{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261003-advance-20261004-r03-relation-0032", "pilot_number": "R0068-34982", "number": 34982, "family_id": "qf-relation-domain-projection", "title": { "zh-Hant": "第 34982 題：二元關係的第一投影", "en": "Question 34982: First Projection of a Binary Relation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261003-advance-20261004-r03-relation-0032", "batch_id": "batch-20261003-advance-20261004-r03-relation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-relation" ], "title_zh": "二元關係的第一投影", "stem_zh": "二元關係的來源 A={-6, -5, -4}、目標 B={-2, -1, 0}，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。", "answer": { "kind": "expression", "value": "∅", "explanation_zh": "Dom(R)=∅。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "二元關係的第一投影" ] }, "variant": { "family_id": "qf-relation-domain-projection", "seed": 202610043032, "parameters": { "A": [ -6, -5, -4 ], "B": [ -2, -1, 0 ], "R": [], "operation": "domain_projection", "semantic_template": "qf-relation-domain-projection/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"domain_projection\"}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication" ], "warning_zh": "僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)", "generated_at": "2026-10-03", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261003-advance-20261004-r03-relation-0032", "batch_id": "batch-20261003-advance-20261004-r03-relation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "308c1bc2bb315cf79716f36c40e892e2ce53fe2a45d85f8a2a45402d942b9f2f", "family_id": "qf-relation-domain-projection", "seed": 202610043032, "semantic_template": "qf-relation-domain-projection/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"domain_projection\"}" }, "fields": { "title": "First Projection of a Binary Relation", "stem": "The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.", "answer": { "kind": "expression", "value": "∅", "explanation": "Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection." }, "skills": [ "First Projection of a Binary Relation" ], "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": "二元關係的來源 A={-6, -5, -4}、目標 B={-2, -1, 0}，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。", "en": "The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-relation-domain-projection, [A, B, R, operation]]", "after": "derived = [domain_projection, no-invented-domain]", "relation": "implies", "rule_id": "R68_BANK_specific_keys_operations_and_numeric_guards", "conditions": [], "explanation": { "zh-Hant": "按來源分bank：legacy raw SAFE/衍生BigInt無界，BASE Number.isInteger不偷加SAFE/百萬界。underscore operation、domain及寬/嚴keys各依本bank；原JSON完整保存。", "en": "Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [relation ORIGINAL source A, [-6, -5, -4]]", "after": "derived = [[-6, -5, -4]]", "relation": "implies", "rule_id": "LDL_RAW_distinct_SAFE_set_empty_allowed", "conditions": [], "explanation": { "zh-Hant": "本LDL set長0..100、元素raw SAFE且不重複，空集合合法；排序只屬衍生表示，原JSON順序不改。不是BASE無上限的另一set協定。", "en": "THIS LDL set has0..100 DISTINCT RAW SAFE labels,with the empty set allowed.Sort only the derived representation;preserve original JSON order.Do not import another BASE set protocol." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [relation ORIGINAL target B, [-2, -1, 0]]", "after": "derived = [[-2, -1, 0]]", "relation": "implies", "rule_id": "LDL_RAW_distinct_SAFE_set_empty_allowed", "conditions": [], "explanation": { "zh-Hant": "本LDL set長0..100、元素raw SAFE且不重複，空集合合法；排序只屬衍生表示，原JSON順序不改。不是BASE無上限的另一set協定。", "en": "THIS LDL set has0..100 DISTINCT RAW SAFE labels,with the empty set allowed.Sort only the derived representation;preserve original JSON order.Do not import another BASE set protocol." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [[-6, -5, -4], [-2, -1, 0], []]", "after": "derived = [是]", "relation": "implies", "rule_id": "Relation_COMPLETE_typed_subset_A_cross_B_NO_function_assumption", "conditions": [], "explanation": { "zh-Hant": "先驗完整R⊆A×B；關係不必是total function、單值、injective或onto。空R與空carrier在型別符合時合法，原raw重複/順序仍保存。", "en": "FIRST qualify the COMPLETE R⊆A×B.A relation need not be total,single-valued,injective or onto.Empty R/carriers are allowed when typed.Keep ORIGINAL raw duplicates/order in the source copy." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [[-6, -5, -4], [-2, -1, 0], []]", "after": "derived = [[], ∅]", "relation": "implies", "rule_id": "Relation_DOMAIN_actual_FIRST_coordinates_not_declared_A", "conditions": [], "explanation": { "zh-Hant": "Dom(R)只收真正出現在pair第一分量的值，每個一次；A可能有未參與的元素，所以不直接答A。R空時domain空。", "en": "Dom(R)collects ONLY first coordinates ACTUALLY appearing in pairs,once each.A may contain unused elements,so do not simply return A.Empty R has empty domain." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向核對本題的原始已知量、論域與本步條件；保留原題/英文/MKO/AST/狀態，不把單向推論倒成等價，也不把有限核對當獨立審查或一般形式證明。", "en": "Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof." } }, "arithmetic": null } ], "math_model": { "result": "∅" }, "conclusion": { "canonical_value": "∅", "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-20261003-advance-20261004-r03-relation-foundations", "batch_path": "public/data/questions/batches/batch-20261003-advance-20261004-r03-relation-foundations/questions.jsonl", "batch_sha256": "dafc3bbe4cc84e28621dd86797ac13842574a4a0040141c8131ecd0112d1b189", "original_line": 32, "original_line_sha256": "e30569022f63a5ceef331d93d9a76368421fd8810b06bfbb6002e5d07ca88177", "question_compact_sha256": "308c1bc2bb315cf79716f36c40e892e2ce53fe2a45d85f8a2a45402d942b9f2f", "english_row_compact_sha256": "a8504a861dd15b88f679c14416d7684f41699cde2bfc4b3cb498c49f1e02d501", "english_manifest_path": "public/data/i18n/editions/batch-20261003-advance-20261004-r03-relation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261003-advance-20261004-r03-relation-foundations.en.jsonl" }, "mko_bindings": { "mko-relation": { "sha256": "3788215401bc9a5610691985883d04b506880cc9bd5e2ce0a98054cd5a5f0774", "english_sha256": "7d0700fb988776905f5d6ce3bf7399bc7a370ea89a38fe4e6b4c9e8509acdfdb", "original_path": "public/data/mko/relation.json", "english_path": "public/data/i18n/editions/mko-relation.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "345142be27643acca9d0fec2422ac7311b85bf142709391cd6b5b3a33c03cf8e", "original_candidate_sha256": "b3649348e8611e90e54f73bbb698840bb0ba6aa8b293b6082cee6a0751cc18d0", "original_candidate_line_sha256": "b3649348e8611e90e54f73bbb698840bb0ba6aa8b293b6082cee6a0751cc18d0" }, "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-20261026-012-p69", "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": "e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed", "question_first_witness_sha256": "8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "0ad2a3db21080309b4235f495a8d79d027914c1d6740ebcd86d256dba34fce0d", "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." } }