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{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261012-r04-set-membership-0003","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限型別查詢的資格與隸屬真假","stem_zh":"明示有限型別Fin(6)；完整特徵表bits=[0, 0, 0, 1, 1, 1]按索引從0給A的0／1隸屬，queries=[0, 1, 102]。依序列 (各查詢是否合型別, 合型別者的隸屬真假或不合型別)。","answer":{"kind":"expression","value":"([是, 是, 否], [假, 假, 不合型別])","explanation_zh":"型別資格與隸屬真假=([是, 是, 否], [假, 假, 不合型別])。先核查型別再讀完整特徵表，型別外不等於隸屬為假；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限型別查詢的資格與隸屬真假"]},"variant":{"family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124003,"parameters":{"n":6,"bits":[0,0,0,1,1,1],"queries":[0,1,102],"operation":"density12_membership_indexed_predicate_values","domain":"finite_typed_membership","semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"nonmember\",\"untyped\"]"}},"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-20261012-r04-set-membership-0021","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限型別查詢的資格與隸屬真假","stem_zh":"明示有限型別Fin(6)；完整特徵表bits=[0, 0, 0, 1, 1, 1]按索引從0給A的0／1隸屬，queries=[0, 1, 5]。依序列 (各查詢是否合型別, 合型別者的隸屬真假或不合型別)。","answer":{"kind":"expression","value":"([是, 是, 是], [假, 假, 真])","explanation_zh":"型別資格與隸屬真假=([是, 是, 是], [假, 假, 真])。先核查型別再讀完整特徵表，型別外不等於隸屬為假；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限型別查詢的資格與隸屬真假"]},"variant":{"family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124021,"parameters":{"n":6,"bits":[0,0,0,1,1,1],"queries":[0,1,5],"operation":"density12_membership_indexed_predicate_values","domain":"finite_typed_membership","semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"nonmember\",\"member\"]"}},"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-20261012-r04-set-membership-0022","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限型別查詢的資格與隸屬真假","stem_zh":"明示有限型別Fin(6)；完整特徵表bits=[0, 0, 0, 1, 1, 1]按索引從0給A的0／1隸屬，queries=[3, 1, 5]。依序列 (各查詢是否合型別, 合型別者的隸屬真假或不合型別)。","answer":{"kind":"expression","value":"([是, 是, 是], [真, 假, 真])","explanation_zh":"型別資格與隸屬真假=([是, 是, 是], [真, 假, 真])。先核查型別再讀完整特徵表，型別外不等於隸屬為假；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限型別查詢的資格與隸屬真假"]},"variant":{"family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124022,"parameters":{"n":6,"bits":[0,0,0,1,1,1],"queries":[3,1,5],"operation":"density12_membership_indexed_predicate_values","domain":"finite_typed_membership","semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"nonmember\",\"member\"]"}},"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-20261012-r04-set-membership-0023","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限型別查詢的資格與隸屬真假","stem_zh":"明示有限型別Fin(6)；完整特徵表bits=[0, 0, 0, 1, 1, 1]按索引從0給A的0／1隸屬，queries=[100, 4, 5]。依序列 (各查詢是否合型別, 合型別者的隸屬真假或不合型別)。","answer":{"kind":"expression","value":"([否, 是, 是], [不合型別, 真, 真])","explanation_zh":"型別資格與隸屬真假=([否, 是, 是], [不合型別, 真, 真])。先核查型別再讀完整特徵表，型別外不等於隸屬為假；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限型別查詢的資格與隸屬真假"]},"variant":{"family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124023,"parameters":{"n":6,"bits":[0,0,0,1,1,1],"queries":[100,4,5],"operation":"density12_membership_indexed_predicate_values","domain":"finite_typed_membership","semantic_template":"qf-density12-membership-indexed-predicate-values/[\"untyped\",\"member\",\"member\"]"}},"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-20261012-r04-set-membership-0024","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限型別查詢的資格與隸屬真假","stem_zh":"明示有限型別Fin(6)；完整特徵表bits=[0, 0, 0, 1, 1, 1]按索引從0給A的0／1隸屬，queries=[0, 4, 5]。依序列 (各查詢是否合型別, 合型別者的隸屬真假或不合型別)。","answer":{"kind":"expression","value":"([是, 是, 是], [假, 真, 真])","explanation_zh":"型別資格與隸屬真假=([是, 是, 是], [假, 真, 真])。先核查型別再讀完整特徵表，型別外不等於隸屬為假；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限型別查詢的資格與隸屬真假"]},"variant":{"family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124024,"parameters":{"n":6,"bits":[0,0,0,1,1,1],"queries":[0,4,5],"operation":"density12_membership_indexed_predicate_values","domain":"finite_typed_membership","semantic_template":"qf-density12-membership-indexed-predicate-values/[\"nonmember\",\"member\",\"member\"]"}},"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-20261012-r04-set-membership-0025","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限型別查詢的資格與隸屬真假","stem_zh":"明示有限型別Fin(1)；完整特徵表bits=[1]按索引從0給A的0／1隸屬，queries=[0, 0, 0]。依序列 (各查詢是否合型別, 合型別者的隸屬真假或不合型別)。","answer":{"kind":"expression","value":"([是, 是, 是], [真, 真, 真])","explanation_zh":"型別資格與隸屬真假=([是, 是, 是], [真, 真, 真])。先核查型別再讀完整特徵表，型別外不等於隸屬為假；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限型別查詢的資格與隸屬真假"]},"variant":{"family_id":"qf-density12-membership-indexed-predicate-values","seed":202610124025,"parameters":{"n":1,"bits":[1],"queries":[0,0,0],"operation":"density12_membership_indexed_predicate_values","domain":"finite_typed_membership","semantic_template":"qf-density12-membership-indexed-predicate-values/[\"member\",\"member\",\"member\"]"}},"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-20261012-r04-set-membership-0026","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[]，x=0。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [0])","explanation_zh":"隸屬資格與插入結果=(假, [0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124026,"parameters":{"A":[],"x":0,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[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-20261012-r04-set-membership-0027","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[0]，x=0。