{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0083", "pilot_number": "R0040-20933", "number": 20933, "family_id": "qf-mod39-incexc-pairwise-only-boundary", "title": { "zh-Hant": "第 20933 題：容斥原理", "en": "Question 20933: A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula" }, "original_question": { "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" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0083", "batch_id": "batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "eab2e1dfd68a421047cbf6d7dd87b64ca0fa869fbe63e98a3e6e6e60678d7e06", "family_id": "qf-mod39-incexc-pairwise-only-boundary", "seed": 202611085083, "semantic_template": "qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[7,8,9],[7,9,10],[7,9,11]]}" }, "fields": { "title": "A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula", "stem": "Use the complete indexed finite integer-set family sets=[[7, 8, 9], [7, 9, 10], [7, 9, 11]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).", "answer": { "kind": "expression", "value": "(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)", "explanation": "Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)." }, "skills": [ "A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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。", "en": "Use the complete indexed finite integer-set family sets=[[7, 8, 9], [7, 9, 10], [7, 9, 11]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=incexc-pairwise-only-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。標籤不自動是數學證據，No保持來源的聲稱範圍。", "en": "Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "ORIGINAL indexed finite set family in SUPPLIED outer index order=[[7, 8, 9], [7, 9, 10], [7, 9, 11]]", "after": "within-set numeric normalization=[[7, 8, 9], [7, 9, 10], [7, 9, 11]];duplicates INSIDE a set rejected;EQUAL sets at DIFFERENT indices RETAINED;integer labels may be negative", "relation": "implies", "rule_id": "incexc_indexed_family_not_deduplicated_or_semantic_reordered", "conditions": [], "explanation": { "zh-Hant": "只對各集合內元素排序；外family位置是容斥index，不能按semantic fingerprint重排或去掉重複集合。空集合仍是合法indexed成員，元素標籤不是cardinality。", "en": "Normalize elements within each set only.Outer family positions are inclusion-exclusion indices and must not be semantically reordered or deduplicated.Empty sets are legitimate indexed members,and element labels are not cardinalities." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "inclusion mask1 signed intersection cardinality := mul(1, 3)", "after": "inclusion mask1 signed intersection cardinality = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask1 signed intersection cardinality，不以小數近似替代。", "en": "Recompute inclusion mask1 signed intersection cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "inclusion mask1 signed intersection cardinality" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "inclusion mask1 alternating prefix sum := sum(0, 3)", "after": "inclusion mask1 alternating prefix sum = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask1 alternating prefix sum，不以小數近似替代。", "en": "Recompute inclusion mask1 alternating prefix sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "3" ], "output": "3", "label": "inclusion mask1 alternating prefix sum" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "next NONEMPTY NUMERIC bitmask=1;ORIGINAL family indices=[0];actual selected original sets=[[7, 8, 9]]", "after": "FULL intersection members=[7, 8, 9];size=3;ODD order sign=1;alternating prefix=3;zero-size terms retained", "relation": "implies", "rule_id": "incexc_every_index_submask_intersection_parity_source_order", "conditions": [], "explanation": { "zh-Hant": "每個非空index子集都出現一次，按數字mask而非集合大小或lex排序。