{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0001","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [], 是, 是, 是, 同時開閉)","explanation_zh":"第二批擴展核對=(4, [], 是, 是, 是, 同時開閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191001,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"subset":[],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0003","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[1]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [1], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [1], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191003,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"subset":[1],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0004","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [0, 1], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191004,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"subset":[0,1],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0005","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [2], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [2], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191005,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"subset":[2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0006","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[0,2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 2], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [0, 2], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191006,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"subset":[0,2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0007","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[1,2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2], 是, 否, 否, 開)","explanation_zh":"第二批擴展核對=(4, [1, 2], 是, 否, 否, 開)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191007,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"subset":[1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0008","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1,2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2], 否, 是, 否, 閉)","explanation_zh":"第二批擴展核對=(4, [0, 1, 2], 否, 是, 否, 閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191008,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"subset":[0,1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0009","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[3]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [3], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [3], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191009,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"subset":[3],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0012","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,3]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 3], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [0, 1, 3], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191012,"parameters":{"n":4,"opens":[[],[3],[2,3],[0,2,3],[0,1,2,3]],"subset":[0,1,3],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0013","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[2,3]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [2, 3], 否, 是, 否, 閉)","explanation_zh":"第二批擴展核對=(4, [2, 3], 否, 是, 否, 閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191013,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,3],[0,1,2,3]],"subset":[2,3],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0014","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[0,2,3]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 2, 3], 否, 是, 否, 閉)","explanation_zh":"第二批擴展核對=(4, [0, 2, 3], 否, 是, 否, 閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191014,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"subset":[0,2,3],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0015","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[1,2,3]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 3], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [1, 2, 3], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191015,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"subset":[1,2,3],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0016","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,2,3]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3], 是, 是, 是, 同時開閉)","explanation_zh":"第二批擴展核對=(4, [0, 1, 2, 3], 是, 是, 是, 同時開閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191016,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"subset":[0,1,2,3],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0017","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [], 是, 是, 是, 同時開閉)","explanation_zh":"第二批擴展核對=(4, [], 是, 是, 是, 同時開閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191017,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"subset":[],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0018","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[0]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0], 否, 是, 否, 閉)","explanation_zh":"第二批擴展核對=(4, [0], 否, 是, 否, 閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191018,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"subset":[0],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0019","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[1]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [1], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [1], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191019,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"subset":[1],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0020","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [0, 1], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191020,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"subset":[0,1],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0021","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [2], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [2], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191021,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"subset":[2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0022","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 2], 否, 是, 否, 閉)","explanation_zh":"第二批擴展核對=(4, [0, 2], 否, 是, 否, 閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191022,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"subset":[0,2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0023","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[1,2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2], 否, 否, 否, 兩者皆非)","explanation_zh":"第二批擴展核對=(4, [1, 2], 否, 否, 否, 兩者皆非)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191023,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"subset":[1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0024","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1,2]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2], 否, 是, 否, 閉)","explanation_zh":"第二批擴展核對=(4, [0, 1, 2], 否, 是, 否, 閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191024,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"subset":[0,1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0025","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-status；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subset\":[]}。