. "Definici\u00F3 inductiva"@ca . . . . . . "D\u00E9finition par r\u00E9currence"@fr . . . "\u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F (\u0442\u0430\u043A\u043E\u0436 \u0456\u043D\u0434\u0443\u043A\u0442\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F) \u0443 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0456\u0439 \u043B\u043E\u0433\u0456\u0446\u0456 \u0442\u0430 \u0456\u043D\u0444\u043E\u0440\u043C\u0430\u0442\u0438\u0446\u0456 \u2014 \u0437\u0430\u0434\u0430\u043D\u043D\u044F \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0456\u0432 \u043C\u043D\u043E\u0436\u0438\u043D \u0447\u0435\u0440\u0435\u0437 \u0456\u043D\u0448\u0456 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438 \u0446\u0456\u0454\u0457 \u0436 \u043C\u043D\u043E\u0436\u0438\u043D\u0438 (Aczel 1978:740). \u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0432\u0441\u0442\u0430\u043D\u043E\u0432\u043B\u044E\u0454 \u0440\u0435\u0437\u0443\u043B\u044C\u0442\u0430\u0442 \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0434\u043B\u044F \u0434\u0435\u044F\u043A\u0438\u0445 \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u0456\u0432 \u0447\u0435\u0440\u0435\u0437 \u0457\u0457 \u0436 \u0440\u0435\u0437\u0443\u043B\u044C\u0442\u0430\u0442 \u0434\u043B\u044F \u0456\u043D\u0448\u0438\u0445 \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u0456\u0432. \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0444\u0430\u043A\u0442\u043E\u0440\u0456\u0430\u043B\u0430 n! \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u043D\u0430\u0441\u0442\u0443\u043F\u043D\u0438\u043C\u0438 \u043F\u0440\u0430\u0432\u0438\u043B\u0430\u043C\u0438: 0! = 1.(n+1)! = (n+1)\u00B7n!. \u0441\u0442\u0432\u0435\u0440\u0434\u0436\u0443\u0454, \u0449\u043E \u0442\u0430\u043A\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043D\u0430\u0441\u043F\u0440\u0430\u0432\u0434\u0456 \u0437\u0430\u0434\u0430\u0454 \u0444\u0443\u043D\u043A\u0446\u0456\u044E. \u0414\u043E\u0432\u0435\u0434\u0435\u043D\u043D\u044F \u0491\u0440\u0443\u043D\u0442\u0443\u0454\u0442\u044C\u0441\u044F \u043D\u0430 \u043C\u0435\u0442\u043E\u0434\u0456 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u043E\u0457 \u0456\u043D\u0434\u0443\u043A\u0446\u0456\u0457."@uk . . "Definici\u00F3n recursiva"@es . . . "Na l\u00F3gica matem\u00E1tica e em ci\u00EAncia da computa\u00E7\u00E3o, uma defini\u00E7\u00E3o recursiva (ou defini\u00E7\u00E3o indutiva) \u00E9 usada para definir um objeto em termos de si pr\u00F3prio (Aczel 1977). Uma defini\u00E7\u00E3o recursiva de uma fun\u00E7\u00E3o define valores das fun\u00E7\u00F5es para algumas entradas em termos dos valores da mesma fun\u00E7\u00E3o para outras entradas. Por exemplo, a fun\u00E7\u00E3o fatorial n! \u00E9 definida pelas regras 0! = 1.(n+1)! = (n+1)\u00B7n!. Uma defini\u00E7\u00E3o indutiva de um conjunto descreve os elementos de um conjunto em termos de outros elementos no conjunto. Por exemplo, uma defini\u00E7\u00E3o do conjunto dos n\u00FAmeros naturais \u00E9:"@pt . . "1047605"^^ . . "\u9012\u5F52\u5B9A\u4E49\u662F\u6570\u7406\u903B\u8F91\u548C\u8BA1\u7B97\u673A\u79D1\u5B66\u7528\u5230\u7684\u4E00\u79CD\u5B9A\u4E49\u65B9\u5F0F\uFF0C\u4F7F\u7528\u88AB\u5B9A\u4E49\u5BF9\u8C61\u7684\u81EA\u8EAB\u6765\u4E3A\u5176\u4E0B\u5B9A\u4E49\uFF08\u7B80\u5355\u8BF4\u5C31\u662F\u81EA\u6211\u590D\u5236\u7684\u5B9A\u4E49\uFF09\u3002\u9012\u5F52\u5B9A\u4E49\u4E0E\u5F52\u7EB3\u5B9A\u4E49\u7C7B\u4F3C\uFF0C\u4F46\u4E5F\u6709\u4E0D\u540C\u4E4B\u5904\u3002\u9012\u5F52\u5B9A\u4E49\u4E2D\u4F7F\u7528\u88AB\u5B9A\u4E49\u5BF9\u8C61\u81EA\u8EAB\u6765\u5B9A\u4E49\uFF0C\u800C\u5F52\u7EB3\u5B9A\u4E49\u662F\u4F7F\u7528\u88AB\u5B9A\u4E49\u5BF9\u8C61\u7684\u5DF2\u7ECF\u5B9A\u4E49\u7684\u90E8\u5206\u6765\u5B9A\u4E49\u5C1A\u672A\u5B9A\u4E49\u7684\u90E8\u5206\u3002\u4E0D\u8FC7\uFF0C\u4F7F\u7528\u9012\u5F52\u5B9A\u4E49\u7684\u51FD\u6570\u6216\u96C6\u5408\uFF0C\u5B83\u4EEC\u7684\u6027\u8D28\u53EF\u4EE5\u7528\u6570\u5B66\u5F52\u7EB3\u6CD5\uFF0C\u901A\u8FC7\u9012\u5F52\u5B9A\u4E49\u7684\u5185\u5BB9\u6765\u8BC1\u660E\u3002"@zh . . "Defini\u00E7\u00E3o recursiva"@pt . "Recursive definition"@en . "En math\u00E9matiques, on parle de d\u00E9finition par r\u00E9currence pour une suite, c'est-\u00E0-dire une fonction d\u00E9finie sur les entiers positifs et \u00E0 valeurs dans un ensemble donn\u00E9. Une fonction est d\u00E9finie par r\u00E9currence quand, pour d\u00E9finir la valeur de la fonction en un entier donn\u00E9, on utilise les valeurs de cette m\u00EAme fonction pour des entiers strictement inf\u00E9rieurs. \u00C0 la diff\u00E9rence d'une d\u00E9finition usuelle, qui peut \u00EAtre vue comme une simple abr\u00E9viation, une d\u00E9finition par r\u00E9currence utilise le nom de l'objet d\u00E9fini (la fonction en l'occurrence) dans la d\u00E9finition m\u00EAme. Le principe de d\u00E9finition par r\u00E9currence assure l'existence et l'unicit\u00E9 de la fonction ainsi d\u00E9finie. Il est distinct de celui du raisonnement par r\u00E9currence, dont il n'est pas cons\u00E9quence sans les autres axiomes de Peano. Richard Dedekind l'identifie et en donne une d\u00E9monstration en 1888 dans son ouvrage Was sind und was sollen die Zahlen ? (\u00AB Que sont et \u00E0 quoi servent les nombres ? \u00BB), qui utilise une axiomatisation des entiers dans un cadre ensembliste. Les d\u00E9finitions par r\u00E9currence se g\u00E9n\u00E9ralisent aux ordinaux et ensembles bien ordonn\u00E9s, et plus g\u00E9n\u00E9ralement aux relations bien fond\u00E9es. On parle \u00E9galement de d\u00E9finition par induction (sur les entiers positifs, sur tel bon ordre, sur les ordinaux, etc.). Elle se g\u00E9n\u00E9ralise aussi aux objets structur\u00E9s (par exemple les arbres binaires ou les termes), on parle alors de r\u00E9currence structurelle ou d'induction structurelle et elle est particuli\u00E8rement utilis\u00E9e en informatique pour d\u00E9finir des fonctions (par exemple la taille)."@fr . . . . . . "In matematica una definizione ricorsiva di un insieme A si ha quando per definire A vengono elencati degli elementi di A e delle regole per costruire nuovi elementi di A a partire da elementi di A. Ad esempio l'insieme P dei numeri pari pu\u00F2 essere definito ricorsivamente dicendo: \n* 2 appartiene a P \n* se un numero n appartiene a P allora anche n+2 appartiene a P Una definizione ricorsiva di una funzione f definita sui numeri naturali si ha quando f viene definita dando esplicitamente il valore che assume su 0 e dando una regola per calcolare il valore della funzione su n a partire dal valore che assume su n-1. Anche in ambiente informatico l'uso della definizione ricorsiva \u00E8 piuttosto comune, soprattutto sotto forma di acronimo ricorsivo: un esempio molto noto \u00E8 GNU = GNU's Not Unix dove si pu\u00F2 notare come il nome \u00E8 la parte in un certo senso meno importante della definizione stessa. Infine, l'induzione matematica pu\u00F2 portare a una specie di definizione ricorsiva, dove per\u00F2 c'\u00E8 un caso speciale al quale tutti gli altri prima o poi devono giungere e che quindi fa terminare la ricorsione. Ad esempio, per calcolare il fattoriale di un numero positivo n, si pu\u00F2 dire il fattoriale di 1 \u00E8 1;il fattoriale di n \u00E8 n volte il fattoriale di (n-1), se n \u00E8 maggiore di 1."@it . . "\u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F (\u0442\u0430\u043A\u043E\u0436 \u0456\u043D\u0434\u0443\u043A\u0442\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F) \u0443 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u0456\u0439 \u043B\u043E\u0433\u0456\u0446\u0456 \u0442\u0430 \u0456\u043D\u0444\u043E\u0440\u043C\u0430\u0442\u0438\u0446\u0456 \u2014 \u0437\u0430\u0434\u0430\u043D\u043D\u044F \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0456\u0432 \u043C\u043D\u043E\u0436\u0438\u043D \u0447\u0435\u0440\u0435\u0437 \u0456\u043D\u0448\u0456 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438 \u0446\u0456\u0454\u0457 \u0436 \u043C\u043D\u043E\u0436\u0438\u043D\u0438 (Aczel 1978:740). \u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0432\u0441\u0442\u0430\u043D\u043E\u0432\u043B\u044E\u0454 \u0440\u0435\u0437\u0443\u043B\u044C\u0442\u0430\u0442 \u0444\u0443\u043D\u043A\u0446\u0456\u0457 \u0434\u043B\u044F \u0434\u0435\u044F\u043A\u0438\u0445 \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u0456\u0432 \u0447\u0435\u0440\u0435\u0437 \u0457\u0457 \u0436 \u0440\u0435\u0437\u0443\u043B\u044C\u0442\u0430\u0442 \u0434\u043B\u044F \u0456\u043D\u0448\u0438\u0445 \u043F\u0430\u0440\u0430\u043C\u0435\u0442\u0440\u0456\u0432. \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u0444\u0443\u043D\u043A\u0446\u0456\u044F \u0444\u0430\u043A\u0442\u043E\u0440\u0456\u0430\u043B\u0430 n! \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u043D\u0430\u0441\u0442\u0443\u043F\u043D\u0438\u043C\u0438 \u043F\u0440\u0430\u0432\u0438\u043B\u0430\u043C\u0438: 0! = 1.(n+1)! = (n+1)\u00B7n!. \u0414\u0430\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u0434\u0456\u0439\u0441\u043D\u0435 \u0434\u043B\u044F \u0432\u0441\u0456\u0445 n, \u0447\u0435\u0440\u0435\u0437 \u0442\u0435, \u0449\u043E \u0432 \u043F\u0440\u043E\u0446\u0435\u0441\u0456 \u0440\u0435\u043A\u0443\u0440\u0441\u0456\u0457 \u0432\u0440\u0435\u0448\u0442\u0456-\u0440\u0435\u0448\u0442 \u0434\u043E\u0441\u044F\u0433\u0430\u0454\u0442\u044C\u0441\u044F \u043F\u043E\u0447\u0430\u0442\u043A\u043E\u0432\u0438\u0439 \u0432\u0430\u0440\u0456\u0430\u043D\u0442 0. \u041E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043C\u043E\u0436\u043D\u0430 \u0442\u0430\u043A\u043E\u0436 \u0440\u043E\u0437\u0443\u043C\u0456\u0442\u0438 \u044F\u043A \u043E\u043F\u0438\u0441 \u043F\u0440\u043E\u0446\u0435\u0434\u0443\u0440\u0438, \u0449\u043E \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454 \u0444\u0443\u043D\u043A\u0446\u0456\u044E n!, \u043F\u043E\u0447\u0438\u043D\u0430\u044E\u0447\u0438 \u0437 n = 0 \u0456 \u043F\u0440\u043E\u0433\u0440\u0435\u0441\u0443\u044E\u0447\u0438 \u0434\u0430\u043B\u0456 \u0434\u043B\u044F n = 1, n = 2, n = 3, \u0456 \u0442.