{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0001","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"325105159768c663d20888fe024f8e528b42c756230c70f332982ddf47b215d7","family_id":"qf-mod29-eqclass-of","seed":202610297001,"semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0002","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1668033d21abeadce46a883e52d00056212b532e68212cdae5d0b27609816dca","family_id":"qf-mod29-eqclass-of","seed":202610297002,"semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0003","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"898619bcd8c8f117d9c6f0baebbfc67dcd0e4dd4dabc72e2166e4e804ad08765","family_id":"qf-mod29-eqclass-of","seed":202610297003,"semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(2, [0, 2, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0004","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b89f8939bcbb2c75726c01c475ce0f59f5d6c5da62a0cbe4832f7361aa3982e7","family_id":"qf-mod29-eqclass-of","seed":202610297004,"semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(3, [2, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0005","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95bea8370fce700cb52fa31891a45ef826ef00229299ef3a6352757d3d5cf0b3","family_id":"qf-mod29-eqclass-of","seed":202610297005,"semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(4, [2, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0006","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3177de002b0ca9af5eccb1af5940d83606c75959fcd964c0b9d3015919e44b41","family_id":"qf-mod29-eqclass-of","seed":202610297006,"semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0007","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3ca91f707ff388b42a47e282345cc2ce701c7be58db03260f4811265b2c4089","family_id":"qf-mod29-eqclass-of","seed":202610297007,"semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(0, [0, 2])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0008","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5664f6bafa08ec5bf8af53bf3c353f31186460c74593654be9fe17866fcf4a5d","family_id":"qf-mod29-eqclass-of","seed":202610297008,"semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(1, [1, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0009","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86fa392756fdfd5a83b7b924d02bf49662929218d227c65b296e08ad34014fa4","family_id":"qf-mod29-eqclass-of","seed":202610297009,"semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(2, [1, 2])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0010","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e20cd4dfb80c2235c82555d3c574628976374bf1b740108fe3a85200584050f0","family_id":"qf-mod29-eqclass-of","seed":202610297010,"semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(3, [3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0011","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c4100c64c9a3acf6dd1d94ae57d282b97be8c83942022b4feb0efd6d7792a53","family_id":"qf-mod29-eqclass-of","seed":202610297011,"semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(4, [3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0012","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca66d3c8595a913157aca81a23e5dc3eec4aa82f2de1a58af0d9b101bef6c341","family_id":"qf-mod29-eqclass-of","seed":202610297012,"semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(5, [0, 3, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0013","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37ea2d5b2fe2be9fb8421c632e1808b9d1eb0a2a090bd73b3d5d91bd99de59f2","family_id":"qf-mod29-eqclass-of","seed":202610297013,"semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(0, [0, 2])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0014","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e9896e311bf7d80add5e51b116f2bcefecb18103b93684a618887d9a48dc6f4","family_id":"qf-mod29-eqclass-of","seed":202610297014,"semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(1, [0, 1])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0015","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd84364a0e447c24bf0081019bc64b2b3050b34500daabcfdc75518b8fdf20d7","family_id":"qf-mod29-eqclass-of","seed":202610297015,"semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(2, [0, 1, 2, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0016","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b3a4462ef571b0424b2c35f1827844e4f3a73bc2428714b7cf661afd9c6f110","family_id":"qf-mod29-eqclass-of","seed":202610297016,"semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(3, [0, 3])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0017","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfe2134185743cbd508d0e544b0945a292eafb067561597b26f7b25a925fd66e","family_id":"qf-mod29-eqclass-of","seed":202610297017,"semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(4, [1, 2, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0018","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b04cb6ed5fc4856c1697b38ec563f298b39510d4e0c9fde01881d2f7a392cf7","family_id":"qf-mod29-eqclass-of","seed":202610297018,"semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0019","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f91a7d494e2278598114d7db8c51c16187129942c172f02bb52e15fed6bc366","family_id":"qf-mod29-eqclass-of","seed":202610297019,"semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(0, [0, 1, 3])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0020","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9810d8583677225544d3b9841e61a91c3ef3930146e013b5ae8f3093f4da6f9a","family_id":"qf-mod29-eqclass-of","seed":202610297020,"semantic_template":"qf-mod29-eqclass-of/{\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(1, [1, 3])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0021","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac747f1530e484d83d3c5a32ef253842865d7154ebf3038ea482dfdcaec41277","family_id":"qf-mod29-eqclass-of","seed":202610297021,"semantic_template":"qf-mod29-eqclass-of/{\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(2, [0, 2, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0022","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"415f5461c9edf9598325861f867a314d0606e021d28ec17bde9f99ab08aeb22d","family_id":"qf-mod29-eqclass-of","seed":202610297022,"semantic_template":"qf-mod29-eqclass-of/{\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(3, [0, 3])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0023","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c882c813aa0de379b610d6ea90412afeb4c8e35e43df899feded487577f3558","family_id":"qf-mod29-eqclass-of","seed":202610297023,"semantic_template":"qf-mod29-eqclass-of/{\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(4, [2, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0024","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5fceb7f9921b41d8d2a2864af7bea2a3d16c98eda76a6c87d96a9c47a6a0335","family_id":"qf-mod29-eqclass-of","seed":202610297024,"semantic_template":"qf-mod29-eqclass-of/{\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(5, [0, 1, 4, 5])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0025","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d44f229f686bf626c784ea08f1e5650fdace04547f758c5c8f3af2ec2580b5d","family_id":"qf-mod29-eqclass-of","seed":202610297025,"semantic_template":"qf-mod29-eqclass-of/{\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"An Equivalence Class Contains Every Related Carrier Element","stem":"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).","answer":{"kind":"expression","value":"(0, [0, 2, 3])","explanation":"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])."},"skills":["An Equivalence Class Contains Every Related Carrier Element"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0026","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c0244ea2c911f4dc7882acf246ca90c0b41c18646d482afa1a1eb3dd4ef1789","family_id":"qf-mod29-eqclass-membership","seed":202610297026,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0027","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e902b55c840405be3226416c2b6fe3e1a577552915bfccaed6aa304e43f9139","family_id":"qf-mod29-eqclass-membership","seed":202610297027,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0028","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a36a29d29104ad7511e76195564663e3b43486db207fa665e3a2bd1c08a6e49a","family_id":"qf-mod29-eqclass-membership","seed":202610297028,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0029","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdfbb214ac1e620d4f1629458ac81ffadcf63e304a7123d21fa3bd9119746c33","family_id":"qf-mod29-eqclass-membership","seed":202610297029,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0030","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67a07bad48ada785a6c9ef4d4d05c1476a750b5f143ec6a86165dc6db2fe4dd2","family_id":"qf-mod29-eqclass-membership","seed":202610297030,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0031","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9103b0ac7d5a81c53dc481100bcd4260b80d100b3a182b5b49ec910b2c7988c","family_id":"qf-mod29-eqclass-membership","seed":202610297031,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0032","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8525debae452127a33f102fc3238a94d3fcf128681c25da2df630fc856504931","family_id":"qf-mod29-eqclass-membership","seed":202610297032,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0033","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7df67fbb010c8fc66ab8815bd7171c4bcd3c06327afd141e68b17a0b46368b56","family_id":"qf-mod29-eqclass-membership","seed":202610297033,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0034","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4730df183219378f418b6337f0ea79a630c0f212ec5486a3e456bb0001e2d966","family_id":"qf-mod29-eqclass-membership","seed":202610297034,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0035","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6eb8cc8205d82d714f777c99cc9a17da6b969c828de72eecf0c072510936f336","family_id":"qf-mod29-eqclass-membership","seed":202610297035,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0036","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17e84cf39f21dae998608e27bfdf759a39028f2744f5c0935df0277a8dfe7901","family_id":"qf-mod29-eqclass-membership","seed":202610297036,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0037","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01460fe96e8cf4dd27eaa1c2c61b1aba4d66059c73d8dde5a097e0d5018d7e9b","family_id":"qf-mod29-eqclass-membership","seed":202610297037,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0038","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6caab2311d7166c51d6bf853f1b1ea8de0be80f953c010a47604937c71208b8","family_id":"qf-mod29-eqclass-membership","seed":202610297038,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0039","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83e4d2aa709eee075ccf17661ccc343748e3491295bf79a9013d35c9aa601540","family_id":"qf-mod29-eqclass-membership","seed":202610297039,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0040","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb4860ccdd9c4e32568c53873f7083b32dd28b8adcb9cf6db6c5d4ba83567a0","family_id":"qf-mod29-eqclass-membership","seed":202610297040,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0041","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23a12bf8e727c212401fef0712cf8d1f3bbde303b02f85af1dd25cd2f35f2028","family_id":"qf-mod29-eqclass-membership","seed":202610297041,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0042","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cad08430cc304df8c2895df26e6ec42ee8a0f38ece3f9d39efba28d230f5e8df","family_id":"qf-mod29-eqclass-membership","seed":202610297042,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0043","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06f4bda20302ec2b08a9024124a97178d6f0ab9044753239ca850e0a507f5d46","family_id":"qf-mod29-eqclass-membership","seed":202610297043,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0044","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9431f8dbdef43dd029de5a6ce2ddd14186d72196f2f6e4911c6a2d96166d2a96","family_id":"qf-mod29-eqclass-membership","seed":202610297044,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0045","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c70cce568973f28c95fbe291aa30cdaa185e7e0db0ef6084d4aea92bca2c1dd","family_id":"qf-mod29-eqclass-membership","seed":202610297045,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":1,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0046","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b3d8ac3f9412235249284ae3bc3cb1f67d3b424dd98f9b1bdd0854a4be639af","family_id":"qf-mod29-eqclass-membership","seed":202610297046,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":2,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0047","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c19c7302fc24a5dcbc26217c3927088714ac7384043a7279af395a13121aab97","family_id":"qf-mod29-eqclass-membership","seed":202610297047,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":3,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0048","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf284eac4c67f0a89d7fd733b9f20ea9b5342122b25791afc849ec2cab7c0653","family_id":"qf-mod29-eqclass-membership","seed":202610297048,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":4,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0049","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4c1d616549ea4667f182ccab64594072e851cbe3f4246d4002fe146cc148c5c","family_id":"qf-mod29-eqclass-membership","seed":202610297049,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":4,\"element\":5,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, Yes)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0050","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1511ede0d9ed2e3b56efe92aa99aba5733d1b900174532067d3c433768611df","family_id":"qf-mod29-eqclass-membership","seed":202610297050,"semantic_template":"qf-mod29-eqclass-membership/{\"candidate\":1,\"element\":0,\"n\":6,\"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]]}"},"fields":{"title":"Relation Membership and Class Membership Must Agree","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, No, No)","explanation":"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)."},"skills":["Relation Membership and Class Membership Must Agree"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0051","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05d238ef8f71b89f5871ce0d57b7e56ba57ed277f263ef0d97150f803293302b","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297051,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, Yes, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0052","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52a1c8b3644714f03cbeac209fd69f7e4b8da7c30a854228fd095c9359e7faba","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297052,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, Yes, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0053","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c80a16d5c86ef68377e3103b55ab70b14e220b08a23a6496ad808a74ca6c450e","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297053,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, No, [0, 2, 3, 4, 5], [1], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0054","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a271cf15d1d1c3aa5c94130dc59bb4e0e4522c6de5535a4ea35cc99b60307b9e","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297054,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, Yes, [2, 3, 4, 5], [2, 3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0055","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e48e7579090419b956a0f35d3aaed243143b37837ee19335184b2c67e9be521","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297055,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, No, [2, 3, 4, 5], [1], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0056","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ede27f2fc9a45a7dc43124e11a5c00e31862a3d962832f32b8a5d4dc15861c08","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297056,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0057","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b9a2a6ad6d055cdc96d5734ef73f932d728ce6c10d472b3c5aefa311e3d3100","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297057,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, No, [0, 2], [1, 3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0058","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d67ad53a99159113cea2af78f539d316c7629b0c0c12bec8abb0e2f28160797","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297058,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, Yes, [1, 3, 4, 5], [1, 3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0059","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9dc36bbd72c838d1483ce178da663ada26fe54bf2c5e960bf85f1bb2ade247a","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297059,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, Yes, [1, 2], [1, 2], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0060","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3664448df35947d2d9c7f990e2937bb3534f5c122307747391dd4ee59f36b6c6","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297060,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, Yes, [3, 4, 5], [3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0061","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"129801188e6930930fcd651a51f8b23025772b9f416a143cebba49cac80edbd7","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297061,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, No, [3, 4, 5], [1, 2], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0062","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17fc2b2118330a021396e0febd4d70ee03e264b6efd424ebb204e4a67d8fba03","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297062,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 3, 4, 5], [0, 3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0063","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"800ba32d3c7494183effa8d0fa947f3c6bab2f6cb5ffd400c1291d14a1e0f9fc","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297063,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, No, [0, 2], [1], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0064","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"909b9e21f7355fae93ad2316eae9b6b5488ad1dcc41cecb388ac168bf853c359","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297064,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, No, [0, 