# Distinct Full Classes, Canonical Projection and Partition Boundaries

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261030-r04-quotient-set-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,count).

View answer and explanation([[0, 1, 2, 3, 4, 5]], 1)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5]], 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0001/)

Quotient set · qf-mod30-quotient-classes · candidate





## 2. The Quotient Elements Are Entire Classes

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. Give (classes,count).

View answer and explanation([[0], [1, 2, 3, 4, 5]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 2, 3, 4, 5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0002/)

Quotient set · qf-mod30-quotient-classes · candidate





## 3. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,count).

View answer and explanation([[0, 2, 3, 4, 5], [1]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 3, 4, 5], [1]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0003/)

Quotient set · qf-mod30-quotient-classes · candidate





## 4. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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. Give (classes,count).

View answer and explanation([[0, 1], [2, 3, 4, 5]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0004/)

Quotient set · qf-mod30-quotient-classes · candidate





## 5. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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. Give (classes,count).

View answer and explanation([[0], [1], [2, 3, 4, 5]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1], [2, 3, 4, 5]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0005/)

Quotient set · qf-mod30-quotient-classes · candidate





## 6. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [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. Give (classes,count).

View answer and explanation([[0, 1, 3, 4, 5], [2]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 3, 4, 5], [2]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0006/)

Quotient set · qf-mod30-quotient-classes · candidate





## 7. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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. Give (classes,count).

View answer and explanation([[0, 2], [1, 3, 4, 5]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2], [1, 3, 4, 5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0007/)

Quotient set · qf-mod30-quotient-classes · candidate





## 8. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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. Give (classes,count).

View answer and explanation([[0], [1, 3, 4, 5], [2]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 3, 4, 5], [2]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0008/)

Quotient set · qf-mod30-quotient-classes · candidate





## 9. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,count).

View answer and explanation([[0, 3, 4, 5], [1, 2]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1, 2]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0009/)

Quotient set · qf-mod30-quotient-classes · candidate





## 10. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,count).

View answer and explanation([[0, 1, 2], [3, 4, 5]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 2], [3, 4, 5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0010/)

Quotient set · qf-mod30-quotient-classes · candidate





## 11. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,count).

View answer and explanation([[0], [1, 2], [3, 4, 5]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 2], [3, 4, 5]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0011/)

Quotient set · qf-mod30-quotient-classes · candidate





## 12. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,count).

View answer and explanation([[0, 3, 4, 5], [1], [2]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1], [2]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0012/)

Quotient set · qf-mod30-quotient-classes · candidate





## 13. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,count).

View answer and explanation([[0, 2], [1], [3, 4, 5]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2], [1], [3, 4, 5]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0013/)

Quotient set · qf-mod30-quotient-classes · candidate





## 14. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,count).

View answer and explanation([[0, 1], [2], [3, 4, 5]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1], [2], [3, 4, 5]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0014/)

Quotient set · qf-mod30-quotient-classes · candidate





## 15. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. 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. Give (classes,count).

View answer and explanation([[0, 1, 2, 4, 5], [3]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 2, 4, 5], [3]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0015/)

Quotient set · qf-mod30-quotient-classes · candidate





## 16. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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. Give (classes,count).

View answer and explanation([[0, 3], [1, 2, 4, 5]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3], [1, 2, 4, 5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0016/)

Quotient set · qf-mod30-quotient-classes · candidate





## 17. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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. Give (classes,count).

View answer and explanation([[0], [1, 2, 4, 5], [3]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 2, 4, 5], [3]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0017/)

Quotient set · qf-mod30-quotient-classes · candidate





## 18. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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. Give (classes,count).

View answer and explanation([[0, 2, 4, 5], [1, 3]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1, 3]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0018/)

Quotient set · qf-mod30-quotient-classes · candidate





## 19. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,count).

View answer and explanation([[0, 1, 3], [2, 4, 5]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 3], [2, 4, 5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0019/)

Quotient set · qf-mod30-quotient-classes · candidate





## 20. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,count).

View answer and explanation([[0], [1, 3], [2, 4, 5]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0], [1, 3], [2, 4, 5]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0020/)

Quotient set · qf-mod30-quotient-classes · candidate





## 21. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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. Give (classes,count).

