{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261029-r07-equivalence-class-0042", "pilot_number": "R0027-14242", "number": 14242, "family_id": "qf-mod29-eqclass-membership", "title": { "zh-Hant": "第 14242 題：等價類", "en": "Question 14242: Relation Membership and Class Membership Must Agree" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261029-r07-equivalence-class-0042", "batch_id": "batch-20261014-advance-20261029-r07-equivalence-class-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-equivalence-class" ], "title_zh": "等價類", "stem_zh": "同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"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]],\"element\":4,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。", "answer": { "kind": "expression", "value": "(4, 1, 是, 是)", "explanation_zh": "Wave4第三層核對=(4, 1, 是, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；class精確等於全部相關元素且代表元不變；任兩classes相等或不交且不從抽樣relation分群。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "等價類" ] }, "variant": { "family_id": "qf-mod29-eqclass-membership", "seed": 202610297042, "parameters": { "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 ] ], "element": 4, "candidate": 1, "domain": "Complete_finite_equivalence_class", "operation": "eqclass-membership", "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]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "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." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "同一完整載體上的已認證等價關係。操作=eqclass-membership；資料={\"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]],\"element\":4,\"candidate\":1}。請依 [element,candidate,related,in_class] 欄序回答。", "en": "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?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "complete carrier Fin(6);raw relation=[[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]]", "after": "sorted distinct ordered relation=[[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]];missing reflexive=[];missing reverses=[];missing transitive triples=[]", "relation": "implies", "rule_id": "equivalence_relation_complete_parent_qualification", "conditions": [], "explanation": { "zh-Hant": "完整核反身、對稱與每個可接續對的傳遞性後才談等價類；重複輸入不默默去重修補，不從抽樣或連通聚類猜等價關係。", "en": "Qualify reflexivity,symmetry and every composable pair's transitivity before forming classes.Do not silently deduplicate invalid inputs or infer equivalence from samples or connected clustering." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "first class:representative=4;carrier candidate=0", "after": "ordered relation pair present=否;class membership=否", "relation": "implies", "rule_id": "equivalence_class_every_carrier_element", "conditions": [], "explanation": { "zh-Hant": "依定義[a]={x:aRx}遍歷完整載體，不能只列幾個見證或依圖形直覺分群；反身性保證代表元本身在內。", "en": "By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "first class:representative=4;carrier candidate=1", "after": "ordered relation pair present=是;class membership=是", "relation": "implies", "rule_id": "equivalence_class_every_carrier_element", "conditions": [], "explanation": { "zh-Hant": "依定義[a]={x:aRx}遍歷完整載體，不能只列幾個見證或依圖形直覺分群；反身性保證代表元本身在內。", "en": "By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "first class:representative=4;carrier candidate=2", "after": "ordered relation pair present=是;class membership=是", "relation": "implies", "rule_id": "equivalence_class_every_carrier_element", "conditions": [], "explanation": { "zh-Hant": "依定義[a]={x:aRx}遍歷完整載體，不能只列幾個見證或依圖形直覺分群；反身性保證代表元本身在內。", "en": "By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "first class:representative=4;carrier candidate=3", "after": "ordered relation pair present=否;class membership=否", "relation": "implies", "rule_id": "equivalence_class_every_carrier_element", "conditions": [], "explanation": { "zh-Hant": "依定義[a]={x:aRx}遍歷完整載體，不能只列幾個見證或依圖形直覺分群；反身性保證代表元本身在內。", "en": "By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "first class:representative=4;carrier candidate=4", "after": "ordered relation pair present=是;class membership=是", "relation": "implies", "rule_id": "equivalence_class_every_carrier_element", "conditions": [], "explanation": { "zh-Hant": "依定義[a]={x:aRx}遍歷完整載體，不能只列幾個見證或依圖形直覺分群；反身性保證代表元本身在內。", "en": "By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "first class:representative=4;carrier candidate=5", "after": "ordered relation pair present=是;class membership=是", "relation": "implies", "rule_id": "equivalence_class_every_carrier_element", "conditions": [], "explanation": { "zh-Hant": "依定義[a]={x:aRx}遍歷完整載體，不能只列幾個見證或依圖形直覺分群；反身性保證代表元本身在內。", "en": "By[a]={x:aRx},inspect the entire carrier,not a few witnesses or visual grouping.Reflexivity includes the representative itself." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "first class:all carrier labels tested with representative=4", "after": "complete class in increasing order=[1, 2, 4, 5]", "relation": "implies", "rule_id": "class_complete_inventory_not_sample", "conditions": [], "explanation": { "zh-Hant": "完整載體已遍歷，清單是此代表元的整個等價類，不是局部樣本。", "en": "The entire carrier has been inspected;this list is the representative's full equivalence class,not a local sample." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "representative=4;candidate=1;whole class=[1, 2, 4, 5]", "after": "related=是;in class=是", "relation": "implies", "rule_id": "relation_lookup_equals_class_membership", "conditions": [], "explanation": { "zh-Hant": "直接查原有序關係對與已算完整類的成員測試，兩者由類的定義相同。", "en": "Compare the original ordered-pair lookup with membership in the complete class;they agree by the class definition." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 4, 1, true, true ] }, "conclusion": { "canonical_value": "(4, 1, 是, 是)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261029-r07-equivalence-class-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261029-r07-equivalence-class-foundations/questions.jsonl", "batch_sha256": "fd6f8798126c4a189ce511e6d9b05c53dd448b08ee855c00f5916688ea13fef0", "original_line": 42, "original_line_sha256": "97a9fb6e628b9e7d7aa372401b77ef1d4e879cd9ee6d967a53b6dcce51cf22fd", "question_compact_sha256": "cad08430cc304df8c2895df26e6ec42ee8a0f38ece3f9d39efba28d230f5e8df", "english_row_compact_sha256": "aab03f4c381963950d12370a554e284fd5ea1ee298e521142b3ed6f852003074", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261029-r07-equivalence-class-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261029-r07-equivalence-class-foundations.en.jsonl" }, "mko_bindings": { "mko-equivalence-class": { "sha256": "6ead5a95cd13a163d636e78db505bb8f7aad6744f9e25afee8148c838a3a44d9", "english_sha256": "6d92bfefd809a7377beec0c3584a0746a97568a9481e11f85ed0bd5ad27ee177", "original_path": "public/data/mko/equivalence-class.json", "english_path": "public/data/i18n/editions/mko-equivalence-class.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a8b6a7c538e8e95c336dfae507795cf07012b597b3dd6104fa3ecad599076f77", "original_candidate_sha256": "35e3c9082a23603e880c1af98c92d67f4ccca3a9fe53243751f7ebd9daa47eb3", "original_candidate_line_sha256": "35e3c9082a23603e880c1af98c92d67f4ccca3a9fe53243751f7ebd9daa47eb3" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-007-p28", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe", "question_first_witness_sha256": "a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "ef9b70ada2bb35d3ba4707a0f727abc19e372013d20874358fe70afff985b46e", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }