第 20520 題：上確界與下確界

原題與已知條件

非空且相應有界的實值集合或明示有界開區間。操作=supinf-finite-extrema；資料={"values":[[20,21],[-21,22],[41,23]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 非空且相應有界的實值集合或明示有界開區間。操作=supinf-finite-extrema；資料={"values":[[20,21],[-21,22],[41,23]]}。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-supremum-and-infimum

逐步推導

source operation=supinf-finite-extrema;original snapshot and mathematics retained

explicit ordered source output fields=[values, supremum, infimum, supremum_attained, infimum_attained, certificate]

ORIGINAL complete ordered nonempty finite real-value list=[20/21, -21/22, 41/23]

initialize upper/lower extremum candidates at ORIGINAL first value;duplicates retained and do not change set bounds

supinf original value1 minus previous maximum := sub(-21/22, 20/21)

supinf original value1 minus previous maximum = -881/462

supinf original value1 minus previous minimum := sub(-21/22, 20/21)

supinf original value1 minus previous minimum = -881/462

ORIGINAL next index=1;value=-21/22;previous extremum values=[20/21, 20/21];differences=[-881/462, -881/462]

prefix maximum=20/21 attained at original index0;prefix minimum=-21/22 attained at original index1

supinf original value2 minus previous maximum := sub(41/23, 20/21)

supinf original value2 minus previous maximum = 401/483

supinf original value2 minus previous minimum := sub(41/23, -21/22)

supinf original value2 minus previous minimum = 1385/506

ORIGINAL next index=2;value=41/23;previous extremum values=[20/21, -21/22];differences=[401/483, 1385/506]

prefix maximum=41/23 attained at original index2;prefix minimum=-21/22 attained at original index1

ALL original values processed;actual maximum witness=[2, 41/23];actual minimum witness=[1, -21/22]

EVERY listed value between-21/22 and41/23;ANY upper bound must>=actual maximum in the set;ANY lower bound must<=actual minimum;sup=41/23,inf=-21/22,BOTH attained

逐步反向核對

• S009 · 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。
• 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 supinf original value2 minus previous minimum，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 supinf original value2 minus previous maximum，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 supinf original value1 minus previous minimum，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 supinf original value1 minus previous maximum，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

([20/21, -21/22, 41/23], 41/23, -21/22, 是, 是, finite_nonempty_set_extrema)

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。