請依 [n,subset,open,closed,clopen,status] 欄序回答。","answer":{"kind":"expression","value":"(3, [], 是, 是, 是, 同時開閉)","explanation_zh":"第二批擴展核對=(3, [], 是, 是, 是, 同時開閉)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-status","seed":202610191025,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"subset":[],"domain":"Fin_topology_open_closed","operation":"openclosed-status","semantic_template":"qf-ext19-openclosed-status/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0026","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [], [0, 1, 2, 3], 是, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [], [0, 1, 2, 3], 是, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191026,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"subset":[],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0027","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[0]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0], [1, 2, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [0], [1, 2, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191027,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,2],[0,1,2,3]],"subset":[0],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0028","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[1]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [1], [0, 2, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [1], [0, 2, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191028,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"subset":[1],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0029","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1], [2, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [0, 1], [2, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191029,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"subset":[0,1],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0030","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [2], [0, 1, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [2], [0, 1, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191030,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"subset":[2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0031","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[0,2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 2], [1, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [0, 2], [1, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191031,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"subset":[0,2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0032","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[1,2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2], [0, 3], 是, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [1, 2], [0, 3], 是, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191032,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"subset":[1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0033","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1,2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2], [3], 否, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [0, 1, 2], [3], 否, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191033,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"subset":[0,1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0034","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[3]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [3], [0, 1, 2], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [3], [0, 1, 2], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191034,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"subset":[3],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0035","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,3]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 3], [1, 2], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [0, 3], [1, 2], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191035,"parameters":{"n":4,"opens":[[],[1],[1,3],[0,1,3],[0,1,2,3]],"subset":[0,3],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0036","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[1,3]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 3], [0, 2], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [1, 3], [0, 2], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191036,"parameters":{"n":4,"opens":[[],[2],[2,3],[0,2,3],[0,1,2,3]],"subset":[1,3],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[1,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0039","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[0,2,3]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 2, 3], [1], 否, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [0, 2, 3], [1], 否, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191039,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"subset":[0,2,3],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[0,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0040","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[1,2,3]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2, 3], [0], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [1, 2, 3], [0], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191040,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"subset":[1,2,3],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0041","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,2,3]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3], [], 是, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [0, 1, 2, 3], [], 是, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191041,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"subset":[0,1,2,3],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0042","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [], [0, 1, 2, 3], 是, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [], [0, 1, 2, 3], 是, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191042,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"subset":[],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0043","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[0]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0], [1, 2, 3], 否, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [0], [1, 2, 3], 否, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191043,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"subset":[0],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[0]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0044","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[1]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [1], [0, 2, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [1], [0, 2, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191044,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"subset":[1],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0045","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1], [2, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [0, 1], [2, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191045,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"subset":[0,1],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0046","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [2], [0, 1, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [2], [0, 1, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191046,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"subset":[2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0047","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 2], [1, 3], 否, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [0, 2], [1, 3], 否, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191047,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"subset":[0,2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0048","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[1,2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [1, 2], [0, 3], 否, 否, 否, 是)","explanation_zh":"第二批擴展核對=(4, [1, 2], [0, 3], 否, 否, 否, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191048,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"subset":[1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0049","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1,2]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(4, [0, 1, 2], [3], 否, 是, 是, 是)","explanation_zh":"第二批擴展核對=(4, [0, 1, 2], [3], 否, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191049,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"subset":[0,1,2],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1,2]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0050","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-complement-duality；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subset\":[]}。請依 [n,subset,complement,open,complement_open,closed,duality] 欄序回答。","answer":{"kind":"expression","value":"(3, [], [0, 1, 2], 是, 是, 是, 是)","explanation_zh":"第二批擴展核對=(3, [], [0, 1, 2], 是, 是, 是, 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-complement-duality","seed":202610191050,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"subset":[],"domain":"Fin_topology_open_closed","operation":"openclosed-complement-duality","semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subset\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0051","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191051,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0052","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191052,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,2],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0053","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191053,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0054","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191054,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0055","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191055,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0056","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191056,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0057","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191057,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0058","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191058,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0059","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191059,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0060","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191060,"parameters":{"n":4,"opens":[[],[1],[1,3],[0,1,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0061","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191061,"parameters":{"n":4,"opens":[[],[2],[2,3],[0,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0062","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191062,"parameters":{"n":4,"opens":[[],[3],[2,3],[0,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0063","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191063,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0064","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191064,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0065","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191065,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0066","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191066,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0067","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191067,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0068","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191068,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0069","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191069,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0070","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191070,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0071","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191071,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0072","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191072,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0073","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191073,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0074","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation_zh":"第二批擴展核對=(4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191074,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0075","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-clopen-inventory；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]]}。請依 [n,opens,closed,clopen] 欄序回答。","answer":{"kind":"expression","value":"(3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]])","explanation_zh":"第二批擴展核對=(3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]])；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191075,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"domain":"Fin_topology_open_closed","operation":"openclosed-clopen-inventory","semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0076","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]],\"mode\":\"open_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 1], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)","explanation_zh":"第二批擴展核對=(4, open_union, [[], [0, 1], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191076,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,2],[0,1,2,3]],"subsets":[[],[0,1],[0,1,2,3]],"mode":"open_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0077","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"subsets\":[[1],[0,1,2]],\"mode\":\"open_finite_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 2]], 是, [1], 是)","explanation_zh":"第二批擴展核對=(4, open_finite_intersection, [[1], [0, 1, 2]], 是, [1], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191077,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,2],[0,1,2,3]],"subsets":[[1],[0,1,2]],"mode":"open_finite_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0078","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 是, [], 是)","explanation_zh":"第二批擴展核對=(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 是, [], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191078,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,1,2],[0,1,2,3]],"subsets":[[],[1,3],[0,1,2,3]],"mode":"closed_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0079","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subsets\":[[2],[0,1,2]],\"mode\":\"closed_finite_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_finite_union, [[2], [0, 1, 2]], 是, [0, 1, 2], 是)","explanation_zh":"第二批擴展核對=(4, closed_finite_union, [[2], [0, 1, 2]], 是, [0, 1, 2], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191079,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,1,3],[0,1,2,3]],"subsets":[[2],[0,1,2]],"mode":"closed_finite_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subsets\":[[2],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0080","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,2],[0,1,2,3]],\"mode\":\"open_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 2], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)","explanation_zh":"第二批擴展核對=(4, open_union, [[], [0, 2], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191080,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,1,2],[0,1,2,3]],"subsets":[[],[0,2],[0,1,2,3]],"mode":"open_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0081","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subsets\":[[1],[0,1,2]],\"mode\":\"open_finite_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 2]], 是, [1], 是)","explanation_zh":"第二批擴展核對=(4, open_finite_intersection, [[1], [0, 1, 2]], 是, [1], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191081,"parameters":{"n":4,"opens":[[],[1],[1,2],[0,1,2],[0,1,2,3]],"subsets":[[1],[0,1,2]],"mode":"open_finite_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0082","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], 是, [], 是)","explanation_zh":"第二批擴展核對=(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], 是, [], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191082,"parameters":{"n":4,"opens":[[],[2],[1,2],[0,1,2],[0,1,2,3]],"subsets":[[],[0,3],[0,1,2,3]],"mode":"closed_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0083","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subsets\":[[2],[0,1,2]],\"mode\":\"closed_finite_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_finite_union, [[2], [0, 1, 2]], 是, [0, 1, 2], 是)","explanation_zh":"第二批擴展核對=(4, closed_finite_union, [[2], [0, 1, 2]], 是, [0, 1, 2], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191083,"parameters":{"n":4,"opens":[[],[3],[1,3],[0,1,3],[0,1,2,3]],"subsets":[[2],[0,1,2]],"mode":"closed_finite_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subsets\":[[2],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0084","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]],\"mode\":\"open_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 3], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)","explanation_zh":"第二批擴展核對=(4, open_union, [[], [0, 3], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191084,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,1,3],[0,1,2,3]],"subsets":[[],[0,3],[0,1,2,3]],"mode":"open_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0085","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"subsets\":[[1],[0,1,3]],\"mode\":\"open_finite_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 3]], 是, [1], 是)","explanation_zh":"第二批擴展核對=(4, open_finite_intersection, [[1], [0, 1, 3]], 