\u0434. \u0441\u0442\u0432\u0435\u0440\u0434\u0436\u0443\u0454, \u0449\u043E \u0442\u0430\u043A\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043D\u0430\u0441\u043F\u0440\u0430\u0432\u0434\u0456 \u0437\u0430\u0434\u0430\u0454 \u0444\u0443\u043D\u043A\u0446\u0456\u044E. \u0414\u043E\u0432\u0435\u0434\u0435\u043D\u043D\u044F \u0491\u0440\u0443\u043D\u0442\u0443\u0454\u0442\u044C\u0441\u044F \u043D\u0430 \u043C\u0435\u0442\u043E\u0434\u0456 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u043E\u0457 \u0456\u043D\u0434\u0443\u043A\u0446\u0456\u0457. \u0406\u043D\u0434\u0443\u043A\u0442\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043C\u043D\u043E\u0436\u0438\u043D\u0438 \u043E\u043F\u0438\u0441\u0443\u0454 \u0457\u0457 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438 \u0447\u0435\u0440\u0435\u0437 \u0456\u043D\u0448\u0456 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438. \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043C\u043D\u043E\u0436\u0438\u043D\u0438 \u043D\u0430\u0442\u0443\u0440\u0430\u043B\u044C\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B N: 1. \n* 1 \u043D\u0430\u043B\u0435\u0436\u0438\u0442\u044C N. 2. \n* \u042F\u043A\u0449\u043E \u0435\u043B\u0435\u043C\u0435\u043D\u0442 n \u043D\u0430\u043B\u0435\u0436\u0438\u0442\u044C N, \u0442\u043E n+1 \u0442\u0430\u043A\u043E\u0436 \u043D\u0430\u043B\u0435\u0436\u0438\u0442\u044C N. 3. \n* N \u2014 \u043F\u0435\u0440\u0435\u0442\u0438\u043D \u0432\u0441\u0456\u0445 \u043C\u043D\u043E\u0436\u0438\u043D, \u0449\u043E \u0437\u0430\u0434\u043E\u0432\u043E\u043B\u044C\u043D\u044F\u044E\u0442\u044C \u0443\u043C\u043E\u0432\u0430\u043C (1) \u0456 (2). \u041C\u043E\u0436\u043D\u0430 \u0441\u043A\u043E\u043D\u0441\u0442\u0440\u0443\u044E\u0432\u0430\u0442\u0438 \u0431\u0430\u0433\u0430\u0442\u043E \u043C\u043D\u043E\u0436\u0438\u043D, \u0449\u043E \u0437\u0430\u0434\u043E\u0432\u043E\u043B\u044C\u043D\u044F\u044E\u0442\u044C (1) \u0456 (2) \u2014 \u043D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u043C\u043D\u043E\u0436\u0438\u043D\u0430 {1, 1.649, 2, 2.649, 3, 3.649, ...}. \u041E\u0434\u043D\u0430\u043A \u0441\u0430\u043C\u0435 \u0443\u043C\u043E\u0432\u0430 (3) \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454 \u043C\u043D\u043E\u0436\u0438\u043D\u0443 \u043D\u0430\u0442\u0443\u0440\u0430\u043B\u044C\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B, \u0432\u0438\u0434\u0430\u043B\u044F\u044E\u0447\u0438 \u0432\u0441\u0456 \u043F\u0456\u0434\u043C\u043D\u043E\u0436\u0438\u043D\u0438, \u0449\u043E \u043C\u0456\u0441\u0442\u044F\u0442\u044C \u043D\u0435\u043D\u0430\u0442\u0443\u0440\u0430\u043B\u044C\u043D\u0456 \u0447\u0438\u0441\u043B\u0430. \u0412\u043B\u0430\u0441\u0442\u0438\u0432\u043E\u0441\u0442\u0456 \u0440\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u043E-\u043E\u0437\u043D\u0430\u0447\u0435\u043D\u0438\u0445 \u0444\u0443\u043D\u043A\u0446\u0456\u0439 \u0456 \u043C\u043D\u043E\u0436\u0438\u043D \u0447\u0430\u0441\u0442\u043E \u043C\u043E\u0436\u043D\u0430 \u0432\u0438\u0432\u0435\u0441\u0442\u0438 \u0437 \u043F\u0440\u0438\u043D\u0446\u0438\u043F\u0443 \u043C\u0430\u0442\u0435\u043C\u0430\u0442\u0438\u0447\u043D\u043E\u0457 \u0456\u043D\u0434\u0443\u043A\u0446\u0456\u0457 (\u044F\u043A\u0438\u0439 \u0441\u043B\u0456\u0434\u0443\u0454 \u0440\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u043E\u043C\u0443 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044E). \u041D\u0430\u043F\u0440\u0438\u043A\u043B\u0430\u0434, \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043D\u0430\u0442\u0443\u0440\u0430\u043B\u044C\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B \u043D\u0430\u0432\u0435\u0434\u0435\u043D\u0435 \u0432\u0438\u0449\u0435 \u043D\u0430\u043F\u0440\u044F\u043C\u0443 \u043C\u0456\u0441\u0442\u0438\u0442\u044C \u043F\u0440\u0438\u043D\u0446\u0438\u043F \u0456\u043D\u0434\u0443\u043A\u0446\u0456\u0457 \u0434\u043B\u044F \u043D\u0430\u0442\u0443\u0440\u0430\u043B\u044C\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B: \u044F\u043A\u0449\u043E \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u0456\u0441\u0442\u044C \u0447\u0438\u043D\u043D\u0430 \u0434\u043B\u044F \u043D\u0430\u0442\u0443\u0440\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u0447\u0438\u0441\u043B\u0430 0, \u0456 \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u0456\u0441\u0442\u044C \u0447\u0438\u043D\u043D\u0430 \u0434\u043B\u044F n+1 \u043A\u043E\u0436\u043D\u043E\u0433\u043E \u0440\u0430\u0437\u0443, \u043A\u043E\u043B\u0438 \u0432\u043E\u043D\u0430 \u0447\u0438\u043D\u043D\u0430 \u0434\u043B\u044F n, \u0442\u043E\u0434\u0456 \u0432\u043B\u0430\u0441\u0442\u0438\u0432\u0456\u0441\u0442\u044C \u0447\u0438\u043D\u043D\u0430 \u0434\u043B\u044F \u0432\u0441\u0456\u0445 \u043D\u0430\u0442\u0443\u0440\u0430\u043B\u044C\u043D\u0438\u0445 \u0447\u0438\u0441\u0435\u043B (Aczel 1978:742)."@uk . . . "\u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u043E\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u0438\u043B\u0438 \u0438\u043D\u0434\u0443\u043A\u0442\u0438\u0432\u043D\u043E\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442 \u0441\u0443\u0449\u043D\u043E\u0441\u0442\u044C \u0432 \u0442\u0435\u0440\u043C\u0438\u043D\u0430\u0445 \u0435\u0451 \u0441\u0430\u043C\u043E\u0439 (\u0442\u043E \u0435\u0441\u0442\u044C \u0440\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u043E), \u0445\u043E\u0442\u044F \u0438 \u043F\u043E\u043B\u0435\u0437\u043D\u044B\u043C \u0441\u043F\u043E\u0441\u043E\u0431\u043E\u043C. \u0414\u043B\u044F \u0442\u043E\u0433\u043E, \u0447\u0442\u043E\u0431\u044B \u044D\u0442\u043E \u0431\u044B\u043B\u043E \u0432\u043E\u0437\u043C\u043E\u0436\u043D\u043E, \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u0432 \u043B\u044E\u0431\u043E\u043C \u0434\u0430\u043D\u043D\u043E\u043C \u0441\u043B\u0443\u0447\u0430\u0435 \u0434\u043E\u043B\u0436\u043D\u043E \u0431\u044B\u0442\u044C \u0444\u0443\u043D\u0434\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u044B\u043C, \u0438\u0437\u0431\u0435\u0433\u0430\u044F ."@ru . . "\u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u043E\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u0438\u043B\u0438 \u0438\u043D\u0434\u0443\u043A\u0442\u0438\u0432\u043D\u043E\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442 \u0441\u0443\u0449\u043D\u043E\u0441\u0442\u044C \u0432 \u0442\u0435\u0440\u043C\u0438\u043D\u0430\u0445 \u0435\u0451 \u0441\u0430\u043C\u043E\u0439 (\u0442\u043E \u0435\u0441\u0442\u044C \u0440\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u043E), \u0445\u043E\u0442\u044F \u0438 \u043F\u043E\u043B\u0435\u0437\u043D\u044B\u043C \u0441\u043F\u043E\u0441\u043E\u0431\u043E\u043C. \u0414\u043B\u044F \u0442\u043E\u0433\u043E, \u0447\u0442\u043E\u0431\u044B \u044D\u0442\u043E \u0431\u044B\u043B\u043E \u0432\u043E\u0437\u043C\u043E\u0436\u043D\u043E, \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u0432 \u043B\u044E\u0431\u043E\u043C \u0434\u0430\u043D\u043D\u043E\u043C \u0441\u043B\u0443\u0447\u0430\u0435 \u0434\u043E\u043B\u0436\u043D\u043E \u0431\u044B\u0442\u044C \u0444\u0443\u043D\u0434\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u044B\u043C, \u0438\u0437\u0431\u0435\u0433\u0430\u044F ."