# Complete Reflexivity, Symmetry, Transitivity and Qualified Classes

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

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

batch-20261013-advance-20261026-r07-equivalence-relation-foundations · 2026-10-13 · 100 questions

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



## 1. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [0], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0001/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 2. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [1], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0002/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 3. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [2], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0003/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 4. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [3], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0004/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 5. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [4], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0005/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 6. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [5], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0006/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 7. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [0], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0007/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 8. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [1], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0008/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 9. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [2], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0009/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 10. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [3], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0010/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 11. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [4], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0011/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 12. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [5], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0012/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 13. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [0], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0013/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 14. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [1], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0014/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 15. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [2], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0015/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 16. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [3], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0016/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 17. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [4], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0017/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 18. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [5], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0018/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 19. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [0], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0019/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 20. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [1], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0020/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 21. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [2], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0021/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 22. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [3], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0022/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 23. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [4], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0023/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 24. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [5], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [5], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0024/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 25. Every Carrier Element Needs Its Diagonal Pair

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all missing diagonal element labels in increasing order,reflexive?).

View answer and explanation(6, [0], No)

Check (x,x) for every carrier label, not merely labels occurring elsewhere in the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0025/)

Equivalence relation · qf-mod26-equiv-reflexivity-audit · candidate





## 26. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [], Yes)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0026/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 27. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 2]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0027/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 28. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0028/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 29. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [], Yes)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0029/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 30. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1], [0, 3]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0030/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 31. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 2]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0031/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 32. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 2], [0, 3]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0032/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 33. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 4]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0033/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 34. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0034/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 35. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 4], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 4]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0035/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 36. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1], [0, 2]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0036/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 37. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 3]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0037/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 38. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 3], [0, 4]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3], [0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0038/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 39. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 2], [0, 3]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2], [0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0039/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 40. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1], [0, 3], [0, 4]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 3], [0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0040/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 41. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [], Yes)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0041/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 42. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1], [0, 5]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0042/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 43. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 2]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0043/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 44. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 2], [0, 5]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 2], [0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0044/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 45. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 3], [0, 5]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3], [0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0045/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 46. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1], [0, 5]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0046/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 47. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5], [2, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 3], [0, 5]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 3], [0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0047/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 48. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 1], [0, 2], [0, 5]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 1], [0, 2], [0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0048/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 49. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 4]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0049/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 50. Record Original Pairs Whose Reverse Is Missing

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,lexicographically ordered original [a,b] pairs for which [b,a] is missing,symmetric?). The records are the original pairs, not their absent reverses.

View answer and explanation(6, [[0, 4], [0, 5]], No)

For every supplied ordered pair check its reverse. An empty relation passes symmetry vacuously but need not pass reflexivity. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[0, 4], [0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0050/)

Equivalence relation · qf-mod26-equiv-symmetry-audit · candidate





## 51. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [], Yes)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0051/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 52. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [], Yes)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0052/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 53. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 2]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0053/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 54. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [], Yes)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0054/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 55. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 3]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0055/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 56. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[3, 0, 2]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[3, 0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0056/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 57. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [1, 0], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 2], [1, 0, 3], [3, 0, 1], [3, 0, 2]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 3], [3, 0, 1], [3, 0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0057/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 58. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [], Yes)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0058/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 59. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 4]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0059/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 60. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[4, 0, 2]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[4, 0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0060/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 61. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 4], [1, 0], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 2], [1, 0, 4], [4, 0, 1], [4, 0, 2]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 4], [4, 0, 1], [4, 0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0061/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 62. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[4, 0, 3]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[4, 0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0062/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 63. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4], [1, 0], [4, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 3], [1, 0, 4], [4, 0, 1], [4, 0, 3]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 3], [1, 0, 4], [4, 0, 1], [4, 0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0063/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 64. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[3, 0, 2], [3, 0, 4]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[3, 0, 2], [3, 0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0064/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 65. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4], [1, 0], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 2], [1, 0, 3], [1, 0, 4], [3, 0, 1], [3, 0, 2], [3, 0, 4]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 3], [1, 0, 4], [3, 0, 1], [3, 0, 2], [3, 0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0065/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 66. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [], Yes)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0066/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 67. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 5], [1, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 5]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0067/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 68. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[5, 0, 2]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[5, 0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0068/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 69. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5], [1, 0], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 2], [1, 0, 5], [5, 0, 1], [5, 0, 2]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 5], [5, 0, 1], [5, 0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0069/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 70. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[5, 0, 3]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[5, 0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0070/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 71. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5], [1, 0], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 3], [1, 0, 5], [5, 0, 1], [5, 0, 3]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 3], [1, 0, 5], [5, 0, 1], [5, 0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0071/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 72. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[3, 0, 2], [3, 0, 5]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[3, 0, 2], [3, 0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0072/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 73. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5], [1, 0], [3, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 2], [1, 0, 3], [1, 0, 5], [3, 0, 1], [3, 0, 2], [3, 0, 5]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 2], [1, 0, 3], [1, 0, 5], [3, 0, 1], [3, 0, 2], [3, 0, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0073/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 74. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4], [0, 5], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[5, 0, 4]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[5, 0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0074/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 75. Record Every Failed Composable-Pair Triple

