第 20914 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":[[13,14,15],[13,15,16]]}。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":[[13,14,15],[13,15,16]]}。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=[[13, 14, 15], [13, 15, 16]]

within-set numeric normalization=[[13, 14, 15], [13, 15, 16]];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=[[13, 14, 15]]

FULL intersection members=[13, 14, 15];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=[[13, 15, 16]]

FULL intersection members=[13, 15, 16];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=[[13, 14, 15], [13, 15, 16]]

FULL intersection members=[13, 15];size=2;EVEN order sign=-1;alternating prefix=4;zero-size terms retained

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

inclusion COMPLETE actual union cardinality = 4

ACTUAL union element=13;ALL containing ORIGINAL family indices=[0, 1];indexed multiplicity r=2

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=14;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=15;ALL containing ORIGINAL family indices=[0, 1];indexed multiplicity r=2

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=16;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

COMPLETE indexed family size=2;ALL3 nonempty index subsets accounted for;ACTUAL union=[13, 14, 15, 16]

each covered element has coefficient1 and any uncovered element coefficient0;FULL alternating sum=4=actual union size=4;empty family yields empty union/no terms/0

逐步反向核對

• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · 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。
• 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

(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, 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。