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Question 12310: Emit Classes Only after All Three Axioms Qualify

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0023-12310 · q-20261013-advance-20261026-r07-equivalence-relation-0085

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, 2], [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, 2], [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.

Step-by-step derivation

S001 · implies

Before

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

After

lexicographically sorted distinct ordered pairs=[[0, 0], [0, 2], [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 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=[0, 2];required reverse=[2, 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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: relation_original_pair_missing_reverse_record · Uses: G001, S009

S011 · 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, S010

S012 · 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, S011

S013 · 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, S012

S014 · 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, S013

S015 · 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, S014

S016 · implies

Before

first original pair=[0, 0];all composable second endpoints=[0, 2, 4]

After

required transitivity rows [a,b,d,aRd]=[[0, 0, 0, Yes], [0, 0, 2, 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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S015

S017 · implies

Before

first original pair=[0, 2];all composable second endpoints=[2]

After

required transitivity rows [a,b,d,aRd]=[[0, 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, S016

S018 · implies

Before

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 dependencies

Rule: relation_every_composable_original_pair · Uses: G001, S017

S019 · 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, S018

S020 · 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, S019

S021 · 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, S020

S022 · 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, S021

S023 · 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, S022

S024 · implies

Before

missing reflexive=[];missing reverses=[[0, 2], [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 dependencies

Rule: 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, 2], [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: 994ae2f61a36f00ff423a1d4da7e81b62869773585c76ff7df6d395d338829f9
  • candidate record SHA-256: adeceb2a97f9b9efeff4a353ec959052a93d3adae3f32f05cfeaaf0571bc13d9
  • semantic review receipt SHA-256: f783e81f44ff0faa13dbf948657e9187244a59a809ebdf96f22cca24966649ff
  • question-first witness SHA-256: 49a50ed0550be55124d6562260c2ccf4fa6d3f316d7ad9d0d32174400e362e18
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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