View answer and explanation([[0, 2, 4, 5], [1], [3]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1], [3]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0021/)

Quotient set · qf-mod30-quotient-classes · candidate





## 22. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,count).

View answer and explanation([[0, 3], [1], [2, 4, 5]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 3], [1], [2, 4, 5]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0022/)

Quotient set · qf-mod30-quotient-classes · candidate





## 23. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,count).

View answer and explanation([[0, 1], [2, 4, 5], [3]], 3)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1], [2, 4, 5], [3]], 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0023/)

Quotient set · qf-mod30-quotient-classes · candidate





## 24. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [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. Give (classes,count).

View answer and explanation([[0, 1, 4, 5], [2, 3]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 1, 4, 5], [2, 3]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0024/)

Quotient set · qf-mod30-quotient-classes · candidate





## 25. The Quotient Elements Are Entire Classes

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [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. Give (classes,count).

View answer and explanation([[0, 2, 3], [1, 4, 5]], 2)

Start at each least unseen carrier label, form its complete equivalence class and mark every element seen. Deduplicate classes, not their individual members across unrelated classes. The exact result, retaining the requested order, is ([[0, 2, 3], [1, 4, 5]], 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0025/)

Quotient set · qf-mod30-quotient-classes · candidate





## 26. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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. element=0. Give (element,class).

View answer and explanation(0, [0, 1, 2, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0026/)

Quotient set · qf-mod30-quotient-projection · candidate





## 27. Canonical Projection Returns a Class, Not a Chosen Label

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. element=1. Give (element,class).

View answer and explanation(1, [1, 2, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [1, 2, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0027/)

Quotient set · qf-mod30-quotient-projection · candidate





## 28. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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. element=2. Give (element,class).

View answer and explanation(2, [0, 2, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [0, 2, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0028/)

Quotient set · qf-mod30-quotient-projection · candidate





## 29. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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. element=3. Give (element,class).

View answer and explanation(3, [2, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [2, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0029/)

Quotient set · qf-mod30-quotient-projection · candidate





## 30. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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. element=4. Give (element,class).

View answer and explanation(4, [2, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [2, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0030/)

Quotient set · qf-mod30-quotient-projection · candidate





## 31. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [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. element=5. Give (element,class).

View answer and explanation(5, [0, 1, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 1, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0031/)

Quotient set · qf-mod30-quotient-projection · candidate





## 32. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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. element=0. Give (element,class).

View answer and explanation(0, [0, 2])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0032/)

Quotient set · qf-mod30-quotient-projection · candidate





## 33. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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. element=1. Give (element,class).

View answer and explanation(1, [1, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [1, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0033/)

Quotient set · qf-mod30-quotient-projection · candidate





## 34. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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. element=2. Give (element,class).

View answer and explanation(2, [1, 2])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0034/)

Quotient set · qf-mod30-quotient-projection · candidate





## 35. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. element=3. Give (element,class).

View answer and explanation(3, [3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0035/)

Quotient set · qf-mod30-quotient-projection · candidate





## 36. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. element=4. Give (element,class).

View answer and explanation(4, [3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0036/)

Quotient set · qf-mod30-quotient-projection · candidate





## 37. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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. element=5. Give (element,class).

View answer and explanation(5, [0, 3, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 3, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0037/)

Quotient set · qf-mod30-quotient-projection · candidate





## 38. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. element=0. Give (element,class).

View answer and explanation(0, [0, 2])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0038/)

Quotient set · qf-mod30-quotient-projection · candidate





## 39. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. element=1. Give (element,class).

View answer and explanation(1, [0, 1])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0039/)

Quotient set · qf-mod30-quotient-projection · candidate





## 40. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. 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. element=2. Give (element,class).

View answer and explanation(2, [0, 1, 2, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [0, 1, 2, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0040/)

Quotient set · qf-mod30-quotient-projection · candidate





## 41. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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. element=3. Give (element,class).

View answer and explanation(3, [0, 3])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0041/)

Quotient set · qf-mod30-quotient-projection · candidate





## 42. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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. element=4. Give (element,class).

View answer and explanation(4, [1, 2, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [1, 2, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0042/)

Quotient set · qf-mod30-quotient-projection · candidate





## 43. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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. element=5. Give (element,class).

View answer and explanation(5, [0, 2, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 2, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0043/)

Quotient set · qf-mod30-quotient-projection · candidate





## 44. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. element=0. Give (element,class).

View answer and explanation(0, [0, 1, 3])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0044/)

Quotient set · qf-mod30-quotient-projection · candidate





## 45. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. element=1. Give (element,class).

View answer and explanation(1, [1, 3])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (1, [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0045/)

Quotient set · qf-mod30-quotient-projection · candidate





## 46. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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. element=2. Give (element,class).

View answer and explanation(2, [0, 2, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (2, [0, 2, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0046/)

Quotient set · qf-mod30-quotient-projection · candidate





## 47. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. element=3. Give (element,class).

View answer and explanation(3, [0, 3])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (3, [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0047/)

Quotient set · qf-mod30-quotient-projection · candidate





## 48. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. element=4. Give (element,class).

View answer and explanation(4, [2, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (4, [2, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0048/)

Quotient set · qf-mod30-quotient-projection · candidate





## 49. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [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. element=5. Give (element,class).

View answer and explanation(5, [0, 1, 4, 5])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (5, [0, 1, 4, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0049/)

Quotient set · qf-mod30-quotient-projection · candidate





## 50. Canonical Projection Returns a Class, Not a Chosen Label

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [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. element=0. Give (element,class).

View answer and explanation(0, [0, 2, 3])

Send the specified element to its full equivalence class. Related representatives project to the same class; no arbitrary representative is the quotient value. The exact result, retaining the requested order, is (0, [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0050/)

Quotient set · qf-mod30-quotient-projection · candidate





## 51. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1, 2, 3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0051/)

Quotient set · qf-mod30-quotient-partition · candidate





## 52. Verify Coverage and Disjointness of the Full Class Collection

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. Give (classes,covers?,disjoint?).

View answer and explanation([[0], [1, 2, 3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 2, 3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0052/)

Quotient set · qf-mod30-quotient-partition · candidate





## 53. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 2, 3, 4, 5], [1]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 3, 4, 5], [1]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0053/)

Quotient set · qf-mod30-quotient-partition · candidate





## 54. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1], [2, 3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0054/)

Quotient set · qf-mod30-quotient-partition · candidate





## 55. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0], [1], [2, 3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1], [2, 3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0055/)

Quotient set · qf-mod30-quotient-partition · candidate





## 56. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1, 3, 4, 5], [2]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 3, 4, 5], [2]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0056/)

Quotient set · qf-mod30-quotient-partition · candidate





## 57. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 2], [1, 3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2], [1, 3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0057/)

Quotient set · qf-mod30-quotient-partition · candidate





## 58. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0], [1, 3, 4, 5], [2]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 3, 4, 5], [2]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0058/)

Quotient set · qf-mod30-quotient-partition · candidate





## 59. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 3, 4, 5], [1, 2]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1, 2]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0059/)

Quotient set · qf-mod30-quotient-partition · candidate





## 60. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1, 2], [3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 2], [3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0060/)

Quotient set · qf-mod30-quotient-partition · candidate





## 61. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0], [1, 2], [3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 2], [3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0061/)

Quotient set · qf-mod30-quotient-partition · candidate





## 62. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 3, 4, 5], [1], [2]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3, 4, 5], [1], [2]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0062/)

Quotient set · qf-mod30-quotient-partition · candidate





## 63. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 2], [1], [3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2], [1], [3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0063/)