奇數階加、偶數階減；空intersection也列0項，partial alternating sum不是已宣稱cardinality。", "en": "Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "inclusion mask2 signed intersection cardinality := mul(1, 3)", "after": "inclusion mask2 signed intersection cardinality = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask2 signed intersection cardinality，不以小數近似替代。", "en": "Recompute inclusion mask2 signed intersection cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "inclusion mask2 signed intersection cardinality" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "inclusion mask2 alternating prefix sum := sum(3, 3)", "after": "inclusion mask2 alternating prefix sum = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask2 alternating prefix sum，不以小數近似替代。", "en": "Recompute inclusion mask2 alternating prefix sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "3" ], "output": "6", "label": "inclusion mask2 alternating prefix sum" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "next NONEMPTY NUMERIC bitmask=2;ORIGINAL family indices=[1];actual selected original sets=[[7, 9, 10]]", "after": "FULL intersection members=[7, 9, 10];size=3;ODD order sign=1;alternating prefix=6;zero-size terms retained", "relation": "implies", "rule_id": "incexc_every_index_submask_intersection_parity_source_order", "conditions": [], "explanation": { "zh-Hant": "每個非空index子集都出現一次，按數字mask而非集合大小或lex排序。奇數階加、偶數階減；空intersection也列0項，partial alternating sum不是已宣稱cardinality。", "en": "Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "inclusion mask3 signed intersection cardinality := mul(-1, 2)", "after": "inclusion mask3 signed intersection cardinality = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask3 signed intersection cardinality，不以小數近似替代。", "en": "Recompute inclusion mask3 signed intersection cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "2" ], "output": "-2", "label": "inclusion mask3 signed intersection cardinality" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "inclusion mask3 alternating prefix sum := sum(6, -2)", "after": "inclusion mask3 alternating prefix sum = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask3 alternating prefix sum，不以小數近似替代。", "en": "Recompute inclusion mask3 alternating prefix sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "6", "-2" ], "output": "4", "label": "inclusion mask3 alternating prefix sum" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "next NONEMPTY NUMERIC bitmask=3;ORIGINAL family indices=[0, 1];actual selected original sets=[[7, 8, 9], [7, 9, 10]]", "after": "FULL intersection members=[7, 9];size=2;EVEN order sign=-1;alternating prefix=4;zero-size terms retained", "relation": "implies", "rule_id": "incexc_every_index_submask_intersection_parity_source_order", "conditions": [], "explanation": { "zh-Hant": "每個非空index子集都出現一次，按數字mask而非集合大小或lex排序。奇數階加、偶數階減；空intersection也列0項，partial alternating sum不是已宣稱cardinality。", "en": "Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "inclusion mask4 signed intersection cardinality := mul(1, 3)", "after": "inclusion mask4 signed intersection cardinality = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask4 signed intersection cardinality，不以小數近似替代。", "en": "Recompute inclusion mask4 signed intersection cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "3" ], "output": "3", "label": "inclusion mask4 signed intersection cardinality" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "inclusion mask4 alternating prefix sum := sum(4, 3)", "after": "inclusion mask4 alternating prefix sum = 7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask4 alternating prefix sum，不以小數近似替代。", "en": "Recompute inclusion mask4 alternating prefix sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4", "3" ], "output": "7", "label": "inclusion mask4 alternating prefix sum" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "next NONEMPTY NUMERIC bitmask=4;ORIGINAL family indices=[2];actual selected original sets=[[7, 9, 11]]", "after": "FULL intersection members=[7, 9, 11];size=3;ODD order sign=1;alternating prefix=7;zero-size terms retained", "relation": "implies", "rule_id": "incexc_every_index_submask_intersection_parity_source_order", "conditions": [], "explanation": { "zh-Hant": "每個非空index子集都出現一次，按數字mask而非集合大小或lex排序。