# Question 12333: Every Carrier Element Needs Its Diagonal Pair [Return to catalogue page 62](https://ocme.evemisslab.com/en/questions/page/62/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261026-r07-equivalence-relation-foundations/#q-20261013-advance-20261026-r07-equivalence-relation-0008) 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. R0023-12333 · q-20261013-advance-20261026-r07-equivalence-relation-0008 ## Original question and givens Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?). - G001 — Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?). ## 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-relation](https://ocme.evemisslab.com/en/mko/mko-equivalence-relation/) ## Step-by-step derivation S001 · impliesBefore raw relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]];carrier Fin(6) After lexicographically sorted distinct ordered pairs=[[0, 0], [0, 4], [2, 2], [3, 3], [4, 4], [5, 5]] Conditions: none List distinct directed pairs lexicographically.Duplicates are invalid;do not silently add reverses,diagonals or a transitive closure. Step identifiers and dependenciesRule: complete_relation_no_duplicate_or_implicit_repair · Uses: G001 S002 · impliesBefore carrier label=0;required diagonal=[0, 0] After present=Yes Conditions: none Every carrier element needs its diagonal pair,not only labels appearing in the relation. Step identifiers and dependenciesRule: relation_every_carrier_diagonal · Uses: G001, S001 S003 · impliesBefore carrier label=1;required diagonal=[1, 1] After present=No Conditions: none Every carrier element needs its diagonal pair,not only labels appearing in the relation. Step identifiers and dependenciesRule: relation_every_carrier_diagonal · Uses: G001, S002 S004 · impliesBefore carrier label=2;required diagonal=[2, 2] After present=Yes Conditions: none Every carrier element needs its diagonal pair,not only labels appearing in the relation. Step identifiers and dependenciesRule: relation_every_carrier_diagonal · Uses: G001, S003 S005 · impliesBefore carrier label=3;required diagonal=[3, 3] After present=Yes Conditions: none Every carrier element needs its diagonal pair,not only labels appearing in the relation. Step identifiers and dependenciesRule: relation_every_carrier_diagonal · Uses: G001, S004 S006 · impliesBefore carrier label=4;required diagonal=[4, 4] After present=Yes Conditions: none Every carrier element needs its diagonal pair,not only labels appearing in the relation. Step identifiers and dependenciesRule: relation_every_carrier_diagonal · Uses: G001, S005 S007 · impliesBefore carrier label=5;required diagonal=[5, 5] After present=Yes Conditions: none Every carrier element needs its diagonal pair,not only labels appearing in the relation. Step identifiers and dependenciesRule: relation_every_carrier_diagonal · Uses: G001, S006 ## Reverse checks - 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 (6, [1], No) 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: e2d8ccead1ce4d878244256cdc168c693817be2f636635dc92276ae323c03b44 - candidate record SHA-256: baacc31f0252d8ec68b6c4061e95042ae4c1ee1ab10309a6a4288c71036ba353 - semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff - question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261026-r07-equivalence-relation-0008.json)