1], [3, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0065","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfcb539af1483e8937e450a8cffe98ea171940c24e297ef9536fc1cd203b753b","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297065,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, Yes, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0066","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"523520abd63c393ab8a76f5a90176deb7deb3ddfa84404df74fe89c9d39814f8","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297066,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, No, [0, 3], [1, 2, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0067","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"825a831f75b15e0a120188438263138972e0a8a7aff831fd5e4a4fe8b7279075","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297067,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, Yes, [1, 2, 4, 5], [1, 2, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0068","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08d8b7df6346b49322020646ae104fbbb5deeeaf3a83d5b2adec30cd6c259793","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297068,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 2, 4, 5], [0, 2, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0069","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21957e6284c46ebb2b626faca5ff48194e79e843fa983be229b848c7dd912cbd","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297069,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, Yes, [0, 1, 3], [0, 1, 3], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0070","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da60a546be6f81740189599dd3b6e109a6220cbb03994a6611ee39df8c5b3d4d","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297070,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(1, 4, No, [1, 3], [2, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0071","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07297b24d72c6a24b0e09f82bb95c550542c8d58c036ce2f40f95c89e882df08","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297071,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(2, 1, No, [0, 2, 4, 5], [1], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0072","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0d51a000a220780873df742edce1665a6da8f3b189a4b9f9e1a78efe1c43c13","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297072,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(3, 4, No, [0, 3], [2, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0073","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be0d611ffb3e4de64f840350a50610072cb6eef4548e426a9f93fff3d6de9610","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297073,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(4, 1, No, [2, 4, 5], [0, 1], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0074","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f66ae90e9d043c2c89b94c55ce1ec8f14fbed0df036bdeefe437daefe14e849d","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297074,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(5, 4, Yes, [0, 1, 4, 5], [0, 1, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0075","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52240ecbda839c313b274b8c6e64afa733506e1ccbb7a5435a9694792f95951c","family_id":"qf-mod29-eqclass-representative-invariance","seed":202610297075,"semantic_template":"qf-mod29-eqclass-representative-invariance/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Representative Invariance Is Conditional on Relatedness","stem":"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?).","answer":{"kind":"expression","value":"(0, 1, No, [0, 2, 3], [1, 4, 5], Yes)","explanation":"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)."},"skills":["Representative Invariance Is Conditional on Relatedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0076","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"543f725c473e8dbfed1856d94dd5a6aefba0a655c4eefff3065ac7f882c1e05c","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297076,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(0, 1, [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0077","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a130a5b6ac14479616634ef727a907d671cda10143b40fdc1fd3862bf6f38511","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297077,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(1, 4, [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0078","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddc7626c3dd95ecfe0b3569dc1b42f9a72f5f7df1203f71fcb68b4012341f9aa","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297078,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(2, 1, [0, 2, 3, 4, 5], [1], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0079","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"570e7f38506d5799abb61bd47cd7553178d01417b0431036bb347a439ecda478","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297079,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(3, 4, [2, 3, 4, 5], [2, 3, 4, 5], [2, 3, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0080","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6f6f2a7f192005e5b61dd96926597fb4ac409c80f55096ba5b18b0ed2adcdd1","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297080,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(4, 1, [2, 3, 4, 5], [1], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0081","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"883da3c05938a53e39dfedc1ef1ba5e8675b1e9fc7e59ac38fbbec8f43d4bb94","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297081,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(5, 4, [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], [0, 1, 3, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0082","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"966a8d9baa90e9fbf48c0c85571269dff2b762d33aad9e73255e359ce110329f","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297082,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(0, 1, [0, 2], [1, 3, 4, 5], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0083","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d2195902416e77ea65285b90e0a97fa9e01af19c9f6eec23f6da1b3146bbb5b","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297083,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(1, 4, [1, 3, 4, 5], [1, 3, 4, 5], [1, 3, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0084","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"882dd22667eab9483ca5c31b774b4ae70a14b6c10bddacc5bc827807d82a9c17","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297084,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(2, 1, [1, 2], [1, 2], [1, 2], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0085","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1920a3bed6a5256dffa6141131a8c6461e9e52b8af2361698d841c39eea1940","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297085,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(3, 4, [3, 4, 5], [3, 4, 5], [3, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0086","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f26a0c4ee40bf1082357a592aaa8a916ce2ebf58db877035607d8e41a46bd051","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297086,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(4, 1, [3, 4, 5], [1, 2], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0087","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a3f3e5fe761cc3c896ff8d923901bd2f1143ea5b6c3fd4b191f62a3445f36d4","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297087,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(5, 4, [0, 3, 4, 5], [0, 3, 4, 5], [0, 3, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0088","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e637cc92d5632fb5c5aed5764ffa6f4e16631993159ffbf27c69612367c54ab3","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297088,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(0, 1, [0, 2], [1], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0089","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830d4b0661fd1326a686bfafeb8ffdb3fc6ea09d755ee83c760f2a2c6d1a344b","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297089,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(1, 4, [0, 1], [3, 4, 5], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0090","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad108fdee97be23f90f8fffe3b51970b88fab3a987ba60711d884f276644a52f","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297090,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(2, 1, [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], [0, 1, 2, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0091","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fb828953d3865421289dbcac8aa8133d85a2b815576b54182a741d753e1346d","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297091,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(3, 4, [0, 3], [1, 2, 4, 5], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0092","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca54e6b1622afe74e95684391ed8ad599b3fa97ea787fa66008e03c2b3859d7e","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297092,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(4, 1, [1, 2, 4, 5], [1, 2, 4, 5], [1, 2, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0093","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb82fceef9443f08ebe5e944132dd2cb48ccc7b41c50b293502ab7dafb7ee027","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297093,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(5, 4, [0, 2, 4, 5], [0, 2, 4, 5], [0, 2, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0094","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a722e36f1917b25e873b81d9bf6b2ed8bf601da314662dfee8993db5408d0195","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297094,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(0, 1, [0, 1, 3], [0, 1, 3], [0, 1, 3], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0095","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee8e4994071a0aac56775a8574c118e62953641495d0bca4f8d70cdac13da82","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297095,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":1,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(1, 4, [1, 3], [2, 4, 5], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0096","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3bfe6026117029dd4c63b4ff614a6e9906d88f5a3862949fc07fad535e98373","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297096,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":2,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(2, 1, [0, 2, 4, 5], [1], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0097","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93cceea71e56ee579f93abe94b61e3e4535a43ec06429700b93102488ed37906","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297097,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":3,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(3, 4, [0, 3], [2, 4, 5], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0098","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2af16102ec9fa170fa190077cf5d0d4543395302b986894625cd18f9adfeb50a","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297098,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":4,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(4, 1, [2, 4, 5], [0, 1], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0099","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91a67e4d09a13ec9aa4668fce23979921d9be1b0750bcdc91ce7c50f0f1d0d2b","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297099,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":5,\"b\":4,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(5, 4, [0, 1, 4, 5], [0, 1, 4, 5], [0, 1, 4, 5], Yes, equal)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261029-r07-equivalence-class-0100","batch_id":"batch-20261014-advance-20261029-r07-equivalence-class-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8abce7d0f4af1e7a50babca663a23c5fc5823587f78f9df94df98847a774d491","family_id":"qf-mod29-eqclass-equal-or-disjoint","seed":202610297100,"semantic_template":"qf-mod29-eqclass-equal-or-disjoint/{\"a\":0,\"b\":1,\"n\":6,\"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]]}"},"fields":{"title":"Two Certified Classes Are Equal or Disjoint","stem":"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.","answer":{"kind":"expression","value":"(0, 1, [0, 2, 3], [1, 4, 5], [], Yes, disjoint)","explanation":"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)."},"skills":["Two Certified Classes Are Equal or Disjoint"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}