{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"knowledge","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":{"page":"2"},"kind":"knowledge","total":302,"pages":13,"page":2,"entries":[{"id":"mko-tends-to-relation","title":{"zh-Hant":"趨近關係","en":"Tends-to relation"},"summary":{"zh-Hant":"符號 x→a 表示變數或索引所描述的狀態朝目標 a 趨近。","en":"The notation x→a says that the state described by a variable or index approaches the target a."},"type":"concept","domain":"analysis","formalization":"formalized_filter_tendsto_core","tags":["極限","分析","趨近","Filter","Lean","Mathlib"]},{"id":"mko-mathematical-induction","title":{"zh-Hant":"數學歸納法","en":"Mathematical induction"},"summary":{"zh-Hant":"數學歸納法以基礎情況與歸納步驟證明自然數索引的整個命題族。","en":"Mathematical induction proves an entire natural-number-indexed family of propositions through a base case and an induction step."},"type":"concept","domain":"history_philosophy_methodology","formalization":"not_formalized","tags":["證明","歸納法","自然數","AI候選","待形式化"]},{"id":"mko-counting-principle","title":{"zh-Hant":"基本計數原理","en":"Basic counting principles"},"summary":{"zh-Hant":"加法與乘法原理把互斥分支或連續選擇的有限數量組合成總數。","en":"The addition and multiplication principles combine finite counts across disjoint branches or successive choices."},"type":"concept","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["組合","計數","加法原理","乘法原理","AI候選"]},{"id":"mko-factorial","title":{"zh-Hant":"階乘","en":"Factorial"},"summary":{"zh-Hant":"自然數 n 的階乘由 0!=1 與後繼遞迴式定義，等於 1 到 n 的正整數乘積。","en":"The factorial of a natural number n is defined by 0!=1 and a successor recurrence, and equals the product of the positive integers from 1 through n."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["組合","階乘","自然數","遞迴","AI候選","待形式化"]},{"id":"mko-absolute-value","title":{"zh-Hant":"絕對值","en":"Absolute value"},"summary":{"zh-Hant":"實數 x 的絕對值是 x 到 0 的非負距離，並依 x 的正負號分段取 x 或 -x。","en":"The absolute value of a real number x is its nonnegative distance from 0, taking x or -x according to the sign of x."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["分析","實數","絕對值","距離","AI候選","待形式化"]},{"id":"mko-fraction","title":{"zh-Hant":"分數","en":"Fraction"},"summary":{"zh-Hant":"分數 p/q 是有理數的一種表示；分子與分母為整數，分母必須非零。","en":"A fraction p/q is a representation of a rational number whose numerator and denominator are integers and whose denominator is nonzero."},"type":"definition","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["分數","AI候選","待形式化"]},{"id":"mko-ratio-proportion","title":{"zh-Hant":"比與比例","en":"Ratio and proportion"},"summary":{"zh-Hant":"本條目把數值比 a:b 解讀為 a/b，把比例定義為兩個合法比值相等。","en":"This entry interprets the numerical ratio a:b as a/b and defines a proportion as the equality of two valid ratios."},"type":"definition","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["比與比例","AI候選","待形式化"]},{"id":"mko-domain-codomain-range","title":{"zh-Hant":"定義域、陪域與值域","en":"Domain, codomain, and range"},"summary":{"zh-Hant":"總函數 f:A→B 的定義域是 A，陪域是 B，值域是實際取得的輸出集合 f[A]。","en":"For a total function f:A→B, A is the domain, B is the codomain, and f[A] is the set of outputs actually attained."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["定義域、陪域與值域","AI候選","待形式化"]},{"id":"mko-composition","title":{"zh-Hant":"函數合成","en":"Function composition"},"summary":{"zh-Hant":"合成 g∘f 先求 f 的輸出，再把它送入 g；合成前須確認中間值可作為 g 的輸入。","en":"The composition g∘f first evaluates f and then supplies its output to g; the intermediate value must be an allowed input of g."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["函數合成","AI候選","待形式化"]},{"id":"mko-sequence","title":{"zh-Hant":"數列","en":"Sequence"},"summary":{"zh-Hant":"本條目採從 0 起始的實數數列 a:N→R；數列項依索引排列，不能任意重排成集合。","en":"This entry uses real sequences a:N→R indexed from 0. Sequence terms are ordered by their indices and cannot be rearranged into an unordered set."},"type":"definition","domain":"analysis","formalization":"not_formalized","tags":["數列","AI候選","待形式化"]},{"id":"mko-permutation-combination","title":{"zh-Hant":"排列與組合","en":"Permutations and combinations"},"summary":{"zh-Hant":"由 n 個相異物件不重複取 k 個，有序選取是排列，無序選取是組合。","en":"When k distinct objects are selected without replacement from n distinct objects, an ordered selection is a permutation and an unordered selection is a combination."},"type":"concept","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["排列與組合","AI候選","待形式化"]},{"id":"mko-coordinate-plane","title":{"zh-Hant":"座標平面","en":"Coordinate plane"},"summary":{"zh-Hant":"固定原點、互相垂直的有向實數軸與单位後，平面點用有序實數對 (x,y) 表示。","en":"After fixing an origin, perpendicular oriented real axes, and units, a point in the plane is represented by an ordered pair of real numbers (x,y)."