Quotient set · qf-mod30-quotient-partition · candidate





## 64. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1], [2], [3, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1], [2], [3, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0064/)

Quotient set · qf-mod30-quotient-partition · candidate





## 65. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. 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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1, 2, 4, 5], [3]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 2, 4, 5], [3]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0065/)

Quotient set · qf-mod30-quotient-partition · candidate





## 66. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 3], [1, 2, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3], [1, 2, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0066/)

Quotient set · qf-mod30-quotient-partition · candidate





## 67. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0], [1, 2, 4, 5], [3]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 2, 4, 5], [3]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0067/)

Quotient set · qf-mod30-quotient-partition · candidate





## 68. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 2, 4, 5], [1, 3]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1, 3]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0068/)

Quotient set · qf-mod30-quotient-partition · candidate





## 69. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1, 3], [2, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 3], [2, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0069/)

Quotient set · qf-mod30-quotient-partition · candidate





## 70. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0], [1, 3], [2, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0], [1, 3], [2, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0070/)

Quotient set · qf-mod30-quotient-partition · candidate





## 71. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 2, 4, 5], [1], [3]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 4, 5], [1], [3]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0071/)

Quotient set · qf-mod30-quotient-partition · candidate





## 72. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 3], [1], [2, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 3], [1], [2, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0072/)

Quotient set · qf-mod30-quotient-partition · candidate





## 73. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1], [2, 4, 5], [3]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1], [2, 4, 5], [3]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0073/)

Quotient set · qf-mod30-quotient-partition · candidate





## 74. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 1, 4, 5], [2, 3]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 1, 4, 5], [2, 3]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0074/)

Quotient set · qf-mod30-quotient-partition · candidate





## 75. Verify Coverage and Disjointness of the Full Class Collection

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [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. Give (classes,covers?,disjoint?).

View answer and explanation([[0, 2, 3], [1, 4, 5]], Yes, Yes)

Flatten the distinct classes and compare with the complete carrier, then check no element occurs in two different classes. These checks supply a partition, not an inherited algebraic operation. The exact result, retaining the requested order, is ([[0, 2, 3], [1, 4, 5]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0075/)

Quotient set · qf-mod30-quotient-partition · candidate





## 76. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(0, 1, Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0076/)

Quotient set · qf-mod30-quotient-representative · candidate





## 77. Representative Invariance Does Not Define Quotient Operations

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).

View answer and explanation(1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0077/)

Quotient set · qf-mod30-quotient-representative · candidate





## 78. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [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=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(2, 1, No, [0, 2, 3, 4, 5], [1], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, No, [0, 2, 3, 4, 5], [1], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0078/)

Quotient set · qf-mod30-quotient-representative · candidate





## 79. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(3, 4, Yes, [2, 3, 4, 5], [2, 3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, Yes, [2, 3, 4, 5], [2, 3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0079/)

Quotient set · qf-mod30-quotient-representative · candidate





## 80. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [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=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(4, 1, No, [2, 3, 4, 5], [1], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, No, [2, 3, 4, 5], [1], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0080/)

Quotient set · qf-mod30-quotient-representative · candidate





## 81. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [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=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(5, 4, Yes, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0081/)

Quotient set · qf-mod30-quotient-representative · candidate





## 82. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(0, 1, No, [0, 2], [1, 3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, No, [0, 2], [1, 3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0082/)

Quotient set · qf-mod30-quotient-representative · candidate





## 83. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [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).

View answer and explanation(1, 4, Yes, [1, 3, 4, 5], [1, 3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, Yes, [1, 3, 4, 5], [1, 3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0083/)

Quotient set · qf-mod30-quotient-representative · candidate





## 84. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(2, 1, Yes, [1, 2], [1, 2], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, Yes, [1, 2], [1, 2], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0084/)

Quotient set · qf-mod30-quotient-representative · candidate





## 85. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 2], [1, 0], [1, 1], [1, 2], [2, 0], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(3, 4, Yes, [3, 4, 5], [3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, Yes, [3, 4, 5], [3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0085/)

Quotient set · qf-mod30-quotient-representative · candidate





## 86. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(4, 1, No, [3, 4, 5], [1, 2], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, No, [3, 4, 5], [1, 2], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0086/)

Quotient set · qf-mod30-quotient-representative · candidate





## 87. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [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=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(5, 4, Yes, [0, 3, 4, 5], [0, 3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 3, 4, 5], [0, 3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0087/)

Quotient set · qf-mod30-quotient-representative · candidate





## 88. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(0, 1, No, [0, 2], [1], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, No, [0, 2], [1], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0088/)

Quotient set · qf-mod30-quotient-representative · candidate





## 89. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [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).

View answer and explanation(1, 4, No, [0, 1], [3, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, No, [0, 1], [3, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0089/)

Quotient set · qf-mod30-quotient-representative · candidate





## 90. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence 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]]. 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=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(2, 1, Yes, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, Yes, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0090/)

Quotient set · qf-mod30-quotient-representative · candidate





## 91. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(3, 4, No, [0, 3], [1, 2, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, No, [0, 3], [1, 2, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0091/)

Quotient set · qf-mod30-quotient-representative · candidate





## 92. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [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=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(4, 1, Yes, [1, 2, 4, 5], [1, 2, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, Yes, [1, 2, 4, 5], [1, 2, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0092/)

Quotient set · qf-mod30-quotient-representative · candidate





## 93. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(5, 4, Yes, [0, 2, 4, 5], [0, 2, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 2, 4, 5], [0, 2, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0093/)

Quotient set · qf-mod30-quotient-representative · candidate





## 94. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(0, 1, Yes, [0, 1, 3], [0, 1, 3], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, Yes, [0, 1, 3], [0, 1, 3], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0094/)

Quotient set · qf-mod30-quotient-representative · candidate





## 95. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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).

View answer and explanation(1, 4, No, [1, 3], [2, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (1, 4, No, [1, 3], [2, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0095/)

Quotient set · qf-mod30-quotient-representative · candidate





## 96. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [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=2, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(2, 1, No, [0, 2, 4, 5], [1], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (2, 1, No, [0, 2, 4, 5], [1], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0096/)

Quotient set · qf-mod30-quotient-representative · candidate





## 97. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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=3, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(3, 4, No, [0, 3], [2, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (3, 4, No, [0, 3], [2, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0097/)

Quotient set · qf-mod30-quotient-representative · candidate





## 98. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [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=4, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(4, 1, No, [2, 4, 5], [0, 1], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (4, 1, No, [2, 4, 5], [0, 1], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0098/)

Quotient set · qf-mod30-quotient-representative · candidate





## 99. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [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=5, b=4. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(5, 4, Yes, [0, 1, 4, 5], [0, 1, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (5, 4, Yes, [0, 1, 4, 5], [0, 1, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0099/)

Quotient set · qf-mod30-quotient-representative · candidate





## 100. Representative Invariance Does Not Define Quotient Operations

Use the same complete carrier Fin(6)={0,…,5} and certified equivalence relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [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=0, b=1. Give (a,b,related?,class_a,class_b,invariance?,no_quotient_operation_claim).

View answer and explanation(0, 1, No, [0, 2, 3], [1, 4, 5], Yes, no_quotient_operation_claim)

Compute both full classes. invariance is the implication aRb implies class_a=class_b; it is vacuously Yes for unrelated representatives, without claiming their classes are equal. Algebraic operations still require a separate well-definedness proof. The exact result, retaining the requested order, is (0, 1, No, [0, 2, 3], [1, 4, 5], Yes, no_quotient_operation_claim).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261030-r04-quotient-set-0100/)

Quotient set · qf-mod30-quotient-representative · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r04-quotient-set-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261030-r04-quotient-set-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261030-r04-quotient-set-foundations/validation-report.json)