{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261003-advance-20261004-r03-relation-0079", "pilot_number": "R0068-34954", "number": 34954, "family_id": "qf-relation-converse-set", "title": { "zh-Hant": "第 34954 題：二元關係的逆關係", "en": "Question 34954: Converse of a Binary Relation" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261003-advance-20261004-r03-relation-0079", "batch_id": "batch-20261003-advance-20261004-r03-relation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-relation" ], "title_zh": "二元關係的逆關係", "stem_zh": "二元關係的來源 A={-9, -8, -7}、目標 B={-5, -4, -3}，R={(-9, -4), (-8, -4)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？", "answer": { "kind": "expression", "value": "{(-4, -9), (-4, -8)}", "explanation_zh": "R⁻¹={(-4, -9), (-4, -8)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "二元關係的逆關係" ] }, "variant": { "family_id": "qf-relation-converse-set", "seed": 202610043079, "parameters": { "A": [ -9, -8, -7 ], "B": [ -5, -4, -3 ], "R": [ [ -9, -4 ], [ -8, -4 ] ], "operation": "converse", "semantic_template": "qf-relation-converse-set/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"converse\"}" } }, "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-0079", "batch_id": "batch-20261003-advance-20261004-r03-relation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "7820f592214397b925248f579df1c220b58b08cb72d578fa5a241695b64af1ae", "family_id": "qf-relation-converse-set", "seed": 202610043079, "semantic_template": "qf-relation-converse-set/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"converse\"}" }, "fields": { "title": "Converse of a Binary Relation", "stem": "The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.", "answer": { "kind": "expression", "value": "{(-4, -9), (-4, -8)}", "explanation": "Exchange every (x,y) for (y,x), giving R⁻¹={(-4, -9), (-4, -8)}. A converse relation is not automatically an inverse function." }, "skills": [ "Converse 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={-9, -8, -7}、目標 B={-5, -4, -3}，R={(-9, -4), (-8, -4)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？", "en": "The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-relation-converse-set, [A, B, R, operation]]", "after": "derived = [converse, 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, [-9, -8, -7]]", "after": "derived = [[-9, -8, -7]]", "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, [-5, -4, -3]]", "after": "derived = [[-5, -4, -3]]", "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 = [0, [-9, -4]]", "after": "derived = [[-9, -4]]", "relation": "implies", "rule_id": "Relation_RAW_EVERY_pair_typed_BEFORE_dedup", "conditions": [], "explanation": { "zh-Hant": "先驗全部raw R的每對恰兩個SAFE座標，再去重；重複同一pair合法，但不能利用dedup藏起未驗cell。", "en": "Type EVERY raw R pair as EXACTLY TWO SAFE coordinates BEFORE deduplication.Identical repeated pairs are legal,but deduplication must not hide an unvalidated cell." }, "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 = [1, [-8, -4]]", "after": "derived = [[-8, -4]]", "relation": "implies", "rule_id": "Relation_RAW_EVERY_pair_typed_BEFORE_dedup", "conditions": [], "explanation": { "zh-Hant": "先驗全部raw R的每對恰兩個SAFE座標，再去重；重複同一pair合法，但不能利用dedup藏起未驗cell。", "en": "Type EVERY raw R pair as EXACTLY TWO SAFE coordinates BEFORE deduplication.Identical repeated pairs are legal,but deduplication must not hide an unvalidated cell." }, "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": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [[-9, -8, -7], [-5, -4, -3], [[-9, -4], [-8, -4]]]", "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": "S007", "uses": [ "G001", "S006" ], "before": "source roles = [-9, -4, A x B]", "after": "derived = [-4, -9, B x A]", "relation": "implies", "rule_id": "Relation_CONVERSE_EACH_pair_swaps_roles_and_carriers", "conditions": [], "explanation": { "zh-Hant": "每個原(x,y)改成(y,x)，source/target角色同步對調；不是僅反轉清單順序。", "en": "Swap EACH ORIGINAL(x,y)to(y,x),exchanging source/target roles too.This is not merely reversing the list order." }, "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": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [-8, -4, A x B]", "after": "derived = [-4, -8, B x A]", "relation": "implies", "rule_id": "Relation_CONVERSE_EACH_pair_swaps_roles_and_carriers", "conditions": [], "explanation": { "zh-Hant": "每個原(x,y)改成(y,x)，source/target角色同步對調；不是僅反轉清單順序。", "en": "Swap EACH ORIGINAL(x,y)to(y,x),exchanging source/target roles too.This is not merely reversing the list order." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "source roles = [[-5, -4, -3], [-9, -8, -7], [[-4, -9], [-4, -8]]]", "after": "derived = [{(-4, -9), (-4, -8)}]", "relation": "implies", "rule_id": "Relation_CONVERSE_numeric_lex_order_not_inverse_FUNCTION", "conditions": [], "explanation": { "zh-Hant": "以第一再第二座標的數值lex序顯示並去重。R⁻¹總是反向關係，不自動是反函數；空R仍空，不加未給雙射前提。", "en": "Display in NUMERIC lexicographic order by first,then second coordinate,without duplicates.R⁻¹ is a converse RELATION,not automatically an inverse FUNCTION.Empty R stays empty;do not invent a bijection premise." }, "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": "{(-4, -9), (-4, -8)}" }, "conclusion": { "canonical_value": "{(-4, -9), (-4, -8)}", "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": 79, "original_line_sha256": "d3b0431c74641d7890b2df116860dae4812a3f0c102f3793a1e7bbf1b601fd2e", "question_compact_sha256": "7820f592214397b925248f579df1c220b58b08cb72d578fa5a241695b64af1ae", "english_row_compact_sha256": "f0f5e8449fca0962c755f4ca40353d68116c3b24d0ee5ce860b6730a8b474450", "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": "07b78ddb5fb6c27b22e1f144cde7629c8472f51aec383d32c2475edfd6d674e8", "original_candidate_line_sha256": "07b78ddb5fb6c27b22e1f144cde7629c8472f51aec383d32c2475edfd6d674e8" }, "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": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "e8dc633a53c040fab447c3520f9f125553aa6c782ce7facd660888cad02f6dd6", "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." } }