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Question 15477: Representative Invariance Does Not Define Quotient Operations

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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R0029-15477 · q-20261014-advance-20261030-r04-quotient-set-0077

Original question and givens

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

  • G001 — Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Reflexivity, symmetry and transitivity are prior complete obligations. The quotient contains all distinct full equivalence classes, not selected representatives. Sort elements within a class increasingly and enumerate classes by their least unseen carrier label. Quotient operations require a separate representative-independence proof. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

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

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

After

set of pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]];reflexive/symmetric/transitive=[Yes, Yes, Yes]

Conditions: none

Read pairs with set semantics;duplicate entries do not change the relation.Check reflexivity,symmetry and transitivity before forming the quotient.Connected components of a non-equivalence relation are not its quotient classes.

Step identifiers and dependencies

Rule: quotient_complete_equivalence_not_connectedness_only · Uses: G001

S002 · implies

Before

least uncovered carrier label=0;all labels related from it=[0]

After

next canonical full class=[0];covered labels=[0]

Conditions: none

List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label.

Step identifiers and dependencies

Rule: quotient_full_classes_in_least_representative_order · Uses: G001, S001

S003 · implies

Before

least uncovered carrier label=1;all labels related from it=[1, 2, 3, 4, 5]

After

next canonical full class=[1, 2, 3, 4, 5];covered labels=[0, 1, 2, 3, 4, 5]

Conditions: none

List every element in the class in carrier order,and order classes by the least not-yet-covered element.A quotient element is the whole class,not merely its representative label.

Step identifiers and dependencies

Rule: quotient_full_classes_in_least_representative_order · Uses: G001, S002

S004 · implies

Before

all complete canonical classes=[[0], [1, 2, 3, 4, 5]];original carrier=[0, 1, 2, 3, 4, 5]

After

cover whole carrier=Yes;pairwise disjoint=Yes;class count=2

Conditions: none

Equivalence classes cover the original carrier and are pairwise disjoint,so each element lies in exactly one class.The empty family on an empty carrier also satisfies empty coverage and disjointness;class counts alone are not the partition test.

Step identifiers and dependencies

Rule: quotient_partition_cover_and_disjoint_not_count_only · Uses: G001, S003

S005 · implies

Before

original representatives a=1;b=4;related=Yes;full classes=[[1, 2, 3, 4, 5], [1, 2, 3, 4, 5]]

After

related implies same class=Yes;no_quotient_operation_claim

Conditions: none

If a~b then their classes agree.When unrelated,the original implication is true without asserting class equality.This is representative independence of the class,not well-definedness of an unspecified quotient operation.

Step identifiers and dependencies

Rule: quotient_representative_implication_no_algebra_operation · Uses: G001, S004

Reverse checks

  • 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

(1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, no_quotient_operation_claim)

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: 646fe4ba27daf1f10da073af8058aa0a9707465d90d3d005ec0ecc6d9e75bdee
  • candidate record SHA-256: 357dabe20ee6a7f47ec4b76fe84d1e1114d0eb167e492a1e05daaf28c9925abd
  • semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
  • question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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