奇數階加、偶數階減；空intersection也列0項，partial alternating sum不是已宣稱cardinality。", "en": "Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "inclusion mask5 signed intersection cardinality := mul(-1, 2)", "after": "inclusion mask5 signed intersection cardinality = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask5 signed intersection cardinality，不以小數近似替代。", "en": "Recompute inclusion mask5 signed intersection cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "2" ], "output": "-2", "label": "inclusion mask5 signed intersection cardinality" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "inclusion mask5 alternating prefix sum := sum(7, -2)", "after": "inclusion mask5 alternating prefix sum = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask5 alternating prefix sum，不以小數近似替代。", "en": "Recompute inclusion mask5 alternating prefix sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "7", "-2" ], "output": "5", "label": "inclusion mask5 alternating prefix sum" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "next NONEMPTY NUMERIC bitmask=5;ORIGINAL family indices=[0, 2];actual selected original sets=[[7, 8, 9], [7, 9, 11]]", "after": "FULL intersection members=[7, 9];size=2;EVEN order sign=-1;alternating prefix=5;zero-size terms retained", "relation": "implies", "rule_id": "incexc_every_index_submask_intersection_parity_source_order", "conditions": [], "explanation": { "zh-Hant": "每個非空index子集都出現一次，按數字mask而非集合大小或lex排序。奇數階加、偶數階減；空intersection也列0項，partial alternating sum不是已宣稱cardinality。", "en": "Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "inclusion mask6 signed intersection cardinality := mul(-1, 2)", "after": "inclusion mask6 signed intersection cardinality = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask6 signed intersection cardinality，不以小數近似替代。", "en": "Recompute inclusion mask6 signed intersection cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-1", "2" ], "output": "-2", "label": "inclusion mask6 signed intersection cardinality" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "inclusion mask6 alternating prefix sum := sum(5, -2)", "after": "inclusion mask6 alternating prefix sum = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask6 alternating prefix sum，不以小數近似替代。", "en": "Recompute inclusion mask6 alternating prefix sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "5", "-2" ], "output": "3", "label": "inclusion mask6 alternating prefix sum" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "next NONEMPTY NUMERIC bitmask=6;ORIGINAL family indices=[1, 2];actual selected original sets=[[7, 9, 10], [7, 9, 11]]", "after": "FULL intersection members=[7, 9];size=2;EVEN order sign=-1;alternating prefix=3;zero-size terms retained", "relation": "implies", "rule_id": "incexc_every_index_submask_intersection_parity_source_order", "conditions": [], "explanation": { "zh-Hant": "每個非空index子集都出現一次，按數字mask而非集合大小或lex排序。奇數階加、偶數階減；空intersection也列0項，partial alternating sum不是已宣稱cardinality。", "en": "Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "inclusion mask7 signed intersection cardinality := mul(1, 2)", "after": "inclusion mask7 signed intersection cardinality = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask7 signed intersection cardinality，不以小數近似替代。", "en": "Recompute inclusion mask7 signed intersection cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1", "2" ], "output": "2", "label": "inclusion mask7 signed intersection cardinality" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "inclusion mask7 alternating prefix sum := sum(3, 2)", "after": "inclusion mask7 alternating prefix sum = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion mask7 alternating prefix sum，不以小數近似替代。", "en": "Recompute inclusion mask7 alternating prefix sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "2" ], "output": "5", "label": "inclusion mask7 alternating prefix sum" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "next NONEMPTY NUMERIC bitmask=7;ORIGINAL family indices=[0, 1, 2];actual selected original sets=[[7, 8, 9], [7, 9, 10], [7, 9, 11]]", "after": "FULL intersection members=[7, 9];size=2;ODD order sign=1;alternating prefix=5;zero-size terms retained", "relation": "implies", "rule_id": "incexc_every_index_submask_intersection_parity_source_order", "conditions": [], "explanation": { "zh-Hant": "每個非空index子集都出現一次，按數字mask而非集合大小或lex排序。