{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","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":{},"kind":"knowledge","total":302,"pages":13,"page":1,"entries":[{"id":"mko-point-line-plane","title":{"zh-Hant":"點、線、面","en":"Points, lines, and planes"},"summary":{"zh-Hant":"點、線與面提供描述位置、方向、共線、共面與入射關係的基礎幾何語言。","en":"Points, lines, and planes provide foundational geometric language for position, direction, collinearity, coplanarity, and incidence."},"type":"concept","domain":"geometry","formalization":"not_formalized","tags":["幾何","點","線","面","AI候選","待形式化"]},{"id":"mko-angle","title":{"zh-Hant":"角","en":"Angle"},"summary":{"zh-Hant":"在初等歐幾里得幾何中，角由具有共同端點的兩條射線形成，並可用度數描述其張開量。","en":"In elementary Euclidean geometry, an angle is formed by two rays with a common endpoint, and its opening can be described in degrees."},"type":"concept","domain":"geometry","formalization":"not_formalized","tags":["幾何","角","度數","AI候選","待形式化"]},{"id":"mko-triangle","title":{"zh-Hant":"三角形","en":"Triangle"},"summary":{"zh-Hant":"在平面中，三個相異且不共線的點與連接它們的三條線段形成非退化三角形。","en":"In the plane, three distinct noncollinear points together with the three segments joining them form a nondegenerate triangle."},"type":"concept","domain":"geometry","formalization":"not_formalized","tags":["幾何","三角形","頂點","邊","AI候選","待形式化"]},{"id":"mko-right-triangle","title":{"zh-Hant":"直角三角形","en":"Right triangle"},"summary":{"zh-Hant":"具有一個內角為直角的三角形。","en":"A triangle having one interior angle that is a right angle."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["幾何","三角形","定義"]},{"id":"mko-euclidean-length","title":{"zh-Hant":"歐幾里得長度","en":"Euclidean length"},"summary":{"zh-Hant":"平面上兩點之間由座標差平方和的正平方根所定義的距離。","en":"The distance between two points in the plane is defined as the nonnegative square root of the sum of squared coordinate differences."},"type":"definition","domain":"geometry","formalization":"not_formalized","tags":["幾何","距離","定義"]},{"id":"mko-euclid-pythagorean-theorem","title":{"zh-Hant":"畢達哥拉斯定理","en":"Pythagorean theorem"},"summary":{"zh-Hant":"直角三角形兩股長度的平方和，等於斜邊長度的平方。","en":"In a right triangle, the sum of the squares of the two leg lengths equals the square of the hypotenuse length."},"type":"theorem","domain":"geometry","formalization":"formalized_declared_side_model","tags":["幾何","三角形","定理","FELRA","Lean","Mathlib","語義橋"]},{"id":"mko-natural-number","title":{"zh-Hant":"自然數","en":"Natural number"},"summary":{"zh-Hant":"自然數是用於計數與離散次序的基本數；本條目採用包含 0 的慣例。","en":"Natural numbers are basic numbers for counting and discrete order; this entry adopts the convention that they include 0."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["自然數","數系","基礎","AI候選","待形式化"]},{"id":"mko-integer","title":{"zh-Hant":"整數","en":"Integer"},"summary":{"zh-Hant":"整數由零、正整數與負整數組成，擴充自然數以表示相反方向與差額。","en":"The integers consist of zero, the positive integers, and the negative integers, extending natural numbers to represent opposite directions and differences."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["整數","數系","負數","AI候選","待形式化"]},{"id":"mko-rational-number","title":{"zh-Hant":"有理數","en":"Rational number"},"summary":{"zh-Hant":"有理數可表示為兩個整數之比，且分母不可為零。","en":"A rational number can be represented as a ratio of two integers with a nonzero denominator."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["有理數","數系","分數","比值","AI候選","待形式化"]},{"id":"mko-real-number","title":{"zh-Hant":"實數","en":"Real number"},"summary":{"zh-Hant":"實數包含有理數，並以完備化補入有理數系統中缺少的極限點。","en":"The real numbers contain the rational numbers and complete the rational system by adding its missing limit points."},"type":"concept","domain":"analysis","formalization":"not_formalized","tags":["實數","數系","完備化","分析","AI候選","待形式化"]},{"id":"mko-arithmetic-operations","title":{"zh-Hant":"四則運算","en":"Arithmetic operations"},"summary":{"zh-Hant":"加、減、乘、除是依指定數系與定義域操作兩個數量的基本運算家族。","en":"Addition, subtraction, multiplication, and division are a family of basic operations on two quantities within a specified number system and domain."},"type":"concept","domain":"arithmetic_number_systems","formalization":"not_formalized","tags":["算術","運算","數系","AI候選","待形式化"]},{"id":"mko-variable-expression","title":{"zh-Hant":"變數與代數式","en":"Variables and algebraic expressions"},"summary":{"zh-Hant":"代數式以變數、常數與已定義運算表示依輸入而定的數學量。","en":"An algebraic expression uses variables, constants, and defined operations to represent a mathematical quantity depending on its inputs."},"type":"concept","domain":"algebra","formalization":"not_formalized","tags":["代數","變數","表達式","AI候選","待形式化"]},{"id":"mko-equation","title":{"zh-Hant":"方程式","en":"Equation"},"summary":{"zh-Hant":"方程式要求兩個表達式在指定變數範圍內取相等值；解是使等式成立的允許值。","en":"An equation requires two expressions to take equal values over a specified variable domain; a solution is an allowed value that makes the equality true."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["代數","方程式","解集","等式","AI候選","待形式化"]},{"id":"mko-linear-equation","title":{"zh-Hant":"一元一次方程式","en":"Linear equation in one variable"},"summary":{"zh-Hant":"一元一次方程式可寫成 ax+b=0，且一次係數 a 必須非零。","en":"A linear equation in one variable can be written as ax+b=0 with a nonzero linear coefficient a."},"type":"definition","domain":"algebra","formalization":"not_formalized","tags":["代數","一次方程式","唯一解","非零係數","AI候選","待形式化"]},{"id":"mko-set","title":{"zh-Hant":"集合","en":"Set"},"summary":{"zh-Hant":"集合把指定脈絡中的若干對象視為一個整體，並以元素隸屬來描述其內容。","en":"A set treats selected objects within a specified context as a whole and describes its contents through element membership."},"type":"concept","domain":"foundations_logic","formalization":"not_formalized","tags":["集合","基礎","外延","AI候選","待形式化"]},{"id":"mko-proposition","title":{"zh-Hant":"命題","en":"Proposition"},"summary":{"zh-Hant":"命題是在指定邏輯脈絡中可被斷言或否定，並可談論真假的有意義陳述。","en":"A proposition is a meaningful statement that can be asserted or denied and discussed as true or false within a specified logical context."},"type":"concept","domain":"foundations_logic","formalization":"not_formalized","tags":["邏輯","命題","基礎","AI候選","待形式化"]},{"id":"mko-logical-connectives","title":{"zh-Hant":"邏輯連接詞","en":"Logical connectives"},"summary":{"zh-Hant":"否定、合取、析取、蘊涵與雙條件用來由既有命題形成複合命題。","en":"Negation, conjunction, disjunction, implication, and biconditional form compound propositions from existing propositions."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["邏輯","連接詞","合取","蘊涵","AI候選","待形式化"]},{"id":"mko-proof","title":{"zh-Hant":"數學證明","en":"Mathematical proof"},"summary":{"zh-Hant":"數學證明是在明示的前提與推理規則下，建立指定結論的可檢查論證。","en":"A mathematical proof is a checkable argument establishing a specified conclusion from explicit premises and rules of inference."},"type":"concept","domain":"history_philosophy_methodology","formalization":"not_formalized","tags":["證明","邏輯","方法論","AI候選","待形式化"]},{"id":"mko-set-membership","title":{"zh-Hant":"集合隸屬","en":"Set membership"},"summary":{"zh-Hant":"符號 x∈A 表示元素 x 屬於集合 A。","en":"The notation x∈A states that the element x belongs to the set A."},"type":"definition","domain":"foundations_logic","formalization":"formalized_lean_set_semantics","tags":["集合","基礎","關係","Lean","Mathlib"]},{"id":"mko-subset","title":{"zh-Hant":"子集合","en":"Subset"},"summary":{"zh-Hant":"A⊆B 表示 A 的每個元素也都是 B 的元素。","en":"A⊆B means that every element of A is also an element of B."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["集合","子集合","包含","關係","AI候選","待形式化"]},{"id":"mko-set-operations","title":{"zh-Hant":"集合運算","en":"Set operations"},"summary":{"zh-Hant":"聯集、交集、差集與相對補集以元素隸屬條件從既有集合構造新集合。","en":"Union, intersection, difference, and relative complement construct new sets from existing sets through element-membership conditions."},"type":"definition","domain":"foundations_logic","formalization":"not_formalized","tags":["集合","聯集","交集","差集","補集","AI候選","待形式化"]},{"id":"mko-sample-space","title":{"zh-Hant":"樣本空間","en":"Sample space"},"summary":{"zh-Hant":"樣本空間是針對一個已指定隨機試驗所列出的所有可能基本結果。","en":"A sample space lists all possible elementary outcomes for one specified random experiment."},"type":"concept","domain":"probability_statistics","formalization":"not_formalized","tags":["機率","樣本空間","結果","AI候選","待形式化"]},{"id":"mko-event","title":{"zh-Hant":"事件","en":"Event"},"summary":{"zh-Hant":"事件描述隨機試驗中關心的一組可能結果。","en":"An event describes a set of outcomes of interest in a random experiment."},"type":"definition","domain":"probability_statistics","formalization":"not_formalized","tags":["機率","事件","樣本空間","子集合","AI候選","待形式化"]},{"id":"mko-function-mapping","title":{"zh-Hant":"函數映射","en":"Function mapping"},"summary":{"zh-Hant":"符號 f:X→Y 表示函數 f 從定義域 X 映射到陪域 Y。","en":"The notation f:X→Y states that the function f maps the domain X into the codomain Y."},"type":"definition","domain":"foundations_logic","formalization":"formalized_total_function_core","tags":["函數","映射","基礎","Lean","Mathlib"]}],"title":"知識物件"}