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Distinct Full Classes, Canonical Projection and Partition Boundaries

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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