{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0001","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([], [])","explanation_zh":"真與假元素分割=([], [])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107001,"parameters":{"U":[],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[],[]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0002","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-2, 0, 2]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([-2, 0, 2], [])","explanation_zh":"真與假元素分割=([-2, 0, 2], [])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107002,"parameters":{"U":[-2,0,2],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[-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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0003","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-1, 0, 1]；可判定述詞P(x)為x=0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0], [-1, 1])","explanation_zh":"真與假元素分割=([0], [-1, 1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107003,"parameters":{"U":[-1,0,1],"predicate":"eq","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-1,1]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0004","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-3, -1, 0, 1, 3]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 3], [-3, -1])","explanation_zh":"真與假元素分割=([0, 1, 3], [-3, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107004,"parameters":{"U":[-3,-1,0,1,3],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,3],[-3,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0005","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-4, -1, 0, 1, 4]；可判定述詞P(x)為x=1。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([1], [-4, -1, 0, 4])","explanation_zh":"真與假元素分割=([1], [-4, -1, 0, 4])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107005,"parameters":{"U":[-4,-1,0,1,4],"predicate":"eq","parameter":1,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-4,-1,0,4]]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0006","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-5, -1, 0, 1, 5]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0], [-5, -1, 1, 5])","explanation_zh":"真與假元素分割=([0], [-5, -1, 1, 5])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107006,"parameters":{"U":[-5,-1,0,1,5],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-5,-1,1,5]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0007","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-6, -1, 0, 1, 6]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 6], [-6, -1])","explanation_zh":"真與假元素分割=([0, 1, 6], [-6, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107007,"parameters":{"U":[-6,-1,0,1,6],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,6],[-6,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0008","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-7, -1, 0, 1, 7]；可判定述詞P(x)為x=1。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([1], [-7, -1, 0, 7])","explanation_zh":"真與假元素分割=([1], [-7, -1, 0, 7])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107008,"parameters":{"U":[-7,-1,0,1,7],"predicate":"eq","parameter":1,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-7,-1,0,7]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0009","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-8, -1, 0, 1, 8]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([-8, 0, 8], [-1, 1])","explanation_zh":"真與假元素分割=([-8, 0, 8], [-1, 1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107009,"parameters":{"U":[-8,-1,0,1,8],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[-8,0,8],[-1,1]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0010","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-9, -1, 0, 1, 9]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 9], [-9, -1])","explanation_zh":"真與假元素分割=([0, 1, 9], [-9, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107010,"parameters":{"U":[-9,-1,0,1,9],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,9],[-9,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0011","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-10, -1, 0, 1, 10]；可判定述詞P(x)為x=1。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([1], [-10, -1, 0, 10])","explanation_zh":"真與假元素分割=([1], [-10, -1, 0, 10])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107011,"parameters":{"U":[-10,-1,0,1,10],"predicate":"eq","parameter":1,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-10,-1,0,10]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0012","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-11, -1, 0, 1, 11]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0], [-11, -1, 1, 11])","explanation_zh":"真與假元素分割=([0], [-11, -1, 1, 11])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107012,"parameters":{"U":[-11,-1,0,1,11],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-11,-1,1,11]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0013","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-12, -1, 0, 1, 12]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 12], [-12, -1])","explanation_zh":"真與假元素分割=([0, 1, 12], [-12, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107013,"parameters":{"U":[-12,-1,0,1,12],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,12],[-12,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0014","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-13, -1, 0, 1, 13]；可判定述詞P(x)為x=1。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([1], [-13, -1, 0, 13])","explanation_zh":"真與假元素分割=([1], [-13, -1, 0, 13])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107014,"parameters":{"U":[-13,-1,0,1,13],"predicate":"eq","parameter":1,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-13,-1,0,13]]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0015","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-14, -1, 0, 1, 14]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([-14, 0, 14], [-1, 1])","explanation_zh":"真與假元素分割=([-14, 0, 14], [-1, 1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107015,"parameters":{"U":[-14,-1,0,1,14],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[-14,0,14],[-1,1]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0016","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-15, -1, 0, 1, 15]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 15], [-15, -1])","explanation_zh":"真與假元素分割=([0, 1, 15], [-15, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107016,"parameters":{"U":[-15,-1,0,1,15],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,15],[-15,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0017","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-16, -1, 0, 1, 16]；可判定述詞P(x)為x=1。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([1], [-16, -1, 0, 16])","explanation_zh":"真與假元素分割=([1], [-16, -1, 0, 16])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107017,"parameters":{"U":[-16,-1,0,1,16],"predicate":"eq","parameter":1,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-16,-1,0,16]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0018","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-17, -1, 0, 1, 17]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0], [-17, -1, 1, 17])","explanation_zh":"真與假元素分割=([0], [-17, -1, 1, 17])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107018,"parameters":{"U":[-17,-1,0,1,17],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-17,-1,1,17]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0019","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-18, -1, 0, 1, 18]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 18], [-18, -1])","explanation_zh":"真與假元素分割=([0, 1, 18], [-18, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107019,"parameters":{"U":[-18,-1,0,1,18],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,18],[-18,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0020","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-19, -1, 0, 1, 19]；可判定述詞P(x)為x=1。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([1], [-19, -1, 0, 19])","explanation_zh":"真與假元素分割=([1], [-19, -1, 0, 19])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107020,"parameters":{"U":[-19,-1,0,1,19],"predicate":"eq","parameter":1,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-19,-1,0,19]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0021","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-20, -1, 0, 1, 20]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([-20, 0, 20], [-1, 1])","explanation_zh":"真與假元素分割=([-20, 0, 20], [-1, 1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107021,"parameters":{"U":[-20,-1,0,1,20],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[-20,0,20],[-1,1]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0022","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-21, -1, 0, 1, 21]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 21], [-21, -1])","explanation_zh":"真與假元素分割=([0, 1, 21], [-21, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107022,"parameters":{"U":[-21,-1,0,1,21],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,21],[-21,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0023","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-22, -1, 0, 1, 22]；可判定述詞P(x)為x=1。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([1], [-22, -1, 0, 22])","explanation_zh":"真與假元素分割=([1], [-22, -1, 0, 22])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107023,"parameters":{"U":[-22,-1,0,1,22],"predicate":"eq","parameter":1,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[1],[-22,-1,0,22]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0024","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-23, -1, 0, 1, 23]；可判定述詞P(x)為2整除x。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0], [-23, -1, 1, 23])","explanation_zh":"真與假元素分割=([0], [-23, -1, 1, 23])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107024,"parameters":{"U":[-23,-1,0,1,23],"predicate":"divides","parameter":2,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0],[-23,-1,1,23]]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0025","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"有限域的真假元素分割","stem_zh":"有限明示論域 U=[-24, -1, 0, 1, 24]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。","answer":{"kind":"expression","value":"([0, 1, 24], [-24, -1])","explanation_zh":"真與假元素分割=([0, 1, 24], [-24, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["有限域的真假元素分割"]},"variant":{"family_id":"qf-density10-quantifier-truth-partition","seed":202610107025,"parameters":{"U":[-24,-1,0,1,24],"predicate":"ge","parameter":0,"operation":"density10_quantifier_truth_partition","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-truth-partition/[[0,1,24],[-24,-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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0026","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(假, 真)","explanation_zh":"否定全稱與否定存在=(假, 真)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107026,"parameters":{"U":[],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0027","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[-1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 真)","explanation_zh":"否定全稱與否定存在=(真, 真)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107027,"parameters":{"U":[-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0028","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(假, 假)","explanation_zh":"否定全稱與否定存在=(假, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107028,"parameters":{"U":[1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0029","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[-2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 真)","explanation_zh":"否定全稱與否定存在=(真, 真)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107029,"parameters":{"U":[-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0030","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107030,"parameters":{"U":[1,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[1,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0031","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(假, 假)","explanation_zh":"否定全稱與否定存在=(假, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107031,"parameters":{"U":[1,2],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[2,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0032","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[-3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 真)","explanation_zh":"否定全稱與否定存在=(真, 真)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107032,"parameters":{"U":[-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0033","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107033,"parameters":{"U":[1,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[1,2]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0034","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107034,"parameters":{"U":[1,2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[2,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0035","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, 3]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(假, 假)","explanation_zh":"否定全稱與否定存在=(假, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107035,"parameters":{"U":[1,2,3],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[3,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0036","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[-4, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 真)","explanation_zh":"否定全稱與否定存在=(真, 真)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107036,"parameters":{"U":[-4,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0037","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107037,"parameters":{"U":[1,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[1,3]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0038","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107038,"parameters":{"U":[1,2,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[2,2]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0039","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, 3, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107039,"parameters":{"U":[1,2,3,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[3,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0040","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, 3, 4]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(假, 假)","explanation_zh":"否定全稱與否定存在=(假, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107040,"parameters":{"U":[1,2,3,4],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[4,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0041","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[-5, -4, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 真)","explanation_zh":"否定全稱與否定存在=(真, 真)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107041,"parameters":{"U":[-5,-4,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[0,5]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0042","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, -4, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107042,"parameters":{"U":[1,-4,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[1,4]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0043","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107043,"parameters":{"U":[1,2,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[2,3]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0044","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, 3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107044,"parameters":{"U":[1,2,3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[3,2]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0045","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, 3, 4, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107045,"parameters":{"U":[1,2,3,4,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[4,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0046","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, 3, 4, 5]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(假, 假)","explanation_zh":"否定全稱與否定存在=(假, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107046,"parameters":{"U":[1,2,3,4,5],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[5,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0047","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[-6, -5, -4, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 真)","explanation_zh":"否定全稱與否定存在=(真, 真)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107047,"parameters":{"U":[-6,-5,-4,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[0,6]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0048","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, -5, -4, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107048,"parameters":{"U":[1,-5,-4,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[1,5]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0049","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, -4, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107049,"parameters":{"U":[1,2,-4,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[2,4]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0050","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"全稱與存在的有序否定值","stem_zh":"有限明示論域 U=[1, 2, 3, -3, -2, -1]；可判定述詞P(x)為x≥0。依序答 (¬∀x∈U P(x), ¬∃x∈U P(x))，用真或假，保留空域約定。","answer":{"kind":"expression","value":"(真, 假)","explanation_zh":"否定全稱與否定存在=(真, 假)。有假元素才否定全稱，沒有真元素才否定存在；空域得(假, 真)，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全稱與存在的有序否定值"]},"variant":{"family_id":"qf-density10-quantifier-negation-pair","seed":202610107050,"parameters":{"U":[1,2,3,-3,-2,-1],"predicate":"ge","parameter":0,"operation":"density10_quantifier_negation_pair","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-negation-pair/[3,3]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0051","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[]；可判定述詞P(x)為x≥0。有序候選=[100, 101, 102]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"域內反例資格=[否, 否, 否]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107051,"parameters":{"U":[],"predicate":"ge","parameter":0,"candidates":[100,101,102],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"outside\",\"outside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0052","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[-1, 101, 102]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"域內反例資格=[是, 否, 否]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107052,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[-1,101,102],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"outside\",\"outside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0053","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[1, 2, 102]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 否, 否]","explanation_zh":"域內反例資格=[否, 否, 否]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107053,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[1,2,102],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"satisfies\",\"outside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0054","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[-1, 2, 102]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[是, 否, 否]","explanation_zh":"域內反例資格=[是, 否, 否]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107054,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[-1,2,102],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"satisfies\",\"outside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0055","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[100, -2, 102]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 是, 否]","explanation_zh":"域內反例資格=[否, 是, 否]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107055,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[100,-2,102],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"violates\",\"outside\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0068","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[1, 101, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 否, 是]","explanation_zh":"域內反例資格=[否, 否, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107068,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[1,101,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"outside\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0069","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[-1, 101, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[是, 否, 是]","explanation_zh":"域內反例資格=[是, 否, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107069,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[-1,101,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"outside\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0070","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[100, 2, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 否, 是]","explanation_zh":"域內反例資格=[否, 否, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107070,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[100,2,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"satisfies\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0071","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[1, 2, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 否, 是]","explanation_zh":"域內反例資格=[否, 否, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107071,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[1,2,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"satisfies\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0072","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[-1, 2, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[是, 否, 是]","explanation_zh":"域內反例資格=[是, 否, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107072,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[-1,2,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"satisfies\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0073","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[100, -2, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 是, 是]","explanation_zh":"域內反例資格=[否, 是, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107073,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[100,-2,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"outside\",\"violates\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0074","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[1, -2, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[否, 是, 是]","explanation_zh":"域內反例資格=[否, 是, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107074,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[1,-2,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"satisfies\",\"violates\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0075","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"三個候選的域內反例資格","stem_zh":"有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[-1, -2, -3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。","answer":{"kind":"expression","value":"[是, 是, 是]","explanation_zh":"域內反例資格=[是, 是, 是]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["三個候選的域內反例資格"]},"variant":{"family_id":"qf-density10-quantifier-candidate-mask","seed":202610107075,"parameters":{"U":[-3,-2,-1,1,2,3],"predicate":"ge","parameter":0,"candidates":[-1,-2,-3],"operation":"density10_quantifier_candidate_mask","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-candidate-mask/[\"violates\",\"violates\",\"violates\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0076","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[]；V=[]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107076,"parameters":{"U":[],"V":[],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[0,0,0]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0077","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[]；V=[-1, 1]，關係P(x,y)為x≤y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 2)","explanation_zh":"有序嵌套枚舉數=(0, 2)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107077,"parameters":{"U":[],"V":[-1,1],"relation":"le","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[[],[-1,1],\"le\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0078","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[1, 2]；V=[]，關係P(x,y)為x≤y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107078,"parameters":{"U":[1,2],"V":[],"relation":"le","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[[1,2],[],\"le\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0079","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-1, 0, 1]；V=[-2, 0, 2]，關係P(x,y)為x≤y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(3, 1)","explanation_zh":"有序嵌套枚舉數=(3, 1)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107079,"parameters":{"U":[-1,0,1],"V":[-2,0,2],"relation":"le","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[[-1,0,1],[-2,0,2],\"le\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0080","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-2, -1, 0, 1, 2]；V=[-1, 0, 1]，關係P(x,y)為x≤y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(4, 0)","explanation_zh":"有序嵌套枚舉數=(4, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107080,"parameters":{"U":[-2,-1,0,1,2],"V":[-1,0,1],"relation":"le","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[[-2,-1,0,1,2],[-1,0,1],\"le\"]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0081","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-1]；V=[100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107081,"parameters":{"U":[-1],"V":[100],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,1,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0082","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0]；V=[0]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 1)","explanation_zh":"有序嵌套枚舉數=(1, 1)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107082,"parameters":{"U":[0],"V":[0],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,1,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0083","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-1]；V=[100, 101]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107083,"parameters":{"U":[-1],"V":[100,101],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,2,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0084","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0]；V=[0, 100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 1)","explanation_zh":"有序嵌套枚舉數=(1, 1)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107084,"parameters":{"U":[0],"V":[0,100],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,2,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0085","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-1]；V=[100, 101, 102]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107085,"parameters":{"U":[-1],"V":[100,101,102],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,3,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0086","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0]；V=[0, 100, 101]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 1)","explanation_zh":"有序嵌套枚舉數=(1, 1)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107086,"parameters":{"U":[0],"V":[0,100,101],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,3,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0087","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-1]；V=[100, 101, 102, 103]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107087,"parameters":{"U":[-1],"V":[100,101,102,103],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,4,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0088","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0]；V=[0, 100, 101, 102]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 1)","explanation_zh":"有序嵌套枚舉數=(1, 1)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107088,"parameters":{"U":[0],"V":[0,100,101,102],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[1,4,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0089","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-2, -1]；V=[100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107089,"parameters":{"U":[-2,-1],"V":[100],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,1,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0090","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0, -1]；V=[0]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 0)","explanation_zh":"有序嵌套枚舉數=(1, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107090,"parameters":{"U":[0,-1],"V":[0],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,1,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0091","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-2, -1]；V=[100, 101]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107091,"parameters":{"U":[-2,-1],"V":[100,101],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,2,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0092","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0, -1]；V=[0, 100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 0)","explanation_zh":"有序嵌套枚舉數=(1, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107092,"parameters":{"U":[0,-1],"V":[0,100],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,2,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0093","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0, 1]；V=[0, 1]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(2, 0)","explanation_zh":"有序嵌套枚舉數=(2, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107093,"parameters":{"U":[0,1],"V":[0,1],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,2,2]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0094","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-2, -1]；V=[100, 101, 102]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107094,"parameters":{"U":[-2,-1],"V":[100,101,102],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,3,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0095","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0, -1]；V=[0, 100, 101]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 0)","explanation_zh":"有序嵌套枚舉數=(1, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107095,"parameters":{"U":[0,-1],"V":[0,100,101],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,3,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0096","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0, 1]；V=[0, 1, 100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(2, 0)","explanation_zh":"有序嵌套枚舉數=(2, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107096,"parameters":{"U":[0,1],"V":[0,1,100],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,3,2]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0097","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-2, -1]；V=[100, 101, 102, 103]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107097,"parameters":{"U":[-2,-1],"V":[100,101,102,103],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,4,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0098","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0, -1]；V=[0, 100, 101, 102]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(1, 0)","explanation_zh":"有序嵌套枚舉數=(1, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107098,"parameters":{"U":[0,-1],"V":[0,100,101,102],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,4,1]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0099","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[0, 1]；V=[0, 1, 100, 101]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(2, 0)","explanation_zh":"有序嵌套枚舉數=(2, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107099,"parameters":{"U":[0,1],"V":[0,1,100,101],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[2,4,2]"}},"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":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r07-quantifier-0100","batch_id":"batch-20261008-advance-20261010-r07-quantifier-foundations","language":"zh-Hant","source_mko_ids":["mko-quantifier"],"title_zh":"兩種有序嵌套枚舉的計數","stem_zh":"有限明示論域 U=[-3, -2, -1]；V=[100]，關係P(x,y)為x=y。依序求 (有某y∈V使P(x,y)的x∈U數, 對每個x∈U皆滿足P(x,y)的y∈V數)。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序嵌套枚舉數=(0, 0)。按指定內層存在或全稱逐項枚舉，空集全稱真、存在假；不交換量詞，不外推無限論域。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩種有序嵌套枚舉的計數"]},"variant":{"family_id":"qf-density10-quantifier-nested-counts","seed":202610107100,"parameters":{"U":[-3,-2,-1],"V":[100],"relation":"eq","operation":"density10_quantifier_nested_counts","domain":"finite_integer_domain","semantic_template":"qf-density10-quantifier-nested-counts/[3,1,0]"}},"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":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}