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5], [1, 0], [5, 0]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,all [a,b,c] witnesses with aRb and bRc but not aRc,transitive?). Preserve the nested lexicographic order of the two supplied relation pairs. Despite the source field name missing_transitive_pairs, the records are triples.

View answer and explanation(6, [[1, 0, 4], [1, 0, 5], [5, 0, 1], [5, 0, 4]], No)

Enumerate all composable ordered relation-pair combinations. A triple is a failure witness, not a newly added pair or a request to close the relation. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [[1, 0, 4], [1, 0, 5], [5, 0, 1], [5, 0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0075/)

Equivalence relation · qf-mod26-equiv-transitivity-audit · candidate





## 76. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [], [], Yes, [[0], [1], [2], [3], [4], [5]])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0], [1], [2], [3], [4], [5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0076/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 77. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered 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]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [], [], Yes, [[0, 1, 2, 3, 4, 5]])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 1, 2, 3, 4, 5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0077/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 78. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [3, 2], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [], [], Yes, [[0, 1], [2, 3], [4, 5]])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 1], [2, 3], [4, 5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0078/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 79. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered 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]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [], [], Yes, [[0, 1, 2], [3, 4, 5]])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 1, 2], [3, 4, 5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0079/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 80. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [2, 0], [2, 2], [3, 1], [3, 3], [4, 4], [4, 5], [5, 4], [5, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [], [], Yes, [[0, 2], [1, 3], [4, 5]])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [], [], Yes, [[0, 2], [1, 3], [4, 5]]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0080/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 81. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 2], [0, 3]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 3]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0081/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 82. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 2], [0, 3]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 3]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0082/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 83. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0083/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 84. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0084/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 85. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 2], [0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0085/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 86. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 2], [0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0086/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 87. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 3], [0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 3], [0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0087/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 88. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 3], [0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 3], [0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0088/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 89. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 2], [0, 3], [0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 3], [0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0089/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 90. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 4]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 2], [0, 3], [0, 4]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 3], [0, 4]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0090/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 91. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0091/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 92. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0092/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 93. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 2], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0093/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 94. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 2], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0094/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 95. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 3], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 3], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0095/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 96. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 3], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 3], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0096/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 97. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 2], [0, 3], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 2], [0, 3], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0097/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 98. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 2], [0, 3], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 2], [0, 3], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 2], [0, 3], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0098/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 99. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 4], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 4], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0099/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate





## 100. Emit Classes Only after All Three Axioms Qualify

Use the full fixed carrier Fin(6)={0,…,5} and complete distinct ordered relation pairs=[[0, 0], [1, 1], [2, 2], [3, 3], [4, 4], [5, 5], [0, 1], [0, 4], [0, 5]]. Check reflexivity, symmetry and transitivity separately on the whole carrier. Equivalence classes may be emitted only after all three qualifications pass; one counterexample refutes its corresponding axiom. Give (n,missing reflexive labels,original pairs with missing reverses,failed transitivity triples,equivalence?,classes). If qualified, visit the least unseen label, list its class increasingly, and repeat; otherwise classes is empty.

View answer and explanation(6, [], [[0, 1], [0, 4], [0, 5]], [], No, [])

Keep the three independent failure lists. Only after all are empty derive the classes from the given relation; do not repair the relation or infer equivalence from a visual grouping. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (6, [], [[0, 1], [0, 4], [0, 5]], [], No, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261026-r07-equivalence-relation-0100/)

Equivalence relation · qf-mod26-equiv-qualification-classes · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r07-equivalence-relation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261026-r07-equivalence-relation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261026-r07-equivalence-relation-foundations/validation-report.json)