{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261012-advance-20261017-r03-relation-0088", "pilot_number": "R0015-08363", "number": 8363, "family_id": "qf-density17-relation-single-valued-total", "title": { "zh-Hant": "第 8363 題：關係是否恰為總函數", "en": "Question 8363: Relation Single-Valuedness versus Total-Function Qualification" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261012-advance-20261017-r03-relation-0088", "batch_id": "batch-20261012-advance-20261017-r03-relation-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-relation" ], "title_zh": "關係是否恰為總函數", "stem_zh": "明示Fin來源與目標，有序關係對。來源Fin(3)=[0, 1, 2]，目標Fin(3)=[0, 1, 2]，關係R的列項=[[1, 0]]。答 ([n,m], 正規關係對, 每個來源纖維大小, 單值資格, 每來源恰一值的總函數資格, 總表或不是總函數)。空來源的逐項條件按真處理。", "answer": { "kind": "expression", "value": "([3, 3], [[1, 0]], [0, 1, 0], 是, 否, 不是總函數)", "explanation_zh": "離散結構核對=([3, 3], [[1, 0]], [0, 1, 0], 是, 否, 不是總函數)；每個來源最多一值才單值，每個來源恰一值才可列總表。；有限精確核對不是一般形式證明；逆關係不自動是反函數；關係不必是總函數；重複列項按集合正規化。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "關係是否恰為總函數" ] }, "variant": { "family_id": "qf-density17-relation-single-valued-total", "seed": 202610173088, "parameters": { "n": 3, "m": 3, "pairs": [ [ 1, 0 ] ], "domain": "Fin_typed_relation", "operation": "relation-single-valued-total", "semantic_template": "qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[1,0]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-12", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261012-advance-20261017-r03-relation-0088", "batch_id": "batch-20261012-advance-20261017-r03-relation-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "af3c565f3909fe0038d23497b219e187fa22c9d78aca1bdc391538ae2c121b53", "family_id": "qf-density17-relation-single-valued-total", "seed": 202610173088, "semantic_template": "qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[1,0]]}" }, "fields": { "title": "Relation Single-Valuedness versus Total-Function Qualification", "stem": "The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).", "answer": { "kind": "expression", "value": "([3, 3], [[1, 0]], [0, 1, 0], Yes, No, not-a-total-function)", "explanation": "At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 0]], [0, 1, 0], Yes, No, not-a-total-function)." }, "skills": [ "Relation Single-Valuedness versus Total-Function Qualification" ], "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": "明示Fin來源與目標，有序關係對。來源Fin(3)=[0, 1, 2]，目標Fin(3)=[0, 1, 2]，關係R的列項=[[1, 0]]。答 ([n,m], 正規關係對, 每個來源纖維大小, 單值資格, 每來源恰一值的總函數資格, 總表或不是總函數)。空來源的逐項條件按真處理。", "en": "The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "types=(3,3);raw ordered pairs=[[1, 0]]", "after": "normalized relation=[[1, 0]]", "relation": "implies", "rule_id": "typed_relation_set_normalization", "conditions": [], "explanation": { "zh-Hant": "先核每一有序對的來源與目標型別，再把重複的同一有序對去重。交換座標是逆關係，不是同一對的排序。關係可非單值或非總。", "en": "Validate both types of every ordered pair,then deduplicate identical pairs.Swapping coordinates forms the converse,not a sorting of one pair.A relation may be multivalued or partial." }, "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": "source=0;pairs=[[1, 0]]", "after": "outgoing values=[];count=0", "relation": "implies", "rule_id": "relation_each_source_value_fiber", "conditions": [], "explanation": { "zh-Hant": "對每個宣告來源收集所有不同目標，未出現的來源纖維為空。", "en": "Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber." }, "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": "source=1;pairs=[[1, 0]]", "after": "outgoing values=[0];count=1", "relation": "implies", "rule_id": "relation_each_source_value_fiber", "conditions": [], "explanation": { "zh-Hant": "對每個宣告來源收集所有不同目標，未出現的來源纖維為空。", "en": "Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber." }, "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": "source=2;pairs=[[1, 0]]", "after": "outgoing values=[];count=0", "relation": "implies", "rule_id": "relation_each_source_value_fiber", "conditions": [], "explanation": { "zh-Hant": "對每個宣告來源收集所有不同目標，未出現的來源纖維為空。", "en": "Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber." }, "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": "source fiber sizes=[0, 1, 0]", "after": "single-valued=是;exactly-one total=否;table/status=不是總函數", "relation": "implies", "rule_id": "partial_single_valued_not_total", "conditions": [], "explanation": { "zh-Hant": "最多一值允許缺來源，恰一值才是總函數；空來源的兩個全稱條件皆真且總表為空。重複同一對不增加纖維大小。", "en": "At most one value permits missing sources;exactly one makes a total function.On an empty source both universal conditions are true and the table is empty.Repeating a pair does not increase a fiber size." }, "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": [ [ 3, 3 ], [ [ 1, 0 ] ], [ 0, 1, 0 ], true, false, "不是總函數" ] }, "conclusion": { "canonical_value": "([3, 3], [[1, 0]], [0, 1, 0], 是, 否, 不是總函數)", "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-20261012-advance-20261017-r03-relation-foundations", "batch_path": "public/data/questions/batches/batch-20261012-advance-20261017-r03-relation-foundations/questions.jsonl", "batch_sha256": "fa93b079fe3332713b9e2dc835c033d2610769938b4950f23e6348ae0d11f894", "original_line": 88, "original_line_sha256": "d22e2efdb4a0a341df20a5304cf17ddf97d46acb5e6b0fd61636fa92f83433d6", "question_compact_sha256": "af3c565f3909fe0038d23497b219e187fa22c9d78aca1bdc391538ae2c121b53", "english_row_compact_sha256": "f0d0a128b4accb08032570312134ef45d2cad375dea5218e348f76093b7f7a28", "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261017-r03-relation-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-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": "ca1078dc467a0f3769865919aadc46c87712d098f1dfbde90f787471e22bcfec", "original_candidate_sha256": "7c0c6b2304d9e074bce2f223e7770adc7ff43e29c46c7d960f76b3ecb2394262", "original_candidate_line_sha256": "7c0c6b2304d9e074bce2f223e7770adc7ff43e29c46c7d960f76b3ecb2394262" }, "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-20261026-005-p16", "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": "6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18", "question_first_witness_sha256": "e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b" }, "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": "e0e6aefa96af3895456bb66b70a89be95811f6d824ecc5619fec193edfd7c64f", "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." } }