Complete Classes, Membership, Conditional Representative Invariance and Disjointness
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261029-r07-equivalence-class-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0. Give (element,class).
View answer and explanation
(0, [0, 1, 2, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
2. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1. Give (element,class).
View answer and explanation
(1, [1, 2, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (1, [1, 2, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
3. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2. Give (element,class).
View answer and explanation
(2, [0, 2, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (2, [0, 2, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
4. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3. Give (element,class).
View answer and explanation
(3, [2, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (3, [2, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
5. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4. Give (element,class).
View answer and explanation
(4, [2, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (4, [2, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
6. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5. Give (element,class).
View answer and explanation
(5, [0, 1, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (5, [0, 1, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
7. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0. Give (element,class).
View answer and explanation
(0, [0, 2])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (0, [0, 2]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
8. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1. Give (element,class).
View answer and explanation
(1, [1, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (1, [1, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
9. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2. Give (element,class).
View answer and explanation
(2, [1, 2])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (2, [1, 2]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
10. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3. Give (element,class).
View answer and explanation
(3, [3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (3, [3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
11. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4. Give (element,class).
View answer and explanation
(4, [3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (4, [3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
12. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5. Give (element,class).
View answer and explanation
(5, [0, 3, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (5, [0, 3, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
13. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0. Give (element,class).
View answer and explanation
(0, [0, 2])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (0, [0, 2]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
14. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1. Give (element,class).
View answer and explanation
(1, [0, 1])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (1, [0, 1]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
15. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2. Give (element,class).
View answer and explanation
(2, [0, 1, 2, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (2, [0, 1, 2, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
16. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3. Give (element,class).
View answer and explanation
(3, [0, 3])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (3, [0, 3]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
17. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4. Give (element,class).
View answer and explanation
(4, [1, 2, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (4, [1, 2, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
18. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5. Give (element,class).
View answer and explanation
(5, [0, 2, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (5, [0, 2, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
19. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0. Give (element,class).
View answer and explanation
(0, [0, 1, 3])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (0, [0, 1, 3]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
20. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1. Give (element,class).
View answer and explanation
(1, [1, 3])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (1, [1, 3]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
21. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2. Give (element,class).
View answer and explanation
(2, [0, 2, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (2, [0, 2, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
22. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3. Give (element,class).
View answer and explanation
(3, [0, 3])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (3, [0, 3]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
23. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4. Give (element,class).
View answer and explanation
(4, [2, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (4, [2, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
24. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5. Give (element,class).
View answer and explanation
(5, [0, 1, 4, 5])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (5, [0, 1, 4, 5]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
25. An Equivalence Class Contains Every Related Carrier Element
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0. Give (element,class).
View answer and explanation
(0, [0, 2, 3])
Filter the entire carrier by the certified relation with this representative. Keep every related element, including the representative by reflexivity. The exact result, retaining the requested order, is (0, [0, 2, 3]).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-of · candidate
26. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(0, 1, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (0, 1, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
27. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(1, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (1, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
28. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(2, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (2, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
29. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(3, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (3, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
30. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(4, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (4, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
31. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(5, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (5, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
32. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(0, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (0, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
33. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(1, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (1, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
34. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(2, 1, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (2, 1, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
35. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(3, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (3, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
36. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(4, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (4, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
37. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(5, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (5, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
38. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(0, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (0, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
39. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(1, 4, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (1, 4, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
40. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(2, 1, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (2, 1, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
41. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(3, 4, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (3, 4, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
42. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(4, 1, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (4, 1, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
43. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(5, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (5, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
44. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(0, 1, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (0, 1, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
45. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=1, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(1, 4, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (1, 4, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
46. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=2, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(2, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (2, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
47. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=3, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(3, 4, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (3, 4, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
48. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=4, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(4, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (4, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
49. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=5, candidate=4. Give (element,candidate,related?,in_class?).
View answer and explanation
(5, 4, Yes, Yes)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (5, 4, Yes, Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
50. Relation Membership and Class Membership Must Agree
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. element=0, candidate=1. Give (element,candidate,related?,in_class?).