@ru . . "In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function n! is defined by the rules 0! = 1.(n + 1)! = (n + 1)\u00B7n!."@en . . . . . . "\u9012\u5F52\u5B9A\u4E49"@zh . "\u518D\u5E30\u7684\u5B9A\u7FA9\uFF08Recursive Definition\uFF09\u306F\u3001\u518D\u5E30\u7684\u306A\u5B9A\u7FA9\u3001\u3059\u306A\u308F\u3061\u3001\u3042\u308B\u3082\u306E\u3092\u5B9A\u7FA9\u3059\u308B\u306B\u3042\u305F\u3063\u3066\u305D\u308C\u81EA\u8EAB\u3092\u5B9A\u7FA9\u306B\u542B\u3080\u3082\u306E\u3092\u8A00\u3046\u3002\u7121\u9650\u5F8C\u9000\u3092\u907F\u3051\u308B\u305F\u3081\u3001\u5B9A\u7FA9\u306B\u542B\u307E\u308C\u308B\u300C\u305D\u308C\u81EA\u8EAB\u300D\u306F\u3088\u304F\u5B9A\u7FA9\u3055\u308C\u3066\u3044\u306A\u3051\u308C\u3070\u306A\u3089\u306A\u3044\u3002\u540C\u7FA9\u8A9E\u3068\u3057\u3066\u5E30\u7D0D\u7684\u5B9A\u7FA9\uFF08Inductive Definition\uFF09\u304C\u3042\u308B\u3002"@ja . "Una definici\u00F3n recursiva (o definici\u00F3n inductiva) en l\u00F3gica matem\u00E1tica y ciencias de la computaci\u00F3n se utiliza para definir los elementos de un conjunto en t\u00E9rminos de otros elementos del conjunto (Aczel 1978:740ff). La definici\u00F3n recursiva de una funci\u00F3n define los valores de las funciones para algunas entradas en t\u00E9rminos de los valores de la misma funci\u00F3n para otras entradas. Por ejemplo, la funci\u00F3n factorial n! est\u00E1 definida por las reglas 0! = 1. (n+1)! = (n+1)\u00B7n!. Esta definici\u00F3n es v\u00E1lida para todos los n, porque la recursividad finalmente alcanza el caso base de 0. Tambi\u00E9n se puede pensar en la definici\u00F3n como un procedimiento que describe c\u00F3mo construir la funci\u00F3n n!, comenzando desde n = 0 y continuando con n = 1, n = 2, n = 3, etc... El teorema de la recursividad establece que tal definici\u00F3n define efectivamente una funci\u00F3n. La prueba utiliza inducci\u00F3n matem\u00E1tica. Una definici\u00F3n inductiva de un conjunto describe los elementos de un conjunto en t\u00E9rminos de otros elementos del conjunto. Por ejemplo, una definici\u00F3n del conjunto de n\u00FAmeros naturales N es: 1. \n* 1 est\u00E1 en N 2. \n* Si un elemento n est\u00E1 en N entonces n+1 est\u00E1 tambi\u00E9n en N. 3. \n* N es la intersecci\u00F3n de todos los conjuntos que satisfacen (1) y (2). Hay muchos conjuntos que satisfacen (1) y (2) - por ejemplo, el conjunto {1, 1.649, 2, 2.649, 3, 3.649, ...} satisface la definici\u00F3n. Sin embargo, la condici\u00F3n (3) especifica el conjunto de n\u00FAmeros naturales eliminando los conjuntos con elementos extra\u00F1os. Las propiedades de las funciones y conjuntos definidos recursivamente se pueden probar a menudo mediante un principio de inducci\u00F3n que sigue la definici\u00F3n recursiva. Por ejemplo, la definici\u00F3n de los n\u00FAmeros naturales presentados aqu\u00ED implica directamente el principio de inducci\u00F3n matem\u00E1tica para los n\u00FAmeros naturales: si una propiedad tiene el n\u00FAmero natural 0, y la propiedad tiene n+1 cuando tiene n, entonces la propiedad tiene todos los n\u00FAmeros naturales (Aczel 1978:742)."