Question 12301: Emit Classes Only after All Three Axioms Qualify

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0023-12301 · q-20261013-advance-20261026-r07-equivalence-relation-0076

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]]. 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]]. 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

Step-by-step derivation

S001 · implies

Before

raw relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]];carrier Fin(6)

After

lexicographically sorted distinct ordered pairs=[[0, 0], [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 dependencies

Rule: complete_relation_no_duplicate_or_implicit_repair · Uses: G001

S002 · implies

Before

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 dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S003

S005 · implies

Before

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 dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S005

S007 · implies

Before

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 dependencies

Rule: relation_every_carrier_diagonal · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S011

S013 · implies

Before

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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S012

S014 · implies

Before

first original pair=[0, 0];all composable second endpoints=[0]

After

required transitivity rows [a,b,d,aRd]=[[0, 0, 0, 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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S013

S015 · implies

Before

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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S014

S016 · implies

Before

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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S015

S017 · implies

Before

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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S016

S018 · implies

Before

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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S017

S019 · implies

Before

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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S018

S020 · implies

Before

least unseen representative=0;complete relation=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]

After

class=[0];seen=[0]

Conditions: none

Only after every axiom passes,derive classes from the original relation.Choose the least unseen representative and increasing class labels.The axioms ensure disjoint-or-equal classes covering the carrier.

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S019

S021 · implies

Before

least unseen representative=1;complete relation=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]

After

class=[1];seen=[0, 1]

Conditions: none

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S020

S022 · implies

Before

least unseen representative=2;complete relation=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]

After

class=[2];seen=[0, 1, 2]

Conditions: none

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S021

S023 · implies

Before

least unseen representative=3;complete relation=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]

After

class=[3];seen=[0, 1, 2, 3]

Conditions: none

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S022

S024 · implies

Before

least unseen representative=4;complete relation=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]

After

class=[4];seen=[0, 1, 2, 3, 4]

Conditions: none

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S023

S025 · implies

Before

least unseen representative=5;complete relation=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]

After

class=[5];seen=[0, 1, 2, 3, 4, 5]

Conditions: none

Step identifiers and dependencies

Rule: equivalence_class_after_all_axioms · Uses: G001, S024

S026 · implies

Before

missing reflexive=[];missing reverses=[];missing transitive triples=[]

After

equivalence=Yes;classes=[[0], [1], [2], [3], [4], [5]]

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 dependencies

Rule: equivalence_three_obligations_no_silent_closure · Uses: G001, S025

Reverse checks

S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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, [], [], [], Yes, [[0], [1], [2], [3], [4], [5]])

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: a4524c5aa04433ddba532510906afdd247d320aa25361411a4065a228cd56ca7
candidate record SHA-256: 6c1f44a10bcb1f14053019687b0810ee97b5324edaab44957b892cdb54545af1
semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
Lean: NOT_RUN · human review: NOT_PERFORMED

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