{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-order-relation"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-order-relation-01","title":{"zh-Hant":"實數的通常順序：加法保序、正分母交叉比較、有限極值與閉區間","en":"The usual order on ℝ: translation, cross-multiplication with positive denominators, finite extrema and closed intervals"},"summary":{"zh-Hant":"證明對所有實數 a、b、c：a<b ⇔ a+c<b+c；q、s>0 時 p/q<r/s ⇔ ps<rq；非空有限實數集合有最小元素，且最小元素唯一。據此比較 M={−5/9, 1/3, −7/12, −1/2}（照此次序列出）的元素，求 M 的最小與最大元素，並判定兩端點與 1/2 是否屬於閉區間 [min M, max M]。","en":"Prove for all reals a, b, c that a<b ⇔ a+c<b+c; that p/q<r/s ⇔ ps<rq when q, s>0; and that a finite nonempty set of reals has a minimum, which is unique. Use these to compare the elements of M={−5/9, 1/3, −7/12, −1/2} (listed in this order), find the minimum and maximum of M, and decide whether the two endpoints and 1/2 belong to the closed interval [min M, max M]."},"mko_ids":["mko-order-relation"]}],"title":"概念推導導讀"}