@es . "\u518D\u5E30\u7684\u5B9A\u7FA9\uFF08Recursive Definition\uFF09\u306F\u3001\u518D\u5E30\u7684\u306A\u5B9A\u7FA9\u3001\u3059\u306A\u308F\u3061\u3001\u3042\u308B\u3082\u306E\u3092\u5B9A\u7FA9\u3059\u308B\u306B\u3042\u305F\u3063\u3066\u305D\u308C\u81EA\u8EAB\u3092\u5B9A\u7FA9\u306B\u542B\u3080\u3082\u306E\u3092\u8A00\u3046\u3002\u7121\u9650\u5F8C\u9000\u3092\u907F\u3051\u308B\u305F\u3081\u3001\u5B9A\u7FA9\u306B\u542B\u307E\u308C\u308B\u300C\u305D\u308C\u81EA\u8EAB\u300D\u306F\u3088\u304F\u5B9A\u7FA9\u3055\u308C\u3066\u3044\u306A\u3051\u308C\u3070\u306A\u3089\u306A\u3044\u3002\u540C\u7FA9\u8A9E\u3068\u3057\u3066\u5E30\u7D0D\u7684\u5B9A\u7FA9\uFF08Inductive Definition\uFF09\u304C\u3042\u308B\u3002"@ja . . "En math\u00E9matiques, on parle de d\u00E9finition par r\u00E9currence pour une suite, c'est-\u00E0-dire une fonction d\u00E9finie sur les entiers positifs et \u00E0 valeurs dans un ensemble donn\u00E9. Une fonction est d\u00E9finie par r\u00E9currence quand, pour d\u00E9finir la valeur de la fonction en un entier donn\u00E9, on utilise les valeurs de cette m\u00EAme fonction pour des entiers strictement inf\u00E9rieurs. \u00C0 la diff\u00E9rence d'une d\u00E9finition usuelle, qui peut \u00EAtre vue comme une simple abr\u00E9viation, une d\u00E9finition par r\u00E9currence utilise le nom de l'objet d\u00E9fini (la fonction en l'occurrence) dans la d\u00E9finition m\u00EAme."@fr . "\u518D\u5E30\u7684\u5B9A\u7FA9"@ja . . . . . . . . . . "En l\u00F2gica matem\u00E0tica i computaci\u00F3, una definici\u00F3 recursiva (o definici\u00F3 inductiva) s'usa per definir un objecte en termes de si mateix (Aczel 1978:740ff). Una definici\u00F3 recursiva d'una funci\u00F3 defineix els valors de la funci\u00F3 per algunes entrades en termes de valors de la mateixa funci\u00F3 donades diferents entrades. Per exemple, la funci\u00F3 factorial n! es defineix per les regles: 0! = 1.(n+1)! = (n+1)\u00B7n!. Aquesta definici\u00F3 \u00E9s v\u00E0lida perqu\u00E8 per tot n, la recursi\u00F3 sempre arriba al cas base 0. Per tant, la definici\u00F3 est\u00E0 ben fundamentada. La definici\u00F3 pot donar-se tamb\u00E9 com un conjunt de regles que descriuen com construir la funci\u00F3 n!, comen\u00E7ant per n = 0 i avan\u00E7at amb n = 1, n = 2, n = 3, etc. Una definici\u00F3 inductiva d'un conjunt descriu els elements d'un conjunt en termes d'altres elements del conjunt. Per exemple, una definici\u00F3 del conjunt N dels nombres naturals \u00E9s: 1. \n* 0 \u00E9s a N. 2. \n* Si un element n \u00E9s en N llavors n+1 \u00E9s en N. 3. \n* N \u00E9s el conjunt m\u00E9s petit que satisf\u00E0 (1) i (2). Hi ha molts conjunts que satisfan (1) i (2); la cl\u00E0usula (3) fa la definici\u00F3 precisa triant el conjunt m\u00E9s petit que esdev\u00E9 N. Les propietats de les funcions i conjunts definides recursivament sovint poden ser provades mitjan\u00E7ant una inducci\u00F3 que segueixi la definici\u00F3 recursiva. Per exemple, la definici\u00F3 dels nombres naturals presentada abans directament implica el principi de la inducci\u00F3 matem\u00E0tica pels nombres naturals: si el nombre 0 t\u00E9 si una propietat donada, i si la mateixa propietat la t\u00E9 tamb\u00E9 un nombre n+1 sempre que tamb\u00E9 la tingui el nombre n, llavors la propietat donada la tenen tots els nombres naturals (Aczel 1978:742)."@ca . . . . . . . . . "\u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F"@uk . . . "\u0420\u0435\u043A\u0443\u0440\u0441\u0438\u0432\u043D\u043E\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435"@ru . . . "In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function n! is defined by the rules 0! = 1.