{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0001","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A=∅、B=∅，函數表為 ∅。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"是","explanation_zh":"單射=是；滿射=是。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051001,"parameters":{"A":[],"B":[],"pairs":[],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[],\"operation\":\"injective\",\"pairs\":[]}"}},"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":"Codex independent producer (inherited model; 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exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0013","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A={-3, -2, -1, 0}、B={-5, -4, -3, -2, -1}，函數表為 {(-3, -1), (-2, -1), (-1, -2), (0, -2)}。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"否","explanation_zh":"單射=否；滿射=否。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051013,"parameters":{"A":[-3,-2,-1,0],"B":[-5,-4,-3,-2,-1],"pairs":[[-3,-1],[-2,-1],[-1,-2],[0,-2]],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[-3,-2,-1,0],\"operation\":\"injective\",\"pairs\":[[-3,-1],[-2,-1],[-1,-2],[0,-2]]}"}},"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":"Codex independent producer (inherited model; 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exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0018","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A={-5, -4, -3, -2, -1, 0}、B={0, 1, 2, 3, 4, 5}，函數表為 {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"是","explanation_zh":"單射=是；滿射=是。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051018,"parameters":{"A":[-5,-4,-3,-2,-1,0],"B":[0,1,2,3,4,5],"pairs":[[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[-5,-4,-3,-2,-1,0],\"operation\":\"injective\",\"pairs\":[[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"}},"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":"Codex independent producer (inherited model; 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exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0021","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A={-5, -4, -3, -2, -1, 0}、B={-7, -6, -5, -4, -3, -2, -1}，函數表為 {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"否","explanation_zh":"單射=否；滿射=否。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051021,"parameters":{"A":[-5,-4,-3,-2,-1,0],"B":[-7,-6,-5,-4,-3,-2,-1],"pairs":[[-5,-1],[-4,-1],[-3,-2],[-2,-2],[-1,-3],[0,-3]],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[-5,-4,-3,-2,-1,0],\"operation\":\"injective\",\"pairs\":[[-5,-1],[-4,-1],[-3,-2],[-2,-2],[-1,-3],[0,-3]]}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0022","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A={-6, -5, -4, -3, -2, -1, 0}、B={0, 1, 2, 3, 4, 5, 6}，函數表為 {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"是","explanation_zh":"單射=是；滿射=是。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051022,"parameters":{"A":[-6,-5,-4,-3,-2,-1,0],"B":[0,1,2,3,4,5,6],"pairs":[[-6,1],[-5,2],[-4,3],[-3,4],[-2,5],[-1,6],[0,0]],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"operation\":\"injective\",\"pairs\":[[-6,1],[-5,2],[-4,3],[-3,4],[-2,5],[-1,6],[0,0]]}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0023","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A={-6, -5, -4, -3, -2, -1, 0}、B={1, 3, 5, 7, 9, 11, 13, 15}，函數表為 {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"是","explanation_zh":"單射=是；滿射=否。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051023,"parameters":{"A":[-6,-5,-4,-3,-2,-1,0],"B":[1,3,5,7,9,11,13,15],"pairs":[[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11],[0,13]],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"operation\":\"injective\",\"pairs\":[[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11],[0,13]]}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0024","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A={-6, -5, -4, -3, -2, -1, 0}、B={0, 1, 2, 3, 4, 5}，函數表為 {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"否","explanation_zh":"單射=否；滿射=是。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051024,"parameters":{"A":[-6,-5,-4,-3,-2,-1,0],"B":[0,1,2,3,4,5],"pairs":[[-6,0],[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"operation\":\"injective\",\"pairs\":[[-6,0],[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0025","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"有限總函數的單射性","stem_zh":"明示總函數 f:A→B，A={-6, -5, -4, -3, -2, -1, 0}、B={-8, -7, -6, -5, -4, -3, -2, -1}，函數表為 {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}。f 是否為單射？只比較不同合法輸入，不以陪域大小作猜測。","answer":{"kind":"short_text","value":"否","explanation_zh":"單射=否；滿射=否。每個輸出纖維至多一個合法輸入才是單射；未覆蓋的陪域元素不會破壞單射。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限總函數的單射性"]},"variant":{"family_id":"qf-injective-table-property","seed":202610051025,"parameters":{"A":[-6,-5,-4,-3,-2,-1,0],"B":[-8,-7,-6,-5,-4,-3,-2,-1],"pairs":[[-6,-1],[-5,-1],[-4,-2],[-3,-2],[-2,-3],[-1,-3],[0,-4]],"operation":"injective","semantic_template":"qf-injective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"operation\":\"injective\",\"pairs\":[[-6,-1],[-5,-1],[-4,-2],[-3,-2],[-2,-3],[-1,-3],[0,-4]]}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}} {"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r01-injective-0026","batch_id":"batch-20261004-advance-20261005-r01-injective-foundations","language":"zh-Hant","source_mko_ids":["mko-injective"],"title_zh":"字典序首個碰撞見證","stem_zh":"明示總函數 f:A→B，A=∅、B=∅，函數表為 ∅。尋找 x