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [0])","explanation_zh":"隸屬資格與插入結果=(真, [0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124027,"parameters":{"A":[0],"x":0,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[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-20261012-r04-set-membership-0028","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-2, 0]，x=2。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-2, 0, 2])","explanation_zh":"隸屬資格與插入結果=(假, [-2, 0, 2])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124028,"parameters":{"A":[-2,0],"x":2,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-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-20261012-r04-set-membership-0029","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-3, 0]，x=-3。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-3, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-3, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124029,"parameters":{"A":[-3,0],"x":-3,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-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-20261012-r04-set-membership-0030","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-4, 0]，x=4。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-4, 0, 4])","explanation_zh":"隸屬資格與插入結果=(假, [-4, 0, 4])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124030,"parameters":{"A":[-4,0],"x":4,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-4,0,4]]"}},"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-20261012-r04-set-membership-0031","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-5, 0]，x=-5。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-5, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-5, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124031,"parameters":{"A":[-5,0],"x":-5,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-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-20261012-r04-set-membership-0032","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-6, 0]，x=6。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-6, 0, 6])","explanation_zh":"隸屬資格與插入結果=(假, [-6, 0, 6])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124032,"parameters":{"A":[-6,0],"x":6,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-6,0,6]]"}},"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-20261012-r04-set-membership-0033","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-7, 0]，x=-7。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-7, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-7, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124033,"parameters":{"A":[-7,0],"x":-7,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-7,0]]"}},"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-20261012-r04-set-membership-0034","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-8, 0]，x=8。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-8, 0, 8])","explanation_zh":"隸屬資格與插入結果=(假, [-8, 0, 8])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124034,"parameters":{"A":[-8,0],"x":8,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-8,0,8]]"}},"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-20261012-r04-set-membership-0035","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-9, 0]，x=-9。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-9, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-9, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124035,"parameters":{"A":[-9,0],"x":-9,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-9,0]]"}},"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-20261012-r04-set-membership-0036","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-10, 0]，x=10。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-10, 0, 10])","explanation_zh":"隸屬資格與插入結果=(假, [-10, 0, 10])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124036,"parameters":{"A":[-10,0],"x":10,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-10,0,10]]"}},"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-20261012-r04-set-membership-0037","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-11, 0]，x=-11。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-11, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-11, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124037,"parameters":{"A":[-11,0],"x":-11,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-11,0]]"}},"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-20261012-r04-set-membership-0038","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-12, 0]，x=12。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-12, 0, 12])","explanation_zh":"隸屬資格與插入結果=(假, [-12, 0, 12])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124038,"parameters":{"A":[-12,0],"x":12,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-12,0,12]]"}},"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-20261012-r04-set-membership-0039","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-13, 0]，x=-13。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-13, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-13, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124039,"parameters":{"A":[-13,0],"x":-13,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-13,0]]"}},"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-20261012-r04-set-membership-0040","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-14, 0]，x=14。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-14, 0, 14])","explanation_zh":"隸屬資格與插入結果=(假, [-14, 0, 14])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124040,"parameters":{"A":[-14,0],"x":14,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-14,0,14]]"}},"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-20261012-r04-set-membership-0041","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-15, 0]，x=-15。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-15, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-15, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124041,"parameters":{"A":[-15,0],"x":-15,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-15,0]]"}},"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-20261012-r04-set-membership-0042","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-16, 0]，x=16。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-16, 0, 16])","explanation_zh":"隸屬資格與插入結果=(假, [-16, 0, 16])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124042,"parameters":{"A":[-16,0],"x":16,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-16,0,16]]"}},"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-20261012-r04-set-membership-0043","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-17, 0]，x=-17。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-17, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-17, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124043,"parameters":{"A":[-17,0],"x":-17,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-17,0]]"}},"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-20261012-r04-set-membership-0044","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-18, 0]，x=18。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-18, 0, 18])","explanation_zh":"隸屬資格與插入結果=(假, [-18, 0, 18])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124044,"parameters":{"A":[-18,0],"x":18,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-18,0,18]]"}},"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-20261012-r04-set-membership-0045","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-19, 0]，x=-19。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-19, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-19, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124045,"parameters":{"A":[-19,0],"x":-19,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-19,0]]"}},"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-20261012-r04-set-membership-0046","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-20, 0]，x=20。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-20, 0, 20])","explanation_zh":"隸屬資格與插入結果=(假, [-20, 0, 20])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124046,"parameters":{"A":[-20,0],"x":20,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-20,0,20]]"}},"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-20261012-r04-set-membership-0047","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-21, 0]，x=-21。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-21, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-21, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124047,"parameters":{"A":[-21,0],"x":-21,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-21,0]]"}},"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-20261012-r04-set-membership-0048","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-22, 0]，x=22。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-22, 0, 22])","explanation_zh":"隸屬資格與插入結果=(假, [-22, 0, 22])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124048,"parameters":{"A":[-22,0],"x":22,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-22,0,22]]"}},"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-20261012-r04-set-membership-0049","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-23, 0]，x=-23。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(真, [-23, 0])","explanation_zh":"隸屬資格與插入結果=(真, [-23, 0])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124049,"parameters":{"A":[-23,0],"x":-23,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[true,[-23,0]]"}},"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-20261012-r04-set-membership-0050","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與插入後集合","stem_zh":"有限整數集合A=[-24, 0]，x=24。依序答 (原本x∈A的真假, 遞增去重A∪{x})。","answer":{"kind":"expression","value":"(假, [-24, 0, 24])","explanation_zh":"隸屬資格與插入結果=(假, [-24, 0, 24])。先查原集合，再插入且去重；已屬者不重列，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與插入後集合"]},"variant":{"family_id":"qf-density12-membership-set-insertion-delta","seed":202610124050,"parameters":{"A":[-24,0],"x":24,"operation":"density12_membership_set_insertion_delta","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-set-insertion-delta/[false,[-24,0,24]]"}},"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-20261012-r04-set-membership-0051","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[0]，x=0。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [])","explanation_zh":"隸屬資格與刪除結果=(真, [])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124051,"parameters":{"A":[0],"x":0,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[]]"}},"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-20261012-r04-set-membership-0052","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[]，x=99。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [])","explanation_zh":"隸屬資格與刪除結果=(假, [])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124052,"parameters":{"A":[],"x":99,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[]]"}},"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-20261012-r04-set-membership-0053","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-2, 0, 2]，x=3。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-2, 0, 2])","explanation_zh":"隸屬資格與刪除結果=(假, [-2, 0, 2])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124053,"parameters":{"A":[-2,0,2],"x":3,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-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-20261012-r04-set-membership-0054","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-3, 0, 3]，x=-3。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 3])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 3])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124054,"parameters":{"A":[-3,0,3],"x":-3,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[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-20261012-r04-set-membership-0055","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-4, 0, 4]，x=5。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-4, 0, 4])","explanation_zh":"隸屬資格與刪除結果=(假, [-4, 0, 4])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124055,"parameters":{"A":[-4,0,4],"x":5,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-4,0,4]]"}},"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-20261012-r04-set-membership-0056","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-5, 0, 5]，x=-5。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 5])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 5])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124056,"parameters":{"A":[-5,0,5],"x":-5,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[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-20261012-r04-set-membership-0057","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-6, 0, 6]，x=7。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-6, 0, 6])","explanation_zh":"隸屬資格與刪除結果=(假, [-6, 0, 6])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124057,"parameters":{"A":[-6,0,6],"x":7,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-6,0,6]]"}},"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-20261012-r04-set-membership-0058","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-7, 0, 7]，x=-7。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 7])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 7])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124058,"parameters":{"A":[-7,0,7],"x":-7,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[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-20261012-r04-set-membership-0059","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-8, 0, 8]，x=9。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-8, 0, 8])","explanation_zh":"隸屬資格與刪除結果=(假, [-8, 0, 8])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124059,"parameters":{"A":[-8,0,8],"x":9,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-8,0,8]]"}},"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-20261012-r04-set-membership-0060","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-9, 0, 9]，x=-9。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 9])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 9])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124060,"parameters":{"A":[-9,0,9],"x":-9,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[0,9]]"}},"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-20261012-r04-set-membership-0061","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-10, 0, 10]，x=11。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-10, 0, 10])","explanation_zh":"隸屬資格與刪除結果=(假, [-10, 0, 10])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124061,"parameters":{"A":[-10,0,10],"x":11,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-10,0,10]]"}},"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-20261012-r04-set-membership-0062","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-11, 0, 11]，x=-11。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 11])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 11])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124062,"parameters":{"A":[-11,0,11],"x":-11,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[0,11]]"}},"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-20261012-r04-set-membership-0063","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-12, 0, 12]，x=13。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-12, 0, 12])","explanation_zh":"隸屬資格與刪除結果=(假, [-12, 0, 12])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124063,"parameters":{"A":[-12,0,12],"x":13,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-12,0,12]]"}},"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-20261012-r04-set-membership-0064","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-13, 0, 13]，x=-13。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 13])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 13])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124064,"parameters":{"A":[-13,0,13],"x":-13,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[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-20261012-r04-set-membership-0065","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-14, 0, 14]，x=15。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-14, 0, 14])","explanation_zh":"隸屬資格與刪除結果=(假, [-14, 0, 14])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124065,"parameters":{"A":[-14,0,14],"x":15,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-14,0,14]]"}},"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-20261012-r04-set-membership-0066","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-15, 0, 15]，x=-15。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 15])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 15])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124066,"parameters":{"A":[-15,0,15],"x":-15,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[0,15]]"}},"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-20261012-r04-set-membership-0067","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-16, 0, 16]，x=17。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-16, 0, 16])","explanation_zh":"隸屬資格與刪除結果=(假, [-16, 0, 16])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124067,"parameters":{"A":[-16,0,16],"x":17,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-16,0,16]]"}},"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-20261012-r04-set-membership-0068","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-17, 0, 17]，x=-17。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 17])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 17])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124068,"parameters":{"A":[-17,0,17],"x":-17,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[0,17]]"}},"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-20261012-r04-set-membership-0069","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-18, 0, 18]，x=19。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-18, 0, 18])","explanation_zh":"隸屬資格與刪除結果=(假, [-18, 0, 18])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124069,"parameters":{"A":[-18,0,18],"x":19,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-18,0,18]]"}},"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-20261012-r04-set-membership-0070","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-19, 0, 19]，x=-19。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 19])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 19])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124070,"parameters":{"A":[-19,0,19],"x":-19,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[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-20261012-r04-set-membership-0071","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-20, 0, 20]，x=21。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-20, 0, 20])","explanation_zh":"隸屬資格與刪除結果=(假, [-20, 0, 20])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124071,"parameters":{"A":[-20,0,20],"x":21,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-20,0,20]]"}},"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-20261012-r04-set-membership-0072","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-21, 0, 21]，x=-21。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 21])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 21])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124072,"parameters":{"A":[-21,0,21],"x":-21,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[0,21]]"}},"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-20261012-r04-set-membership-0073","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-22, 0, 22]，x=23。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-22, 0, 22])","explanation_zh":"隸屬資格與刪除結果=(假, [-22, 0, 22])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124073,"parameters":{"A":[-22,0,22],"x":23,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-22,0,22]]"}},"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-20261012-r04-set-membership-0074","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-23, 0, 23]，x=-23。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(真, [0, 23])","explanation_zh":"隸屬資格與刪除結果=(真, [0, 23])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124074,"parameters":{"A":[-23,0,23],"x":-23,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[true,[0,23]]"}},"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-20261012-r04-set-membership-0075","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"原隸屬資格與刪除後集合","stem_zh":"有限整數集合A=[-24, 0, 24]，x=25。依序答 (原本x∈A的真假, 遞增A\\{x})，保留空集。","answer":{"kind":"expression","value":"(假, [-24, 0, 24])","explanation_zh":"隸屬資格與刪除結果=(假, [-24, 0, 24])。依原集合判隸屬，再排除等於x者；不屬A的候選不改集合，有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["原隸屬資格與刪除後集合"]},"variant":{"family_id":"qf-density12-membership-deletion-residual","seed":202610124075,"parameters":{"A":[-24,0,24],"x":25,"operation":"density12_membership_deletion_residual","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-deletion-residual/[false,[-24,0,24]]"}},"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-20261012-r04-set-membership-0076","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([], [])","explanation_zh":"有限謂詞外延與真假=([], [])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124076,"parameters":{"U":[],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[],[]]"}},"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-20261012-r04-set-membership-0077","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-2, 0, 2]，謂詞P(x)為2整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-2, 0, 2], [真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([-2, 0, 2], [真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124077,"parameters":{"U":[-2,0,2],"predicate":"divides","parameter":2,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-2,0,2],[true,true,true]]"}},"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-20261012-r04-set-membership-0078","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-1, 0, 1]，謂詞P(x)為x=0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0], [假, 真, 假])","explanation_zh":"有限謂詞外延與真假=([0], [假, 真, 假])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124078,"parameters":{"U":[-1,0,1],"predicate":"eq","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0],[false,true,false]]"}},"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-20261012-r04-set-membership-0079","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-3, -1, 0, 1, 3]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 3], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 3], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124079,"parameters":{"U":[-3,-1,0,1,3],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,3],[false,false,true,true,true]]"}},"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-20261012-r04-set-membership-0080","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-4, -1, 0, 1, 4]，謂詞P(x)為x=4。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([4], [假, 假, 假, 假, 真])","explanation_zh":"有限謂詞外延與真假=([4], [假, 假, 假, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124080,"parameters":{"U":[-4,-1,0,1,4],"predicate":"eq","parameter":4,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[4],[false,false,false,false,true]]"}},"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-20261012-r04-set-membership-0081","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-5, -1, 0, 1, 5]，謂詞P(x)為5整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-5, 0, 5], [真, 假, 真, 假, 真])","explanation_zh":"有限謂詞外延與真假=([-5, 0, 5], [真, 假, 真, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124081,"parameters":{"U":[-5,-1,0,1,5],"predicate":"divides","parameter":5,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-5,0,5],[true,false,true,false,true]]"}},"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-20261012-r04-set-membership-0082","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-6, -1, 0, 1, 6]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 6], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 6], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124082,"parameters":{"U":[-6,-1,0,1,6],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,6],[false,false,true,true,true]]"}},"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-20261012-r04-set-membership-0083","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-7, -1, 0, 1, 7]，謂詞P(x)為x=7。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([7], [假, 假, 假, 假, 真])","explanation_zh":"有限謂詞外延與真假=([7], [假, 假, 假, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124083,"parameters":{"U":[-7,-1,0,1,7],"predicate":"eq","parameter":7,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[7],[false,false,false,false,true]]"}},"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-20261012-r04-set-membership-0084","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-8, -1, 0, 1, 8]，謂詞P(x)為8整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-8, 0, 8], [真, 假, 真, 假, 真])","explanation_zh":"有限謂詞外延與真假=([-8, 0, 8], [真, 假, 真, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124084,"parameters":{"U":[-8,-1,0,1,8],"predicate":"divides","parameter":8,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-8,0,8],[true,false,true,false,true]]"}},"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-20261012-r04-set-membership-0085","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-9, -1, 0, 1, 9]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 9], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 9], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124085,"parameters":{"U":[-9,-1,0,1,9],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,9],[false,false,true,true,true]]"}},"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-20261012-r04-set-membership-0086","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-10, -1, 0, 1, 10]，謂詞P(x)為x=10。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([10], [假, 假, 假, 假, 真])","explanation_zh":"有限謂詞外延與真假=([10], [假, 假, 假, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124086,"parameters":{"U":[-10,-1,0,1,10],"predicate":"eq","parameter":10,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[10],[false,false,false,false,true]]"}},"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-20261012-r04-set-membership-0087","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-11, -1, 0, 1, 11]，謂詞P(x)為11整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-11, 0, 11], [真, 假, 真, 假, 真])","explanation_zh":"有限謂詞外延與真假=([-11, 0, 11], [真, 假, 真, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124087,"parameters":{"U":[-11,-1,0,1,11],"predicate":"divides","parameter":11,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-11,0,11],[true,false,true,false,true]]"}},"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-20261012-r04-set-membership-0088","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-12, -1, 0, 1, 12]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 12], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 12], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124088,"parameters":{"U":[-12,-1,0,1,12],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,12],[false,false,true,true,true]]"}},"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-20261012-r04-set-membership-0089","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-13, -1, 0, 1, 13]，謂詞P(x)為x=13。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([13], [假, 假, 假, 假, 真])","explanation_zh":"有限謂詞外延與真假=([13], [假, 假, 假, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124089,"parameters":{"U":[-13,-1,0,1,13],"predicate":"eq","parameter":13,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[13],[false,false,false,false,true]]"}},"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-20261012-r04-set-membership-0090","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-14, -1, 0, 1, 14]，謂詞P(x)為14整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-14, 0, 14], [真, 假, 真, 假, 真])","explanation_zh":"有限謂詞外延與真假=([-14, 0, 14], [真, 假, 真, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124090,"parameters":{"U":[-14,-1,0,1,14],"predicate":"divides","parameter":14,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-14,0,14],[true,false,true,false,true]]"}},"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-20261012-r04-set-membership-0091","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-15, -1, 0, 1, 15]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 15], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 15], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124091,"parameters":{"U":[-15,-1,0,1,15],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,15],[false,false,true,true,true]]"}},"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-20261012-r04-set-membership-0092","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-16, -1, 0, 1, 16]，謂詞P(x)為x=16。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([16], [假, 假, 假, 假, 真])","explanation_zh":"有限謂詞外延與真假=([16], [假, 假, 假, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124092,"parameters":{"U":[-16,-1,0,1,16],"predicate":"eq","parameter":16,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[16],[false,false,false,false,true]]"}},"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-20261012-r04-set-membership-0093","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-17, -1, 0, 1, 17]，謂詞P(x)為17整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-17, 0, 17], [真, 假, 真, 假, 真])","explanation_zh":"有限謂詞外延與真假=([-17, 0, 17], [真, 假, 真, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124093,"parameters":{"U":[-17,-1,0,1,17],"predicate":"divides","parameter":17,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-17,0,17],[true,false,true,false,true]]"}},"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-20261012-r04-set-membership-0094","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-18, -1, 0, 1, 18]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 18], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 18], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124094,"parameters":{"U":[-18,-1,0,1,18],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,18],[false,false,true,true,true]]"}},"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-20261012-r04-set-membership-0095","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-19, -1, 0, 1, 19]，謂詞P(x)為x=19。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([19], [假, 假, 假, 假, 真])","explanation_zh":"有限謂詞外延與真假=([19], [假, 假, 假, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124095,"parameters":{"U":[-19,-1,0,1,19],"predicate":"eq","parameter":19,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[19],[false,false,false,false,true]]"}},"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-20261012-r04-set-membership-0096","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-20, -1, 0, 1, 20]，謂詞P(x)為20整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-20, 0, 20], [真, 假, 真, 假, 真])","explanation_zh":"有限謂詞外延與真假=([-20, 0, 20], [真, 假, 真, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124096,"parameters":{"U":[-20,-1,0,1,20],"predicate":"divides","parameter":20,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-20,0,20],[true,false,true,false,true]]"}},"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-20261012-r04-set-membership-0097","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-21, -1, 0, 1, 21]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 21], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 21], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124097,"parameters":{"U":[-21,-1,0,1,21],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,21],[false,false,true,true,true]]"}},"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-20261012-r04-set-membership-0098","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-22, -1, 0, 1, 22]，謂詞P(x)為x=22。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([22], [假, 假, 假, 假, 真])","explanation_zh":"有限謂詞外延與真假=([22], [假, 假, 假, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124098,"parameters":{"U":[-22,-1,0,1,22],"predicate":"eq","parameter":22,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[22],[false,false,false,false,true]]"}},"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-20261012-r04-set-membership-0099","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-23, -1, 0, 1, 23]，謂詞P(x)為23整除x。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([-23, 0, 23], [真, 假, 真, 假, 真])","explanation_zh":"有限謂詞外延與真假=([-23, 0, 23], [真, 假, 真, 假, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124099,"parameters":{"U":[-23,-1,0,1,23],"predicate":"divides","parameter":23,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[-23,0,23],[true,false,true,false,true]]"}},"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-20261012-r04-set-membership-0100","batch_id":"batch-20261008-advance-20261012-r04-set-membership-foundations","language":"zh-Hant","source_mko_ids":["mko-set-membership"],"title_zh":"有限謂詞的外延與排序真假","stem_zh":"有限整數集合U=[-24, -1, 0, 1, 24]，謂詞P(x)為x≥0。先正規化U為遞增集合，依序答 (滿足P的完整外延A, 對齊排序U的真假列表)。","answer":{"kind":"expression","value":"([0, 1, 24], [假, 假, 真, 真, 真])","explanation_zh":"有限謂詞外延與真假=([0, 1, 24], [假, 假, 真, 真, 真])。逐一測明示U的可判定謂詞，整除使用非零除數；有限隸屬非全部集合論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限謂詞的外延與排序真假"]},"variant":{"family_id":"qf-density12-membership-predicate-extension","seed":202610124100,"parameters":{"U":[-24,-1,0,1,24],"predicate":"ge","parameter":0,"operation":"density12_membership_predicate_extension","domain":"finite_integer_membership","semantic_template":"qf-density12-membership-predicate-extension/[[0,1,24],[false,false,true,true,true]]"}},"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"}}