{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261012-advance-20261016-r07-surjective-0084", "pilot_number": "R0014-07784", "number": 7784, "family_id": "qf-density16-surj-two-target-preimages", "title": { "zh-Hant": "第 7784 題：兩個目標的原像與資格", "en": "Question 7784: Two Target-Type Guards, Complete Fibers, and Least Preimages" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261012-advance-20261016-r07-surjective-0084", "batch_id": "batch-20261012-advance-20261016-r07-surjective-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-surjective" ], "title_zh": "兩個目標的原像與資格", "stem_zh": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[0, 0, 1, 0, 0]。有序候選目標queries=[0, 1]。求 (原images, 兩個陪域資格, 兩個纖維或不合陪域, 各最小原像或不存在/不合陪域)。", "answer": { "kind": "expression", "value": "([0, 0, 1, 0, 0], [是, 是], [[0, 1, 3, 4], [2]], [0, 2])", "explanation_zh": "有限結構核對=([0, 0, 1, 0, 0], [是, 是], [[0, 1, 3, 4], [2]], [0, 2])。合型目標才掃完整表；合法空纖維答不存在，不和域外資格混同。有限精確核對不是一般形式證明。先資格後求值，空函數條件依宣告陪域。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "兩個目標的原像與資格" ] }, "variant": { "family_id": "qf-density16-surj-two-target-preimages", "seed": 202610167084, "parameters": { "n": 5, "m": 3, "images": [ 0, 0, 1, 0, 0 ], "queries": [ 0, 1 ], "domain": "Fin_total_table", "operation": "surj-two-target-preimages", "semantic_template": "qf-density16-surj-two-target-preimages/{\"images\":[0,0,1,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}" } }, "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-20261016-r07-surjective-0084", "batch_id": "batch-20261012-advance-20261016-r07-surjective-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "083f24d5f7e95ace803f94a1f24f51c12a3728618d25407f3d3469fcf045d574", "family_id": "qf-density16-surj-two-target-preimages", "seed": 202610167084, "semantic_template": "qf-density16-surj-two-target-preimages/{\"images\":[0,0,1,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}" }, "fields": { "title": "Two Target-Type Guards, Complete Fibers, and Least Preimages", "stem": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).", "answer": { "kind": "expression", "value": "([0, 0, 1, 0, 0], [Yes, Yes], [[0, 1, 3, 4], [2]], [0, 2])", "explanation": "Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [Yes, Yes], [[0, 1, 3, 4], [2]], [0, 2])." }, "skills": [ "Two Target-Type Guards, Complete Fibers, and Least Preimages" ], "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": "Fin型別從0起，完整總表；Fin(5)=[0, 1, 2, 3, 4]→Fin(3)=[0, 1, 2]，images=[0, 0, 1, 0, 0]。有序候選目標queries=[0, 1]。求 (原images, 兩個陪域資格, 兩個纖維或不合陪域, 各最小原像或不存在/不合陪域)。", "en": "Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "n=5;m=3;images=[0, 0, 1, 0, 0]", "after": "all original inputs and outputs form a typed total table;labels start at0", "relation": "implies", "rule_id": "finite_surjection_typed_scope", "conditions": [], "explanation": { "zh-Hant": "先核整個原表為Fin(n)到Fin(m)的總函數，包含不會被候選查詢的格；目標資格通過後才談其原像與滿射。", "en": "First validate the entire original table as a total function from Fin(n) to Fin(m),including cells not used by a query.Only then evaluate preimages for eligible targets and surjectivity." }, "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": "query0=0;codomain Fin(3);original images=[0, 0, 1, 0, 0]", "after": "eligible=是;fiber/status=[0, 1, 3, 4];least/status=0", "relation": "implies", "rule_id": "preimage_query_type_and_empty_distinction", "conditions": [], "explanation": { "zh-Hant": "先查0≤目標