{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0001","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[0,1,2],[0,2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085001,"parameters":{"sets":[[0,1,2],[0,2,3]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[0,1,2],[0,2,3]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0002","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[1,2,3],[1,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085002,"parameters":{"sets":[[1,2,3],[1,3,4]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[1,2,3],[1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0003","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[2,3,4],[2,4,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085003,"parameters":{"sets":[[2,3,4],[2,4,5]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[2,3,4],[2,4,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0004","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[3,4,5],[3,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085004,"parameters":{"sets":[[3,4,5],[3,5,6]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[3,4,5],[3,5,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0005","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[4,5,6],[4,6,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085005,"parameters":{"sets":[[4,5,6],[4,6,7]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[4,5,6],[4,6,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0006","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[5,6,7],[5,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085006,"parameters":{"sets":[[5,6,7],[5,7,8]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[5,6,7],[5,7,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0007","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[6,7,8],[6,8,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085007,"parameters":{"sets":[[6,7,8],[6,8,9]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[6,7,8],[6,8,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0008","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[7,8,9],[7,9,10]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085008,"parameters":{"sets":[[7,8,9],[7,9,10]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[7,8,9],[7,9,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0009","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[8,9,10],[8,10,11]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085009,"parameters":{"sets":[[8,9,10],[8,10,11]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[8,10,11],[8,9,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0010","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[9,10,11],[9,11,12]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085010,"parameters":{"sets":[[9,10,11],[9,11,12]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[9,10,11],[9,11,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0011","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[10,11,12],[10,12,13]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085011,"parameters":{"sets":[[10,11,12],[10,12,13]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[10,11,12],[10,12,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0012","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[11,12,13],[11,13,14]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085012,"parameters":{"sets":[[11,12,13],[11,13,14]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[11,12,13],[11,13,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0013","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[12,13,14],[12,14,15]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085013,"parameters":{"sets":[[12,13,14],[12,14,15]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[12,13,14],[12,14,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0014","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[13,14,15],[13,15,16]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085014,"parameters":{"sets":[[13,14,15],[13,15,16]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[13,14,15],[13,15,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0015","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[14,15,16],[14,16,17]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085015,"parameters":{"sets":[[14,15,16],[14,16,17]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[14,15,16],[14,16,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0016","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[15,16,17],[15,17,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085016,"parameters":{"sets":[[15,16,17],[15,17,18]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[15,16,17],[15,17,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0017","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[16,17,18],[16,18,19]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085017,"parameters":{"sets":[[16,17,18],[16,18,19]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[16,17,18],[16,18,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0018","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[17,18,19],[17,19,20]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085018,"parameters":{"sets":[[17,18,19],[17,19,20]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[17,18,19],[17,19,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0019","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[18,19,20],[18,20,21]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085019,"parameters":{"sets":[[18,19,20],[18,20,21]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[18,19,20],[18,20,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0020","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[19,20,21],[19,21,22]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085020,"parameters":{"sets":[[19,20,21],[19,21,22]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[19,20,21],[19,21,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0021","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[20,21,22],[20,22,23]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085021,"parameters":{"sets":[[20,21,22],[20,22,23]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[20,21,22],[20,22,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0022","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[21,22,23],[21,23,24]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085022,"parameters":{"sets":[[21,22,23],[21,23,24]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[21,22,23],[21,23,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0023","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[22,23,24],[22,24,25]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085023,"parameters":{"sets":[[22,23,24],[22,24,25]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[22,23,24],[22,24,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0024","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[23,24,25],[23,25,26]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085024,"parameters":{"sets":[[23,24,25],[23,25,26]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[23,24,25],[23,25,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0025","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-two-set-formula；資料={\"sets\":[[24,25,26],[24,26,27]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, 是)","explanation_zh":"Wave6第二層核對=([3, 3], 2, 4, 4, 是)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-two-set-formula","seed":202611085025,"parameters":{"sets":[[24,25,26],[24,26,27]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-two-set-formula","semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[24,25,26],[24,26,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0026","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[0,1,2],[0,2,3],[0,2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085026,"parameters":{"sets":[[0,1,2],[0,2,3],[0,2,4]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[0,1,2],[0,2,3],[0,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0027","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[1,2,3],[1,3,4],[1,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085027,"parameters":{"sets":[[1,2,3],[1,3,4],[1,3,5]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[1,2,3],[1,3,4],[1,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0028","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085028,"parameters":{"sets":[[2,3,4],[2,4,5],[2,4,6]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0029","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[3,4,5],[3,5,6],[3,5,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085029,"parameters":{"sets":[[3,4,5],[3,5,6],[3,5,7]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[3,4,5],[3,5,6],[3,5,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0030","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[4,5,6],[4,6,7],[4,6,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085030,"parameters":{"sets":[[4,5,6],[4,6,7],[4,6,8]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[4,5,6],[4,6,7],[4,6,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0031","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[5,6,7],[5,7,8],[5,7,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085031,"parameters":{"sets":[[5,6,7],[5,7,8],[5,7,9]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[5,6,7],[5,7,8],[5,7,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0032","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[6,7,8],[6,8,9],[6,8,10]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085032,"parameters":{"sets":[[6,7,8],[6,8,9],[6,8,10]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[6,7,8],[6,8,10],[6,8,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0033","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[7,8,9],[7,9,10],[7,9,11]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085033,"parameters":{"sets":[[7,8,9],[7,9,10],[7,9,11]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[7,8,9],[7,9,10],[7,9,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0034","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[8,9,10],[8,10,11],[8,10,12]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085034,"parameters":{"sets":[[8,9,10],[8,10,11],[8,10,12]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[8,10,11],[8,10,12],[8,9,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0035","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[9,10,11],[9,11,12],[9,11,13]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085035,"parameters":{"sets":[[9,10,11],[9,11,12],[9,11,13]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[9,10,11],[9,11,12],[9,11,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0036","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085036,"parameters":{"sets":[[10,11,12],[10,12,13],[10,12,14]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0037","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[11,12,13],[11,13,14],[11,13,15]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085037,"parameters":{"sets":[[11,12,13],[11,13,14],[11,13,15]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[11,12,13],[11,13,14],[11,13,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0038","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[12,13,14],[12,14,15],[12,14,16]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085038,"parameters":{"sets":[[12,13,14],[12,14,15],[12,14,16]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[12,13,14],[12,14,15],[12,14,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0039","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[13,14,15],[13,15,16],[13,15,17]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085039,"parameters":{"sets":[[13,14,15],[13,15,16],[13,15,17]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[13,14,15],[13,15,16],[13,15,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0040","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085040,"parameters":{"sets":[[14,15,16],[14,16,17],[14,16,18]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0041","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[15,16,17],[15,17,18],[15,17,19]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085041,"parameters":{"sets":[[15,16,17],[15,17,18],[15,17,19]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[15,16,17],[15,17,18],[15,17,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0042","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[16,17,18],[16,18,19],[16,18,20]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085042,"parameters":{"sets":[[16,17,18],[16,18,19],[16,18,20]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[16,17,18],[16,18,19],[16,18,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0043","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[17,18,19],[17,19,20],[17,19,21]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085043,"parameters":{"sets":[[17,18,19],[17,19,20],[17,19,21]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[17,18,19],[17,19,20],[17,19,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0044","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085044,"parameters":{"sets":[[18,19,20],[18,20,21],[18,20,22]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0045","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[19,20,21],[19,21,22],[19,21,23]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085045,"parameters":{"sets":[[19,20,21],[19,21,22],[19,21,23]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[19,20,21],[19,21,22],[19,21,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0046","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[20,21,22],[20,22,23],[20,22,24]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085046,"parameters":{"sets":[[20,21,22],[20,22,23],[20,22,24]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[20,21,22],[20,22,23],[20,22,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0047","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[21,22,23],[21,23,24],[21,23,25]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085047,"parameters":{"sets":[[21,22,23],[21,23,24],[21,23,25]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[21,22,23],[21,23,24],[21,23,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0048","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085048,"parameters":{"sets":[[22,23,24],[22,24,25],[22,24,26]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0049","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[23,24,25],[23,25,26],[23,25,27]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085049,"parameters":{"sets":[[23,24,25],[23,25,26],[23,25,27]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[23,24,25],[23,25,26],[23,25,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0050","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-three-set-formula；資料={\"sets\":[[24,25,26],[24,26,27],[24,26,28]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)","explanation_zh":"Wave6第二層核對=([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, odd_intersections_positive_even_intersections_negative)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-three-set-formula","seed":202611085050,"parameters":{"sets":[[24,25,26],[24,26,27],[24,26,28]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-three-set-formula","semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[24,25,26],[24,26,27],[24,26,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0051","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[0,1,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085051,"parameters":{"sets":[[0,1,2]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[0,1,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0052","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[1,2,3],[1,3,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085052,"parameters":{"sets":[[1,2,3],[1,3,4]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[1,2,3],[1,3,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0053","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085053,"parameters":{"sets":[[2,3,4],[2,4,5],[2,4,6]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0054","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[3,4,5],[3,5,6],[3,5,7],[3,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085054,"parameters":{"sets":[[3,4,5],[3,5,6],[3,5,7],[3,8]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[3,4,5],[3,5,6],[3,5,7],[3,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0055","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[4,5,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085055,"parameters":{"sets":[[4,5,6]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[4,5,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0056","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[5,6,7],[5,7,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085056,"parameters":{"sets":[[5,6,7],[5,7,8]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[5,6,7],[5,7,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0057","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[6,7,8],[6,8,9],[6,8,10]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085057,"parameters":{"sets":[[6,7,8],[6,8,9],[6,8,10]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[6,7,8],[6,8,10],[6,8,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0058","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[7,8,9],[7,9,10],[7,9,11],[7,12]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085058,"parameters":{"sets":[[7,8,9],[7,9,10],[7,9,11],[7,12]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[7,12],[7,8,9],[7,9,10],[7,9,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0059","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[8,9,10]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085059,"parameters":{"sets":[[8,9,10]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[8,9,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0060","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[9,10,11],[9,11,12]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085060,"parameters":{"sets":[[9,10,11],[9,11,12]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[9,10,11],[9,11,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0061","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085061,"parameters":{"sets":[[10,11,12],[10,12,13],[10,12,14]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0062","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[11,12,13],[11,13,14],[11,13,15],[11,16]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085062,"parameters":{"sets":[[11,12,13],[11,13,14],[11,13,15],[11,16]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[11,12,13],[11,13,14],[11,13,15],[11,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0063","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[12,13,14]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085063,"parameters":{"sets":[[12,13,14]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[12,13,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0064","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[13,14,15],[13,15,16]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085064,"parameters":{"sets":[[13,14,15],[13,15,16]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[13,14,15],[13,15,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0065","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085065,"parameters":{"sets":[[14,15,16],[14,16,17],[14,16,18]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0066","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[15,16,17],[15,17,18],[15,17,19],[15,20]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085066,"parameters":{"sets":[[15,16,17],[15,17,18],[15,17,19],[15,20]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[15,16,17],[15,17,18],[15,17,19],[15,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0067","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[16,17,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085067,"parameters":{"sets":[[16,17,18]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[16,17,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0068","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[17,18,19],[17,19,20]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085068,"parameters":{"sets":[[17,18,19],[17,19,20]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[17,18,19],[17,19,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0069","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085069,"parameters":{"sets":[[18,19,20],[18,20,21],[18,20,22]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0070","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[19,20,21],[19,21,22],[19,21,23],[19,24]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085070,"parameters":{"sets":[[19,20,21],[19,21,22],[19,21,23],[19,24]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[19,20,21],[19,21,22],[19,21,23],[19,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0071","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[20,21,22]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085071,"parameters":{"sets":[[20,21,22]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[20,21,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0072","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[21,22,23],[21,23,24]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085072,"parameters":{"sets":[[21,22,23],[21,23,24]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[21,22,23],[21,23,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0073","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085073,"parameters":{"sets":[[22,23,24],[22,24,25],[22,24,26]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0074","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[23,24,25],[23,25,26],[23,25,27],[23,28]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085074,"parameters":{"sets":[[23,24,25],[23,25,26],[23,25,27],[23,28]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[23,24,25],[23,25,26],[23,25,27],[23,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0075","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-general-alternating-sum；資料={\"sets\":[[24,25,26]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)","explanation_zh":"Wave6第二層核對=(1, [[[0], 1, 3]], 3, 3, 是, all_nonempty_index_subsets_included)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085075,"parameters":{"sets":[[24,25,26]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-general-alternating-sum","semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[24,25,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0076","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[0,1,2],[0,2,3],[0,2,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085076,"parameters":{"sets":[[0,1,2],[0,2,3],[0,2,4]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[0,1,2],[0,2,3],[0,2,4]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0077","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[1,2,3],[1,3,4],[1,3,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085077,"parameters":{"sets":[[1,2,3],[1,3,4],[1,3,5]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[1,2,3],[1,3,4],[1,3,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0078","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085078,"parameters":{"sets":[[2,3,4],[2,4,5],[2,4,6]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0079","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[3,4,5],[3,5,6],[3,5,7]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085079,"parameters":{"sets":[[3,4,5],[3,5,6],[3,5,7]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[3,4,5],[3,5,6],[3,5,7]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0080","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[4,5,6],[4,6,7],[4,6,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085080,"parameters":{"sets":[[4,5,6],[4,6,7],[4,6,8]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[4,5,6],[4,6,7],[4,6,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0081","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[5,6,7],[5,7,8],[5,7,9]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085081,"parameters":{"sets":[[5,6,7],[5,7,8],[5,7,9]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[5,6,7],[5,7,8],[5,7,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0082","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[6,7,8],[6,8,9],[6,8,10]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085082,"parameters":{"sets":[[6,7,8],[6,8,9],[6,8,10]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[6,7,8],[6,8,10],[6,8,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0083","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[7,8,9],[7,9,10],[7,9,11]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085083,"parameters":{"sets":[[7,8,9],[7,9,10],[7,9,11]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[7,8,9],[7,9,10],[7,9,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0084","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[8,9,10],[8,10,11],[8,10,12]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085084,"parameters":{"sets":[[8,9,10],[8,10,11],[8,10,12]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[8,10,11],[8,10,12],[8,9,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0085","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[9,10,11],[9,11,12],[9,11,13]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085085,"parameters":{"sets":[[9,10,11],[9,11,12],[9,11,13]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[9,10,11],[9,11,12],[9,11,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0086","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085086,"parameters":{"sets":[[10,11,12],[10,12,13],[10,12,14]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0087","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[11,12,13],[11,13,14],[11,13,15]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085087,"parameters":{"sets":[[11,12,13],[11,13,14],[11,13,15]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[11,12,13],[11,13,14],[11,13,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0088","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[12,13,14],[12,14,15],[12,14,16]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085088,"parameters":{"sets":[[12,13,14],[12,14,15],[12,14,16]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[12,13,14],[12,14,15],[12,14,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0089","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[13,14,15],[13,15,16],[13,15,17]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085089,"parameters":{"sets":[[13,14,15],[13,15,16],[13,15,17]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[13,14,15],[13,15,16],[13,15,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0090","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085090,"parameters":{"sets":[[14,15,16],[14,16,17],[14,16,18]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0091","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[15,16,17],[15,17,18],[15,17,19]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085091,"parameters":{"sets":[[15,16,17],[15,17,18],[15,17,19]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[15,16,17],[15,17,18],[15,17,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0092","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[16,17,18],[16,18,19],[16,18,20]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085092,"parameters":{"sets":[[16,17,18],[16,18,19],[16,18,20]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[16,17,18],[16,18,19],[16,18,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0093","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[17,18,19],[17,19,20],[17,19,21]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085093,"parameters":{"sets":[[17,18,19],[17,19,20],[17,19,21]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[17,18,19],[17,19,20],[17,19,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0094","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085094,"parameters":{"sets":[[18,19,20],[18,20,21],[18,20,22]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0095","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[19,20,21],[19,21,22],[19,21,23]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085095,"parameters":{"sets":[[19,20,21],[19,21,22],[19,21,23]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[19,20,21],[19,21,22],[19,21,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0096","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[20,21,22],[20,22,23],[20,22,24]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085096,"parameters":{"sets":[[20,21,22],[20,22,23],[20,22,24]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[20,21,22],[20,22,23],[20,22,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0097","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[21,22,23],[21,23,24],[21,23,25]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085097,"parameters":{"sets":[[21,22,23],[21,23,24],[21,23,25]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[21,22,23],[21,23,24],[21,23,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0098","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085098,"parameters":{"sets":[[22,23,24],[22,24,25],[22,24,26]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0099","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[23,24,25],[23,25,26],[23,25,27]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085099,"parameters":{"sets":[[23,24,25],[23,25,26],[23,25,27]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[23,24,25],[23,25,26],[23,25,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0100","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","language":"zh-Hant","source_mko_ids":["mko-inclusion-exclusion-principle"],"title_zh":"容斥原理","stem_zh":"完整有限indexed set family與全部非空index subsets。操作=incexc-pairwise-only-boundary；資料={\"sets\":[[24,25,26],[24,26,27],[24,26,28]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation_zh":"Wave6第二層核對=(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)；依完整明示資料獨立重算；odd-order intersections正、even-order負，重複集合與empty intersections保留indexed語義；pairwise-only對三集合以上不充分；截斷只能是Bonferroni bound而非exact equality。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["容斥原理"]},"variant":{"family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085100,"parameters":{"sets":[[24,25,26],[24,26,27],[24,26,28]],"domain":"Complete_finite_set_inclusion_exclusion","operation":"incexc-pairwise-only-boundary","semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[24,25,26],[24,26,27],[24,26,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-step orbit、finite-prefix monotonicity、ambient-relative topology混同、Chebyshev equality、pairwise-only容斥、無Euclid-lemma uniqueness、loopful digraph、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}