Complete Reflexivity, Symmetry, Transitivity and Qualified Classes
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Equivalence relation · qf-mod26-equiv-qualification-classes · candidate
Canonical question JSONL · English JSONL · Validation report