# Question 12312: Emit Classes Only after All Three Axioms Qualify [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-0087) 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. R0023-12312 · q-20261013-advance-20261026-r07-equivalence-relation-0087 ## Original question and givens Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [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,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty. - G001 — Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [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,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty. ## 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], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4]];carrier Fin(6) After lexicographically sorted distinct ordered pairs=[[0, 0], [0, 3], [0, 4], [1, 1], [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=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, 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 S008 · impliesBefore original pair=[0, 0];required reverse=[0, 0] After reverse present=Yes Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S007 S009 · impliesBefore original pair=[0, 3];required reverse=[3, 0] After reverse present=No Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S008 S010 · impliesBefore original pair=[0, 4];required reverse=[4, 0] After reverse present=No Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S009 S011 · impliesBefore original pair=[1, 1];required reverse=[1, 1] After reverse present=Yes Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S010 S012 · impliesBefore original pair=[2, 2];required reverse=[2, 2] After reverse present=Yes Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S011 S013 · impliesBefore original pair=[3, 3];required reverse=[3, 3] After reverse present=Yes Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S012 S014 · impliesBefore original pair=[4, 4];required reverse=[4, 4] After reverse present=Yes Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S013 S015 · impliesBefore original pair=[5, 5];required reverse=[5, 5] After reverse present=Yes Conditions: none Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric. Step identifiers and dependenciesRule: relation_original_pair_missing_reverse_record · Uses: G001, S014 S016 · impliesBefore first original pair=[0, 0];all composable second endpoints=[0, 3, 4] After required transitivity rows [a,b,d,aRd]=[[0, 0, 0, Yes], [0, 0, 3, Yes], [0, 0, 4, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S015 S017 · impliesBefore first original pair=[0, 3];all composable second endpoints=[3] After required transitivity rows [a,b,d,aRd]=[[0, 3, 3, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S016 S018 · impliesBefore first original pair=[0, 4];all composable second endpoints=[4] After required transitivity rows [a,b,d,aRd]=[[0, 4, 4, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S017 S019 · impliesBefore first original pair=[1, 1];all composable second endpoints=[1] After required transitivity rows [a,b,d,aRd]=[[1, 1, 1, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S018 S020 · impliesBefore first original pair=[2, 2];all composable second endpoints=[2] After required transitivity rows [a,b,d,aRd]=[[2, 2, 2, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S019 S021 · impliesBefore first original pair=[3, 3];all composable second endpoints=[3] After required transitivity rows [a,b,d,aRd]=[[3, 3, 3, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S020 S022 · impliesBefore first original pair=[4, 4];all composable second endpoints=[4] After required transitivity rows [a,b,d,aRd]=[[4, 4, 4, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S021 S023 · impliesBefore first original pair=[5, 5];all composable second endpoints=[5] After required transitivity rows [a,b,d,aRd]=[[5, 5, 5, Yes]] Conditions: none List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence. Step identifiers and dependenciesRule: relation_every_composable_original_pair · Uses: G001, S022 S024 · impliesBefore missing reflexive=[];missing reverses=[[0, 3], [0, 4]];missing transitive triples=[] After equivalence=No;classes=[] Conditions: none Emit classes only when all three obligations pass.On any failure leave classes empty,without repairing the relation by connected grouping. Step identifiers and dependenciesRule: equivalence_three_obligations_no_silent_closure · Uses: G001, S023 ## Reverse checks - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (6, [], [[0, 3], [0, 4]], [], 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: c4b2537707fc90caf4c0e07b28387381b46aa82d36944013da4a34f7bc523d84 - candidate record SHA-256: 8781ab41c35fadbc660ccb4943352f36bf2627221d796a7610ac9a8061b81fd4 - 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-0087.json)