Question 14240: Relation Membership and Class Membership Must Agree

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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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R0027-14240 · q-20261014-advance-20261029-r07-equivalence-class-0040

Original question and givens

Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2, candidate=1. Give (element,candidate,related?,in_class?).

G001 — Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2, candidate=1. Give (element,candidate,related?,in_class?).

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-class

Step-by-step derivation

S001 · implies

Before

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

After

sorted distinct ordered relation=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]];missing reflexive=[];missing reverses=[];missing transitive triples=[]

Conditions: none

Qualify reflexivity,symmetry and every composable pair's transitivity before forming classes.Do not silently deduplicate invalid inputs or infer equivalence from samples or connected clustering.

Step identifiers and dependencies

Rule: equivalence_relation_complete_parent_qualification · Uses: G001

S002 · implies

Before

first class:representative=2;carrier candidate=0

After

ordered relation pair present=Yes;class membership=Yes

Conditions: none

By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.

Step identifiers and dependencies

Rule: equivalence_class_every_carrier_element · Uses: G001, S001

S003 · implies

Before

first class:representative=2;carrier candidate=1

After

ordered relation pair present=Yes;class membership=Yes

Conditions: none

By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.

Step identifiers and dependencies

Rule: equivalence_class_every_carrier_element · Uses: G001, S002

S004 · implies

Before

first class:representative=2;carrier candidate=2

After

ordered relation pair present=Yes;class membership=Yes

Conditions: none

By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.

Step identifiers and dependencies

Rule: equivalence_class_every_carrier_element · Uses: G001, S003

S005 · implies

Before

first class:representative=2;carrier candidate=3

After

ordered relation pair present=No;class membership=No

Conditions: none

By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.

Step identifiers and dependencies

Rule: equivalence_class_every_carrier_element · Uses: G001, S004

S006 · implies

Before

first class:representative=2;carrier candidate=4

After

ordered relation pair present=Yes;class membership=Yes

Conditions: none

By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.

Step identifiers and dependencies

Rule: equivalence_class_every_carrier_element · Uses: G001, S005

S007 · implies

Before

first class:representative=2;carrier candidate=5

After

ordered relation pair present=Yes;class membership=Yes

Conditions: none

By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself.

Step identifiers and dependencies

Rule: equivalence_class_every_carrier_element · Uses: G001, S006

S008 · implies

Before

first class:all carrier labels tested with representative=2

After

complete class in increasing order=[0, 1, 2, 4, 5]

Conditions: none

The entire carrier has been inspected;this list is the representative's full equivalence class,not a local sample.

Step identifiers and dependencies

Rule: class_complete_inventory_not_sample · Uses: G001, S007

S009 · implies

Before

representative=2;candidate=1;whole class=[0, 1, 2, 4, 5]

After

related=Yes;in class=Yes

Conditions: none

Compare the original ordered-pair lookup with membership in the complete class;they agree by the class definition.

Step identifiers and dependencies

Rule: relation_lookup_equals_class_membership · Uses: G001, S008

Reverse checks

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

(2, 1, Yes, Yes)

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: 4bb4860ccdd9c4e32568c53873f7083b32dd28b8adcb9cf6db6c5d4ba83567a0
candidate record SHA-256: 5e168b14a20803e8d6618c9ae4c2850f3b16bc1bedb1cde63167d3678500a421
semantic review receipt SHA-256: e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe
question-first witness SHA-256: a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da
Lean: NOT_RUN · human review: NOT_PERFORMED

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