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["座標平面","AI候選","待形式化"]},{"id":"mko-power-root","title":{"zh-Hant":"自然數冪與主平方根","en":"Natural-number powers and principal square roots"},"summary":{"zh-Hant":"本輪限定實數底數的自然數冪，以及非負實數的主平方根，分開冪運算、根值與解集。","en":"This entry covers natural-number powers of real bases and principal square roots of nonnegative real numbers, keeping powers, root values, and solution sets distinct."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["自然數冪與主平方根","AI候選","待形式化"]},{"id":"mko-polynomial","title":{"zh-Hant":"一元有理係數多項式","en":"Univariate polynomials over the rationals"},"summary":{"zh-Hant":"把多項式視為有限支撐的有理係數資料；合併同次項後再讀取係數與次數。","en":"A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["一元有理係數多項式","AI候選","待形式化"]},{"id":"mko-factorization","title":{"zh-Hant":"有理係數多項式因式分解","en":"Polynomial factorization over the rationals"},"summary":{"zh-Hant":"因式分解是在指定係數域中把多項式寫成因子乘積；驗收比較全部係數而不是若干代入值。","en":"Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["有理係數多項式因式分解","AI候選","待形式化"]},{"id":"mko-quadratic-equation","title":{"zh-Hant":"一元實二次方程式","en":"Quadratic equations over the reals"},"summary":{"zh-Hant":"對非零二次係數的實方程，用判別式區分實根數，將方程式、根集合與重根分開。","en":"For a real equation with a nonzero quadratic coefficient, the discriminant classifies its distinct real roots. The equation, root set, and repeated roots are kept distinct."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["一元實二次方程式","AI候選","待形式化"]},{"id":"mko-linear-function","title":{"zh-Hant":"仿射函數與一次函數","en":"Affine functions and degree-one functions"},"summary":{"zh-Hant":"實函數 f(x)=mx+b 稱仿射函數；m≠0 時為一次函數，m=0 時為常數函數。","en":"A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["仿射函數與一次函數","AI候選","待形式化"]},{"id":"mko-function-graph","title":{"zh-Hant":"實值函數圖形","en":"Graphs of real-valued functions"},"summary":{"zh-Hant":"固定直角座標系後，實值函數的圖形是全部合法輸入與其唯一輸出組成的有序對集合。","en":"In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["實值函數圖形","AI候選","待形式化"]},{"id":"mko-probability","title":{"zh-Hant":"有限樣本空間上的機率","en":"Probability on finite sample spaces"},"summary":{"zh-Hant":"在非空有限樣本空間上指定非負且總和為一的權重，事件機率為其成員權重之和。","en":"Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["有限樣本空間上的機率","AI候選","待形式化"]},{"id":"mko-quantifier","title":{"zh-Hant":"量詞與明示論域","en":"Quantifiers and explicit domains"},"summary":{"zh-Hant":"全稱與存在量詞必須綁定論域；古典真值語義下，空論域的全稱為真、存在為假。","en":"Universal and existential quantifiers must bind a domain. Under classical truth semantics, a universal statement over the empty domain is true and an existential statement is false."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["量詞與明示論域","AI候選","待形式化"]},{"id":"mko-counterexample","title":{"zh-Hant":"反例與論域資格","en":"Counterexamples and domain eligibility"},"summary":{"zh-Hant":"反例必須屬於被聲稱的論域且使全稱述詞失敗；域外值和找不到反例都不能自動證明原命題。","en":"A counterexample must belong to the claimed domain and falsify the universal predicate. An outside value or failure to find a counterexample does not by itself prove the original statement."},"type":"definition","domain":"history_philosophy_methodology","formalization":"not_formalized","tags":["反例與論域資格","AI候選","待形式化"]},{"id":"mko-relation","title":{"zh-Hant":"二元關係與集合投影","en":"Binary relations and set projections"},"summary":{"zh-Hant":"二元關係是指定 A×B 中有序對的集合，不要求每個輸入唯一或必有輸出。","en":"A binary relation is a set of ordered pairs in a specified A×B. An input need not have a unique output or any output."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["二元關係與集合投影","AI候選","待形式化"]},{"id":"mko-graph","title":{"zh-Hant":"有限簡單無向圖","en":"Finite simple undirected graphs"},"summary":{"zh-Hant":"本輪限定有限簡單無向圖，邊是兩個相異頂點的無序集合，度數只計相鄰頂點。","en":"This entry covers finite simple undirected graphs. An edge is an unordered set of two distinct vertices, and degree counts distinct neighbors."},"type":"definition","domain":"combinatorics_discrete","formalization":"not_formalized","tags":["有限簡單無向圖","AI候選","待形式化"]},{"id":"mko-vector","title":{"zh-Hant":"固定基底下的平面向量","en":"Plane vectors in a fixed basis"},"summary":{"zh-Hant":"本輪把平面向量限制為固定基底下的 R² 有序對，逐分量定義加法與純量乘法。","en":"This entry represents plane vectors as ordered pairs in R² in a fixed basis, defining addition and scalar multiplication componentwise."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["固定基底下的平面向量","AI候選","待形式化"]}],"title":"Knowledge objects"}