第 20931 題：容斥原理

原題與已知條件

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。

• G001 — 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。

定義與適用範圍

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。

• mko-inclusion-exclusion-principle

逐步推導

source operation=incexc-pairwise-only-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary]

ORIGINAL indexed finite set family in SUPPLIED outer index order=[[5, 6, 7], [5, 7, 8], [5, 7, 9]]

within-set numeric normalization=[[5, 6, 7], [5, 7, 8], [5, 7, 9]];duplicates INSIDE a set rejected;EQUAL sets at DIFFERENT indices RETAINED;integer labels may be negative

inclusion mask1 signed intersection cardinality := mul(1, 3)

inclusion mask1 signed intersection cardinality = 3

inclusion mask1 alternating prefix sum := sum(0, 3)

inclusion mask1 alternating prefix sum = 3

next NONEMPTY NUMERIC bitmask=1;ORIGINAL family indices=[0];actual selected original sets=[[5, 6, 7]]

FULL intersection members=[5, 6, 7];size=3;ODD order sign=1;alternating prefix=3;zero-size terms retained

inclusion mask2 signed intersection cardinality := mul(1, 3)

inclusion mask2 signed intersection cardinality = 3

inclusion mask2 alternating prefix sum := sum(3, 3)

inclusion mask2 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=2;ORIGINAL family indices=[1];actual selected original sets=[[5, 7, 8]]

FULL intersection members=[5, 7, 8];size=3;ODD order sign=1;alternating prefix=6;zero-size terms retained

inclusion mask3 signed intersection cardinality := mul(-1, 2)

inclusion mask3 signed intersection cardinality = -2

inclusion mask3 alternating prefix sum := sum(6, -2)

inclusion mask3 alternating prefix sum = 4

next NONEMPTY NUMERIC bitmask=3;ORIGINAL family indices=[0, 1];actual selected original sets=[[5, 6, 7], [5, 7, 8]]

FULL intersection members=[5, 7];size=2;EVEN order sign=-1;alternating prefix=4;zero-size terms retained

inclusion mask4 signed intersection cardinality := mul(1, 3)

inclusion mask4 signed intersection cardinality = 3

inclusion mask4 alternating prefix sum := sum(4, 3)

inclusion mask4 alternating prefix sum = 7

next NONEMPTY NUMERIC bitmask=4;ORIGINAL family indices=[2];actual selected original sets=[[5, 7, 9]]

FULL intersection members=[5, 7, 9];size=3;ODD order sign=1;alternating prefix=7;zero-size terms retained

inclusion mask5 signed intersection cardinality := mul(-1, 2)

inclusion mask5 signed intersection cardinality = -2

inclusion mask5 alternating prefix sum := sum(7, -2)

inclusion mask5 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=5;ORIGINAL family indices=[0, 2];actual selected original sets=[[5, 6, 7], [5, 7, 9]]

FULL intersection members=[5, 7];size=2;EVEN order sign=-1;alternating prefix=5;zero-size terms retained

inclusion mask6 signed intersection cardinality := mul(-1, 2)

inclusion mask6 signed intersection cardinality = -2

inclusion mask6 alternating prefix sum := sum(5, -2)

inclusion mask6 alternating prefix sum = 3

next NONEMPTY NUMERIC bitmask=6;ORIGINAL family indices=[1, 2];actual selected original sets=[[5, 7, 8], [5, 7, 9]]

FULL intersection members=[5, 7];size=2;EVEN order sign=-1;alternating prefix=3;zero-size terms retained

inclusion mask7 signed intersection cardinality := mul(1, 2)

inclusion mask7 signed intersection cardinality = 2

inclusion mask7 alternating prefix sum := sum(3, 2)

inclusion mask7 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=7;ORIGINAL family indices=[0, 1, 2];actual selected original sets=[[5, 6, 7], [5, 7, 8], [5, 7, 9]]

FULL intersection members=[5, 7];size=2;ODD order sign=1;alternating prefix=5;zero-size terms retained

inclusion COMPLETE actual union cardinality := sum(1, 1, 1, 1, 1)

inclusion COMPLETE actual union cardinality = 5

ACTUAL union element=5;ALL containing ORIGINAL family indices=[0, 1, 2];indexed multiplicity r=3

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

ACTUAL union element=6;ALL containing ORIGINAL family indices=[0];indexed multiplicity r=1

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

ACTUAL union element=7;ALL containing ORIGINAL family indices=[0, 1, 2];indexed multiplicity r=3

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

ACTUAL union element=8;ALL containing ORIGINAL family indices=[1];indexed multiplicity r=1

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

ACTUAL union element=9;ALL containing ORIGINAL family indices=[2];indexed multiplicity r=1

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

COMPLETE indexed family size=3;ALL7 nonempty index subsets accounted for;ACTUAL union=[5, 6, 7, 8, 9]

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

inclusion pairwise truncation singleton sum := sum(3, 3, 3)

inclusion pairwise truncation singleton sum = 9

inclusion pairwise truncation ALL pair intersections := sum(2, 2, 2)

inclusion pairwise truncation ALL pair intersections = 6

inclusion singles MINUS pairs truncation := sub(9, 6)

inclusion singles MINUS pairs truncation = 3

inclusion true union MINUS pairwise truncation := sub(5, 3)

inclusion true union MINUS pairwise truncation = 2

ALL singleton sizes sum=9;ALL pair intersection sizes=[2, 2, 2];pair sum=6;actual union size=5

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

逐步反向核對

• S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · premise_and_result_audit — 以精確有理數重算 inclusion true union MINUS pairwise truncation，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 inclusion singles MINUS pairs truncation，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 inclusion pairwise truncation ALL pair intersections，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 inclusion pairwise truncation singleton sum，不以小數近似替代。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 inclusion COMPLETE actual union cardinality，不以小數近似替代。
• S023 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · premise_and_result_audit — 以精確有理數重算 inclusion mask7 alternating prefix sum，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 inclusion mask7 signed intersection cardinality，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 inclusion mask6 alternating prefix sum，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 inclusion mask6 signed intersection cardinality，不以小數近似替代。
• S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · premise_and_result_audit — 以精確有理數重算 inclusion mask5 alternating prefix sum，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 inclusion mask5 signed intersection cardinality，不以小數近似替代。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 以精確有理數重算 inclusion mask4 alternating prefix sum，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 inclusion mask4 signed intersection cardinality，不以小數近似替代。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 以精確有理數重算 inclusion mask3 alternating prefix sum，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 inclusion mask3 signed intersection cardinality，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 inclusion mask2 alternating prefix sum，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 inclusion mask2 signed intersection cardinality，不以小數近似替代。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 以精確有理數重算 inclusion mask1 alternating prefix sum，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 inclusion mask1 signed intersection cardinality，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(3, 5, 否, 否, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。