奇數階加、偶數階減；空intersection也列0項，partial alternating sum不是已宣稱cardinality。", "en": "Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "inclusion COMPLETE actual union cardinality := sum(1, 1, 1, 1, 1)", "after": "inclusion COMPLETE actual union cardinality = 5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion COMPLETE actual union cardinality，不以小數近似替代。", "en": "Recompute inclusion COMPLETE actual union cardinality with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1", "1", "1", "1" ], "output": "5", "label": "inclusion COMPLETE actual union cardinality" } }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "ACTUAL union element=7;ALL containing ORIGINAL family indices=[0, 1, 2];indexed multiplicity r=3", "after": "only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1", "relation": "implies", "rule_id": "incexc_each_element_full_coefficient_not_one_sum_coincidence", "conditions": [], "explanation": { "zh-Hant": "固定一個containing index，非空且不含它的S與S加此index成對，正負號相反；只有原本未列的空S對應正singleton未被消去，故係數1。相等集合在不同index會增加r，不能去重。", "en": "Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "ACTUAL union element=8;ALL containing ORIGINAL family indices=[0];indexed multiplicity r=1", "after": "only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1", "relation": "implies", "rule_id": "incexc_each_element_full_coefficient_not_one_sum_coincidence", "conditions": [], "explanation": { "zh-Hant": "固定一個containing index，非空且不含它的S與S加此index成對，正負號相反；只有原本未列的空S對應正singleton未被消去，故係數1。相等集合在不同index會增加r，不能去重。", "en": "Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "ACTUAL union element=9;ALL containing ORIGINAL family indices=[0, 1, 2];indexed multiplicity r=3", "after": "only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1", "relation": "implies", "rule_id": "incexc_each_element_full_coefficient_not_one_sum_coincidence", "conditions": [], "explanation": { "zh-Hant": "固定一個containing index，非空且不含它的S與S加此index成對，正負號相反；只有原本未列的空S對應正singleton未被消去，故係數1。相等集合在不同index會增加r，不能去重。", "en": "Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S028", "uses": [ "G001", "S027" ], "before": "ACTUAL union element=10;ALL containing ORIGINAL family indices=[1];indexed multiplicity r=1", "after": "only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1", "relation": "implies", "rule_id": "incexc_each_element_full_coefficient_not_one_sum_coincidence", "conditions": [], "explanation": { "zh-Hant": "固定一個containing index，非空且不含它的S與S加此index成對，正負號相反；只有原本未列的空S對應正singleton未被消去，故係數1。相等集合在不同index會增加r，不能去重。", "en": "Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S029", "uses": [ "G001", "S028" ], "before": "ACTUAL union element=11;ALL containing ORIGINAL family indices=[2];indexed multiplicity r=1", "after": "only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1", "relation": "implies", "rule_id": "incexc_each_element_full_coefficient_not_one_sum_coincidence", "conditions": [], "explanation": { "zh-Hant": "固定一個containing index，非空且不含它的S與S加此index成對，正負號相反；只有原本未列的空S對應正singleton未被消去，故係數1。相等集合在不同index會增加r，不能去重。", "en": "Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S030", "uses": [ "G001", "S029" ], "before": "COMPLETE indexed family size=3;ALL7 nonempty index subsets accounted for;ACTUAL union=[7, 8, 9, 10, 11]", "after": "each covered element has coefficient1 and any uncovered element coefficient0;FULL alternating sum=5=actual union size=5;empty family yields empty union/no terms/0", "relation": "implies", "rule_id": "incexc_full_pointwise_identity_all_finite_indexed_sets", "conditions": [], "explanation": { "zh-Hant": "逐元素係數證書全和才得到整個有限union的大小，不是觀察一次數字相等當一般證明。未覆蓋點沒有任何containing index而貢獻0；empty family亦直接是0。", "en": "Sum the pointwise coefficient certificates to get the entire finite union size,rather than inferring a theorem from one numerical coincidence.Uncovered points have no containing indices and contribute0;the empty family gives0 directly." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S031", "uses": [ "G001", "S030" ], "before": "inclusion pairwise truncation singleton sum := sum(3, 3, 3)", "after": "inclusion pairwise truncation singleton sum = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion pairwise truncation singleton sum，不以小數近似替代。", "en": "Recompute inclusion pairwise truncation singleton sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3", "3", "3" ], "output": "9", "label": "inclusion pairwise truncation singleton sum" } }, { "id": "S032", "uses": [ "G001", "S031" ], "before": "inclusion pairwise truncation ALL pair intersections := sum(2, 2, 2)", "after": "inclusion pairwise truncation ALL pair intersections = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion pairwise truncation ALL pair intersections，不以小數近似替代。", "en": "Recompute inclusion pairwise truncation ALL pair intersections with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2", "2", "2" ], "output": "6", "label": "inclusion pairwise truncation ALL pair intersections" } }, { "id": "S033", "uses": [ "G001", "S032" ], "before": "inclusion singles MINUS pairs truncation := sub(9, 6)", "after": "inclusion singles MINUS pairs truncation = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion singles MINUS pairs truncation，不以小數近似替代。", "en": "Recompute inclusion singles MINUS pairs truncation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "9", "6" ], "output": "3", "label": "inclusion singles MINUS pairs truncation" } }, { "id": "S034", "uses": [ "G001", "S033" ], "before": "inclusion true union MINUS pairwise truncation := sub(5, 3)", "after": "inclusion true union MINUS pairwise truncation = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色及已核中間量精確代值，保留原索引、集合位置、方向、正負號與適用條件。", "en": "Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 inclusion true union MINUS pairwise truncation，不以小數近似替代。", "en": "Recompute inclusion true union MINUS pairwise truncation with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "5", "3" ], "output": "2", "label": "inclusion true union MINUS pairwise truncation" } }, { "id": "S035", "uses": [ "G001", "S034" ], "before": "ALL singleton sizes sum=9;ALL pair intersection sizes=[2, 2, 2];pair sum=6;actual union size=5", "after": "pairwise truncation=3;numerically equals THIS union=否;GENERAL pairwise-only exactness claim=否;Bonferroni lower-bound gap=2>=0,not a cardinality assertion for the truncation", "relation": "implies", "rule_id": "incexc_pairwise_No_not_numerical_inequality_or_nonnegative_count", "conditions": [], "explanation": { "zh-Hant": "每點若r=0貢獻都0，r≥1時truncation係數r−C(r,2)≤1（r=1,2相等；r≥3較小），故全和是union的下界，可為負。這次數字可能碰巧相等，最後No仍只否決一般exact公式，不改numerically_equal旗標。", "en": "At r=0 both contributions are0;for r≥1 the truncated coefficient r−C(r,2)≤1,with equality for r=1,2 and a strict gap for r≥3.So the total is a lower bound and may be negative.This instance may coincide numerically;the final No still withholds a general exact formula without negating the equality flag." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 3, 5, false, false, "pairwise_only_truncation_not_exact_when_higher_intersections_contribute" ] }, "conclusion": { "canonical_value": "(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations/questions.jsonl", "batch_sha256": "21896a22cd4ea8b64b7f5e20f7e77886f087b925919cf710ad8a71a6979d3b55", "original_line": 83, "original_line_sha256": "377d88f6e40a949963c928301e70860a7130f543e561ab71fa19d1dc1db0253b", "question_compact_sha256": "eab2e1dfd68a421047cbf6d7dd87b64ca0fa869fbe63e98a3e6e6e60678d7e06", "english_row_compact_sha256": "a231aa62ff8ba8d8b8b7063ef1081d843bbb43aa0eb3bfbc634863065d1ec375", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations.en.jsonl" }, "mko_bindings": { "mko-inclusion-exclusion-principle": { "sha256": "2eb0f4de0d9b0d93d979ced97fcaadb56ea9045d427c0918e35505bde5f2a649", "english_sha256": "0aa23ad5de1c87cf782892d43f6bbc7c43d34b422d9ee2ebfaa54cd4c4fd80cf", "original_path": "public/data/mko/inclusion-exclusion-principle.json", "english_path": "public/data/i18n/editions/mko-inclusion-exclusion-principle.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "f1ed3239c9185524aa4f0839ab4cfbe8f7d74200b283ea09e850a792f0178a3b", "original_candidate_sha256": "aa11dd9377d37828c04380361b1d9f88c686d262e20b6617d7e91e577a10d53a", "original_candidate_line_sha256": "aa11dd9377d37828c04380361b1d9f88c686d262e20b6617d7e91e577a10d53a" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-008-p41", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d", "question_first_witness_sha256": "e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 35, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "9e0187c6b9c6e3942761896cde1f55437fcbfc276e2674697aa09fded46e6fb7", "arithmetic_steps": 19 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }