第 20908 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-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。

• G001 — 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。

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-general-alternating-sum;original snapshot and mathematics retained

explicit ordered source output fields=[family_size, terms, union_size, alternating_sum, equal, certificate]

ORIGINAL indexed finite set family in SUPPLIED outer index order=[[7, 8, 9], [7, 9, 10], [7, 9, 11], [7, 12]]

within-set numeric normalization=[[7, 8, 9], [7, 9, 10], [7, 9, 11], [7, 12]];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=[[7, 8, 9]]

FULL intersection members=[7, 8, 9];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=[[7, 9, 10]]

FULL intersection members=[7, 9, 10];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=[[7, 8, 9], [7, 9, 10]]

FULL intersection members=[7, 9];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=[[7, 9, 11]]

FULL intersection members=[7, 9, 11];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=[[7, 8, 9], [7, 9, 11]]

FULL intersection members=[7, 9];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=[[7, 9, 10], [7, 9, 11]]

FULL intersection members=[7, 9];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=[[7, 8, 9], [7, 9, 10], [7, 9, 11]]

FULL intersection members=[7, 9];size=2;ODD order sign=1;alternating prefix=5;zero-size terms retained

inclusion mask8 signed intersection cardinality := mul(1, 2)

inclusion mask8 signed intersection cardinality = 2

inclusion mask8 alternating prefix sum := sum(5, 2)

inclusion mask8 alternating prefix sum = 7

next NONEMPTY NUMERIC bitmask=8;ORIGINAL family indices=[3];actual selected original sets=[[7, 12]]

FULL intersection members=[7, 12];size=2;ODD order sign=1;alternating prefix=7;zero-size terms retained

inclusion mask9 signed intersection cardinality := mul(-1, 1)

inclusion mask9 signed intersection cardinality = -1

inclusion mask9 alternating prefix sum := sum(7, -1)

inclusion mask9 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=9;ORIGINAL family indices=[0, 3];actual selected original sets=[[7, 8, 9], [7, 12]]

FULL intersection members=[7];size=1;EVEN order sign=-1;alternating prefix=6;zero-size terms retained

inclusion mask10 signed intersection cardinality := mul(-1, 1)

inclusion mask10 signed intersection cardinality = -1

inclusion mask10 alternating prefix sum := sum(6, -1)

inclusion mask10 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=10;ORIGINAL family indices=[1, 3];actual selected original sets=[[7, 9, 10], [7, 12]]

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

inclusion mask11 signed intersection cardinality := mul(1, 1)

inclusion mask11 signed intersection cardinality = 1

inclusion mask11 alternating prefix sum := sum(5, 1)

inclusion mask11 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=11;ORIGINAL family indices=[0, 1, 3];actual selected original sets=[[7, 8, 9], [7, 9, 10], [7, 12]]

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

inclusion mask12 signed intersection cardinality := mul(-1, 1)

inclusion mask12 signed intersection cardinality = -1

inclusion mask12 alternating prefix sum := sum(6, -1)

inclusion mask12 alternating prefix sum = 5

next NONEMPTY NUMERIC bitmask=12;ORIGINAL family indices=[2, 3];actual selected original sets=[[7, 9, 11], [7, 12]]

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

inclusion mask13 signed intersection cardinality := mul(1, 1)

inclusion mask13 signed intersection cardinality = 1

inclusion mask13 alternating prefix sum := sum(5, 1)

inclusion mask13 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=13;ORIGINAL family indices=[0, 2, 3];actual selected original sets=[[7, 8, 9], [7, 9, 11], [7, 12]]

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

inclusion mask14 signed intersection cardinality := mul(1, 1)

inclusion mask14 signed intersection cardinality = 1

inclusion mask14 alternating prefix sum := sum(6, 1)

inclusion mask14 alternating prefix sum = 7

next NONEMPTY NUMERIC bitmask=14;ORIGINAL family indices=[1, 2, 3];actual selected original sets=[[7, 9, 10], [7, 9, 11], [7, 12]]

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

inclusion mask15 signed intersection cardinality := mul(-1, 1)

inclusion mask15 signed intersection cardinality = -1

inclusion mask15 alternating prefix sum := sum(7, -1)

inclusion mask15 alternating prefix sum = 6

next NONEMPTY NUMERIC bitmask=15;ORIGINAL family indices=[0, 1, 2, 3];actual selected original sets=[[7, 8, 9], [7, 9, 10], [7, 9, 11], [7, 12]]

FULL intersection members=[7];size=1;EVEN order sign=-1;alternating prefix=6;zero-size terms retained

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

inclusion COMPLETE actual union cardinality = 6

ACTUAL union element=7;ALL containing ORIGINAL family indices=[0, 1, 2, 3];indexed multiplicity r=4

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=[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=9;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=10;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=11;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

ACTUAL union element=12;ALL containing ORIGINAL family indices=[3];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=4;ALL15 nonempty index subsets accounted for;ACTUAL union=[7, 8, 9, 10, 11, 12]

each covered element has coefficient1 and any uncovered element coefficient0;FULL alternating sum=6=actual union size=6;empty family yields empty union/no terms/0

逐步反向核對

• S055 · 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。
• S054 · 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。
• S053 · 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。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 inclusion mask15 alternating prefix sum，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 inclusion mask15 signed intersection cardinality，不以小數近似替代。
• S044 · 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。
• S043 · premise_and_result_audit — 以精確有理數重算 inclusion mask14 alternating prefix sum，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 inclusion mask14 signed intersection cardinality，不以小數近似替代。
• S041 · 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。
• S040 · premise_and_result_audit — 以精確有理數重算 inclusion mask13 alternating prefix sum，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 inclusion mask13 signed intersection cardinality，不以小數近似替代。
• S038 · 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。
• S037 · premise_and_result_audit — 以精確有理數重算 inclusion mask12 alternating prefix sum，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 inclusion mask12 signed intersection cardinality，不以小數近似替代。
• 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 mask11 alternating prefix sum，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 inclusion mask11 signed intersection cardinality，不以小數近似替代。
• S032 · 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。
• S031 · premise_and_result_audit — 以精確有理數重算 inclusion mask10 alternating prefix sum，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 inclusion mask10 signed intersection cardinality，不以小數近似替代。
• 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 — 以精確有理數重算 inclusion mask9 alternating prefix sum，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 inclusion mask9 signed intersection cardinality，不以小數近似替代。
• 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 — 以精確有理數重算 inclusion mask8 alternating prefix sum，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 inclusion mask8 signed intersection 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

(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。