Successor function (original) (raw)
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n + 1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession.
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| dbo:abstract | En mathématiques, la fonction successeur est une fonction récursive primitive S telle que S(n) = n+1 pour tout entier naturel n. Par exemple, S(1) = 2 et S(2) = 3. La fonction successeur apparaît dans les axiomes de Peano qui définissent les entiers naturels. Elle n'y est pas définie à partir de l'opération d'addition, mais est une opération primitive qui sert à définir les entiers naturels à partir de 0, mais aussi les autres opérations sur les entiers naturels, dont l'addition. Par exemple, 1 est S(0), et l'addition sur les entiers est définie récursivement par: Par exemple. 5 + 2 = 5 + S(1) = S(5) + 1 = 6 + 1 = 6 + S(0) = S(6) + 0 = 7 + 0 = 7 Pour construire les nombres entiers en théorie des ensembles, une approche classique consiste à définir le nombre 0 par l'ensemble vide {}, et le successeur S(x) par x ∪ { x }. L'axiome de l'infini garantit alors l'existence d'un ensemble ℕ qui contient 0 et qui est clos par successeur, pris comme définition de l'ensemble des nombres entiers naturels. La fonction successeur est le niveau 0 de la hiérarchie infinie des hyperopérations (utilisées pour construire l'addition, la multiplication, l'exponentiation, etc.). (fr) In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n + 1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession. (en) 수학에서 다음수 함수(영어: successor function)또는 다음수 연산(영어: successor operation)은 자연수 n에 대해서 S(n) = n+1인 원시 재귀 함수 S이다.예를 들어, S(1) = 2이고 S(2) = 3이다. 다음수 연산은 또한 0차 하이퍼 연산의 맥락으로 제레이션(영어: zeration)으로 알려져 있다: H0(a, b) = 1 + b. (ko) 数学の分野における後者関数（こうしゃかんすう、後続者関数、英: successor function）、もしくは後者演算 (successor operation)は原始再帰関数のひとつである。後者関数 S は任意の自然数 n にその後者（後継、後続者）n + 1 を割り当てる: S(n) ≔ n + 1 (∀n)。例えばS(1) = 2 であり S(2) = 3 である。0-番目のハイパー演算 H0(a, b) ≔ 1 + b としての後者演算は「ゼレーション」("zeration") とも呼ばれる。 (ja) Em matemática, a função sucessora ou operação sucessora é uma Função recursiva primitiva tal que para cada número natural .Por exemplo, e . Operações sucessoras são também conhecidas como zeração no contexto de zeroth hiperoperação: . (pt) Successor (synonym efterföljare), begrepp inom logik. Successorn till ett naturligt tal är det minsta tal som är större än . Annorlunda uttryckt är successorn till a dess omedelbara efterföljare, därav namnet. (sv) 在 数学、 後繼函數 或 後繼運算 是一个 原始递归函数 S 使得 S(n)= n+1，n 為自然数。例如， S(1)=2和 S(2)=3。后继函數在西方国家也称为zeration，因為它是第零类超運算：H0(a, b)=1+ b。 (zh) |
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| rdfs:comment | In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n + 1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession. (en) 수학에서 다음수 함수(영어: successor function)또는 다음수 연산(영어: successor operation)은 자연수 n에 대해서 S(n) = n+1인 원시 재귀 함수 S이다.예를 들어, S(1) = 2이고 S(2) = 3이다. 다음수 연산은 또한 0차 하이퍼 연산의 맥락으로 제레이션(영어: zeration)으로 알려져 있다: H0(a, b) = 1 + b. (ko) 数学の分野における後者関数（こうしゃかんすう、後続者関数、英: successor function）、もしくは後者演算 (successor operation)は原始再帰関数のひとつである。後者関数 S は任意の自然数 n にその後者（後継、後続者）n + 1 を割り当てる: S(n) ≔ n + 1 (∀n)。例えばS(1) = 2 であり S(2) = 3 である。0-番目のハイパー演算 H0(a, b) ≔ 1 + b としての後者演算は「ゼレーション」("zeration") とも呼ばれる。 (ja) Em matemática, a função sucessora ou operação sucessora é uma Função recursiva primitiva tal que para cada número natural .Por exemplo, e . Operações sucessoras são também conhecidas como zeração no contexto de zeroth hiperoperação: . (pt) Successor (synonym efterföljare), begrepp inom logik. Successorn till ett naturligt tal är det minsta tal som är större än . Annorlunda uttryckt är successorn till a dess omedelbara efterföljare, därav namnet. (sv) 在 数学、 後繼函數 或 後繼運算 是一个 原始递归函数 S 使得 S(n)= n+1，n 為自然数。例如， S(1)=2和 S(2)=3。后继函數在西方国家也称为zeration，因為它是第零类超運算：H0(a, b)=1+ b。 (zh) En mathématiques, la fonction successeur est une fonction récursive primitive S telle que S(n) = n+1 pour tout entier naturel n. Par exemple, S(1) = 2 et S(2) = 3. La fonction successeur apparaît dans les axiomes de Peano qui définissent les entiers naturels. Elle n'y est pas définie à partir de l'opération d'addition, mais est une opération primitive qui sert à définir les entiers naturels à partir de 0, mais aussi les autres opérations sur les entiers naturels, dont l'addition. Par exemple, 1 est S(0), et l'addition sur les entiers est définie récursivement par: (fr) |
| rdfs:label | Fonction successeur (fr) 다음수 함수 (ko) 後者関数 (ja) Função sucessora (pt) Successor function (en) Successor (sv) 後繼函數 (zh) |
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