是, [1], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191085,"parameters":{"n":4,"opens":[[],[1],[1,3],[0,1,3],[0,1,2,3]],"subsets":[[1],[0,1,3]],"mode":"open_finite_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"subsets\":[[1],[0,1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0086","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], 是, [], 是)","explanation_zh":"第二批擴展核對=(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], 是, [], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191086,"parameters":{"n":4,"opens":[[],[2],[2,3],[0,2,3],[0,1,2,3]],"subsets":[[],[0,1],[0,1,2,3]],"mode":"closed_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0087","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"subsets\":[[1],[0,1,2]],\"mode\":\"closed_finite_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_finite_union, [[1], [0, 1, 2]], 是, [0, 1, 2], 是)","explanation_zh":"第二批擴展核對=(4, closed_finite_union, [[1], [0, 1, 2]], 是, [0, 1, 2], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191087,"parameters":{"n":4,"opens":[[],[3],[2,3],[0,2,3],[0,1,2,3]],"subsets":[[1],[0,1,2]],"mode":"closed_finite_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0088","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]],\"mode\":\"open_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 1], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)","explanation_zh":"第二批擴展核對=(4, open_union, [[], [0, 1], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191088,"parameters":{"n":4,"opens":[[],[0],[0,1],[0,1,3],[0,1,2,3]],"subsets":[[],[0,1],[0,1,2,3]],"mode":"open_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0089","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subsets\":[[1],[0,1,3]],\"mode\":\"open_finite_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 3]], 是, [1], 是)","explanation_zh":"第二批擴展核對=(4, open_finite_intersection, [[1], [0, 1, 3]], 是, [1], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191089,"parameters":{"n":4,"opens":[[],[1],[0,1],[0,1,3],[0,1,2,3]],"subsets":[[1],[0,1,3]],"mode":"open_finite_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subsets\":[[1],[0,1,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0090","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 是, [], 是)","explanation_zh":"第二批擴展核對=(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], 是, [], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191090,"parameters":{"n":4,"opens":[[],[2],[0,2],[0,2,3],[0,1,2,3]],"subsets":[[],[1,3],[0,1,2,3]],"mode":"closed_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0091","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subsets\":[[1],[0,1,2]],\"mode\":\"closed_finite_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_finite_union, [[1], [0, 1, 2]], 是, [0, 1, 2], 是)","explanation_zh":"第二批擴展核對=(4, closed_finite_union, [[1], [0, 1, 2]], 是, [0, 1, 2], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191091,"parameters":{"n":4,"opens":[[],[3],[0,3],[0,2,3],[0,1,2,3]],"subsets":[[1],[0,1,2]],"mode":"closed_finite_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0092","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,2],[0,1,2,3]],\"mode\":\"open_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 2], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)","explanation_zh":"第二批擴展核對=(4, open_union, [[], [0, 2], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191092,"parameters":{"n":4,"opens":[[],[0],[0,2],[0,2,3],[0,1,2,3]],"subsets":[[],[0,2],[0,1,2,3]],"mode":"open_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,2],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0093","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subsets\":[[1],[1,2,3]],\"mode\":\"open_finite_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [1, 2, 3]], 是, [1], 是)","explanation_zh":"第二批擴展核對=(4, open_finite_intersection, [[1], [1, 2, 3]], 是, [1], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191093,"parameters":{"n":4,"opens":[[],[1],[1,2],[1,2,3],[0,1,2,3]],"subsets":[[1],[1,2,3]],"mode":"open_finite_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subsets\":[[1],[1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0094","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], 是, [], 是)","explanation_zh":"第二批擴展核對=(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], 是, [], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191094,"parameters":{"n":4,"opens":[[],[2],[1,2],[1,2,3],[0,1,2,3]],"subsets":[[],[0,3],[0,1,2,3]],"mode":"closed_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0095","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subsets\":[[0],[0,1,2]],\"mode\":\"closed_finite_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_finite_union, [[0], [0, 1, 2]], 是, [0, 1, 2], 是)","explanation_zh":"第二批擴展核對=(4, closed_finite_union, [[0], [0, 1, 2]], 是, [0, 1, 2], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191095,"parameters":{"n":4,"opens":[[],[3],[1,3],[1,2,3],[0,1,2,3]],"subsets":[[0],[0,1,2]],"mode":"closed_finite_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subsets\":[[0],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0096","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]],\"mode\":\"open_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 3], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)","explanation_zh":"第二批擴展核對=(4, open_union, [[], [0, 3], [0, 1, 2, 3]], 是, [0, 1, 2, 3], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191096,"parameters":{"n":4,"opens":[[],[0],[0,3],[0,2,3],[0,1,2,3]],"subsets":[[],[0,3],[0,1,2,3]],"mode":"open_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0097","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subsets\":[[1],[1,2,3]],\"mode\":\"open_finite_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [1, 2, 3]], 是, [1], 是)","explanation_zh":"第二批擴展核對=(4, open_finite_intersection, [[1], [1, 2, 3]], 是, [1], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191097,"parameters":{"n":4,"opens":[[],[1],[1,3],[1,2,3],[0,1,2,3]],"subsets":[[1],[1,2,3]],"mode":"open_finite_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subsets\":[[1],[1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0098","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]],\"mode\":\"closed_intersection\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], 是, [], 是)","explanation_zh":"第二批擴展核對=(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], 是, [], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191098,"parameters":{"n":4,"opens":[[],[2],[2,3],[1,2,3],[0,1,2,3]],"subsets":[[],[0,1],[0,1,2,3]],"mode":"closed_intersection","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0099","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subsets\":[[0],[0,1,2]],\"mode\":\"closed_finite_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(4, closed_finite_union, [[0], [0, 1, 2]], 是, [0, 1, 2], 是)","explanation_zh":"第二批擴展核對=(4, closed_finite_union, [[0], [0, 1, 2]], 是, [0, 1, 2], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191099,"parameters":{"n":4,"opens":[[],[3],[2,3],[1,2,3],[0,1,2,3]],"subsets":[[0],[0,1,2]],"mode":"closed_finite_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subsets\":[[0],[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261013-advance-20261019-r01-open-closed-set-0100","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","language":"zh-Hant","source_mko_ids":["mko-open-closed-set"],"title_zh":"開閉集有限核對","stem_zh":"完整有限拓撲。操作=openclosed-family-closure；資料={\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subsets\":[[],[0,1]],\"mode\":\"open_union\"}。請依 [n,mode,selected,all_qualified,result,result_qualified] 欄序回答。","answer":{"kind":"expression","value":"(3, open_union, [[], [0, 1]], 是, [0, 1], 是)","explanation_zh":"第二批擴展核對=(3, open_union, [[], [0, 1]], 是, [0, 1], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；closed不等於not open；補集相對同一底集合。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["開閉集有限核對"]},"variant":{"family_id":"qf-ext19-openclosed-family-closure","seed":202610191100,"parameters":{"n":3,"opens":[[],[0],[0,1],[0,1,2]],"subsets":[[],[0,1]],"mode":"open_union","domain":"Fin_topology_open_closed","operation":"openclosed-family-closure","semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subsets\":[[],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-13","contract_version":"ocme-question-factory-v0.1"}}