an unspecified quotient operation. Step identifiers and dependenciesRule: quotient_representative_implication_no_algebra_operation · Uses: G001, S004 ## Reverse checks - 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 (0, 1, No, [0, 2], [1, 3, 4, 5], Yes, no_quotient_operation_claim) 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: 4456759aec5099de201210168011c52a70c594e963afee67241b9a2361859013 - candidate record SHA-256: 988b211e6204fffddbc2553874ed3ce41fad9c362a150d696b99cef0314dca8a - semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725 - question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r04-quotient-set-0082.json)