View answer and explanation
(0, 1, No, No)
Look up the ordered relation pair and independently read candidate membership in the full class. Both flags use the same carrier and relation. The exact result, retaining the requested order, is (0, 1, No, No).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-membership · candidate
51. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(0, 1, Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (0, 1, Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
52. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
53. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(2, 1, No, [0, 2, 3, 4, 5], [1], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (2, 1, No, [0, 2, 3, 4, 5], [1], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
54. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(3, 4, Yes, [2, 3, 4, 5], [2, 3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (3, 4, Yes, [2, 3, 4, 5], [2, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
55. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(4, 1, No, [2, 3, 4, 5], [1], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (4, 1, No, [2, 3, 4, 5], [1], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
56. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(5, 4, Yes, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (5, 4, Yes, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
57. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(0, 1, No, [0, 2], [1, 3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (0, 1, No, [0, 2], [1, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
58. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(1, 4, Yes, [1, 3, 4, 5], [1, 3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (1, 4, Yes, [1, 3, 4, 5], [1, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
59. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(2, 1, Yes, [1, 2], [1, 2], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (2, 1, Yes, [1, 2], [1, 2], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
60. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(3, 4, Yes, [3, 4, 5], [3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (3, 4, Yes, [3, 4, 5], [3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
61. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(4, 1, No, [3, 4, 5], [1, 2], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (4, 1, No, [3, 4, 5], [1, 2], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
62. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(5, 4, Yes, [0, 3, 4, 5], [0, 3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (5, 4, Yes, [0, 3, 4, 5], [0, 3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
63. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(0, 1, No, [0, 2], [1], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (0, 1, No, [0, 2], [1], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
64. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(1, 4, No, [0, 1], [3, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (1, 4, No, [0, 1], [3, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
65. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(2, 1, Yes, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (2, 1, Yes, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
66. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(3, 4, No, [0, 3], [1, 2, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (3, 4, No, [0, 3], [1, 2, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
67. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(4, 1, Yes, [1, 2, 4, 5], [1, 2, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (4, 1, Yes, [1, 2, 4, 5], [1, 2, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
68. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(5, 4, Yes, [0, 2, 4, 5], [0, 2, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (5, 4, Yes, [0, 2, 4, 5], [0, 2, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
69. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(0, 1, Yes, [0, 1, 3], [0, 1, 3], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (0, 1, Yes, [0, 1, 3], [0, 1, 3], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
70. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(1, 4, No, [1, 3], [2, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (1, 4, No, [1, 3], [2, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
71. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(2, 1, No, [0, 2, 4, 5], [1], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (2, 1, No, [0, 2, 4, 5], [1], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
72. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(3, 4, No, [0, 3], [2, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (3, 4, No, [0, 3], [2, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
73. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(4, 1, No, [2, 4, 5], [0, 1], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (4, 1, No, [2, 4, 5], [0, 1], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
74. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(5, 4, Yes, [0, 1, 4, 5], [0, 1, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (5, 4, Yes, [0, 1, 4, 5], [0, 1, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
75. Representative Invariance Is Conditional on Relatedness
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,related?,class_a,class_b,invariance?).
View answer and explanation
(0, 1, No, [0, 2, 3], [1, 4, 5], Yes)
Compute both full classes and report the implication aRb implies class_a=class_b. When the representatives are unrelated the implication passes vacuously; that does not assert their two classes are equal. The exact result, retaining the requested order, is (0, 1, No, [0, 2, 3], [1, 4, 5], Yes).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-representative-invariance · candidate
76. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(0, 1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (0, 1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
77. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(1, 4, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (1, 4, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
78. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(2, 1, [0, 2, 3, 4, 5], [1], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (2, 1, [0, 2, 3, 4, 5], [1], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
79. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(3, 4, [2, 3, 4, 5], [2, 3, 4, 5], [2, 3, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (3, 4, [2, 3, 4, 5], [2, 3, 4, 5], [2, 3, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
80. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 2], [3, 3], [3, 4], [3, 5], [4, 2], [4, 3], [4, 4], [4, 5], [5, 2], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(4, 1, [2, 3, 4, 5], [1], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (4, 1, [2, 3, 4, 5], [1], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
81. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 0], [3, 1], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(5, 4, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (5, 4, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
82. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [1, 3], [1, 4], [1, 5], [2, 0], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(0, 1, [0, 2], [1, 3, 4, 5], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (0, 1, [0, 2], [1, 3, 4, 5], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
83. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [1, 4], [1, 5], [2, 2], [3, 1], [3, 3], [3, 4], [3, 5], [4, 1], [4, 3], [4, 4], [4, 5], [5, 1], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(1, 4, [1, 3, 4, 5], [1, 3, 4, 5], [1, 3, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (1, 4, [1, 3, 4, 5], [1, 3, 4, 5], [1, 3, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
84. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [1, 2], [2, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(2, 1, [1, 2], [1, 2], [1, 2], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (2, 1, [1, 2], [1, 2], [1, 2], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
85. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete 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]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(3, 4, [3, 4, 5], [3, 4, 5], [3, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (3, 4, [3, 4, 5], [3, 4, 5], [3, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
86. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [2, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(4, 1, [3, 4, 5], [1, 2], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (4, 1, [3, 4, 5], [1, 2], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
87. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [0, 4], [0, 5], [1, 1], [2, 2], [3, 0], [3, 3], [3, 4], [3, 5], [4, 0], [4, 3], [4, 4], [4, 5], [5, 0], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(5, 4, [0, 3, 4, 5], [0, 3, 4, 5], [0, 3, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (5, 4, [0, 3, 4, 5], [0, 3, 4, 5], [0, 3, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
88. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [1, 1], [2, 0], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(0, 1, [0, 2], [1], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (0, 1, [0, 2], [1], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
89. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [3, 3], [3, 4], [3, 5], [4, 3], [4, 4], [4, 5], [5, 3], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(1, 4, [0, 1], [3, 4, 5], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (1, 4, [0, 1], [3, 4, 5], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
90. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 2], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 1], [4, 2], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(2, 1, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (2, 1, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
91. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(3, 4, [0, 3], [1, 2, 4, 5], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (3, 4, [0, 3], [1, 2, 4, 5], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
92. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 2], [1, 4], [1, 5], [2, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 1], [4, 2], [4, 4], [4, 5], [5, 1], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(4, 1, [1, 2, 4, 5], [1, 2, 4, 5], [1, 2, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (4, 1, [1, 2, 4, 5], [1, 2, 4, 5], [1, 2, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
93. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [1, 3], [2, 0], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(5, 4, [0, 2, 4, 5], [0, 2, 4, 5], [0, 2, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (5, 4, [0, 2, 4, 5], [0, 2, 4, 5], [0, 2, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
94. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 3], [1, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 0], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(0, 1, [0, 1, 3], [0, 1, 3], [0, 1, 3], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (0, 1, [0, 1, 3], [0, 1, 3], [0, 1, 3], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
95. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [1, 1], [1, 3], [2, 2], [2, 4], [2, 5], [3, 1], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=1, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(1, 4, [1, 3], [2, 4, 5], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (1, 4, [1, 3], [2, 4, 5], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
96. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 4], [0, 5], [1, 1], [2, 0], [2, 2], [2, 4], [2, 5], [3, 3], [4, 0], [4, 2], [4, 4], [4, 5], [5, 0], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=2, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(2, 1, [0, 2, 4, 5], [1], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (2, 1, [0, 2, 4, 5], [1], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
97. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 3], [1, 1], [2, 2], [2, 4], [2, 5], [3, 0], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=3, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(3, 4, [0, 3], [2, 4, 5], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (3, 4, [0, 3], [2, 4, 5], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
98. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [1, 0], [1, 1], [2, 2], [2, 4], [2, 5], [3, 3], [4, 2], [4, 4], [4, 5], [5, 2], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=4, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(4, 1, [2, 4, 5], [0, 1], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (4, 1, [2, 4, 5], [0, 1], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
99. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 1], [0, 4], [0, 5], [1, 0], [1, 1], [1, 4], [1, 5], [2, 2], [2, 3], [3, 2], [3, 3], [4, 0], [4, 1], [4, 4], [4, 5], [5, 0], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=5, b=4. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(5, 4, [0, 1, 4, 5], [0, 1, 4, 5], [0, 1, 4, 5], Yes, equal)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (5, 4, [0, 1, 4, 5], [0, 1, 4, 5], [0, 1, 4, 5], Yes, equal).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
100. Two Certified Classes Are Equal or Disjoint
Use the complete carrier Fin(6)={0,…,5} and complete relation pairs=[[0, 0], [0, 2], [0, 3], [1, 1], [1, 4], [1, 5], [2, 0], [2, 2], [2, 3], [3, 0], [3, 2], [3, 3], [4, 1], [4, 4], [4, 5], [5, 1], [5, 4], [5, 5]]. Before forming classes, certify reflexivity, symmetry and transitivity on this same carrier; sampled pairs or visual clustering do not suffice. A class contains exactly every related carrier element, listed increasingly. Classes are equal or disjoint; no quotient operation or representative-independent algebra is constructed. a=0, b=1. Give (a,b,class_a,class_b,intersection,equal_or_disjoint?,classification). Classification is equal or disjoint.
View answer and explanation
(0, 1, [0, 2, 3], [1, 4, 5], [], Yes, disjoint)
Compare the complete classes and their increasing intersection. A nonempty intersection between unequal classes would contradict the already required equivalence axioms, not define an additional class type. The exact result, retaining the requested order, is (0, 1, [0, 2, 3], [1, 4, 5], [], Yes, disjoint).
Reviewed step-by-step derivation
Equivalence class · qf-mod29-eqclass-equal-or-disjoint · candidate
Canonical question JSONL · English JSONL · Validation report