(n + 1)! = (n + 1)\u00B7n!. This definition is valid for each natural number n, because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function n!, starting from n = 0 and proceeding onwards with n = 1, n = 2, n = 3 etc. The recursion theorem states that such a definition indeed defines a function that is unique. The proof uses mathematical induction. An inductive definition of a set describes the elements in a set in terms of other elements in the set. For example, one definition of the set N of natural numbers is: 1. \n* 1 is in N. 2. \n* If an element n is in N then n + 1 is in N. 3. \n* N is the intersection of all sets satisfying (1) and (2). There are many sets that satisfy (1) and (2) \u2013 for example, the set {1, 1.649, 2, 2.649, 3, 3.649, ...} satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members. Note that this definition assumes that N is contained in a larger set (such as the set of real numbers) \u2014 in which the operation + is defined. Properties of recursively defined functions and sets can often be proved by an induction principle that follows the recursive definition. For example, the definition of the natural numbers presented here directly implies the principle of mathematical induction for natural numbers: if a property holds of the natural number 0 (or 1), and the property holds of n+1 whenever it holds of n, then the property holds of all natural numbers (Aczel 1977:742)."@en . . "In matematica una definizione ricorsiva di un insieme A si ha quando per definire A vengono elencati degli elementi di A e delle regole per costruire nuovi elementi di A a partire da elementi di A. Ad esempio l'insieme P dei numeri pari pu\u00F2 essere definito ricorsivamente dicendo: \n* 2 appartiene a P \n* se un numero n appartiene a P allora anche n+2 appartiene a P Anche in ambiente informatico l'uso della definizione ricorsiva \u00E8 piuttosto comune, soprattutto sotto forma di acronimo ricorsivo: un esempio molto noto \u00E8 GNU = GNU's Not Unix"@it . . . "1095408984"^^ . 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"\u062A\u0639\u0631\u064A\u0641 \u0627\u0644\u0634\u064A\u0621 \u0628\u0646\u0641\u0633\u0647 \u0647\u0648 \u062A\u0639\u0631\u064A\u0641\u0647 \u0628\u0645\u0627 \u0627\u0634\u062A\u0645\u0644 \u0639\u0644\u0649 \u0646\u0641\u0633 \u0627\u0644\u0645\u062D\u062F\u0648\u062F\u060C \u0643\u0645\u0627 \u0641\u064A \u0642\u0648\u0644\u0646\u0627: \u0627\u0644\u0625\u0646\u0633\u0627\u0646 \u062D\u064A\u0648\u0627\u0646 \u0628\u0634\u0631\u060C \u0641\u0625\u0646 \u062A\u0639\u0631\u064A\u0641 \u0627\u0644\u0625\u0646\u0633\u0627\u0646 \u0645\u0634\u062A\u0645\u0644 \u0639\u0644\u0649 \u0627\u0644\u0628\u0634\u0631\u060C \u0648\u0647\u0648 \u0646\u0641\u0633 \u0627\u0644\u0625\u0646\u0633\u0627\u0646."@ar . "\u9012\u5F52\u5B9A\u4E49\u662F\u6570\u7406\u903B\u8F91\u548C\u8BA1\u7B97\u673A\u79D1\u5B66\u7528\u5230\u7684\u4E00\u79CD\u5B9A\u4E49\u65B9\u5F0F\uFF0C\u4F7F\u7528\u88AB\u5B9A\u4E49\u5BF9\u8C61\u7684\u81EA\u8EAB\u6765\u4E3A\u5176\u4E0B\u5B9A\u4E49\uFF08\u7B80\u5355\u8BF4\u5C31\u662F\u81EA\u6211\u590D\u5236\u7684\u5B9A\u4E49\uFF09\u3002\u9012\u5F52\u5B9A\u4E49\u4E0E\u5F52\u7EB3\u5B9A\u4E49\u7C7B\u4F3C\uFF0C\u4F46\u4E5F\u6709\u4E0D\u540C\u4E4B\u5904\u3002\u9012\u5F52\u5B9A\u4E49\u4E2D\u4F7F\u7528\u88AB\u5B9A\u4E49\u5BF9\u8C61\u81EA\u8EAB\u6765\u5B9A\u4E49\uFF0C\u800C\u5F52\u7EB3\u5B9A\u4E49\u662F\u4F7F\u7528\u88AB\u5B9A\u4E49\u5BF9\u8C61\u7684\u5DF2\u7ECF\u5B9A\u4E49\u7684\u90E8\u5206\u6765\u5B9A\u4E49\u5C1A\u672A\u5B9A\u4E49\u7684\u90E8\u5206\u3002\u4E0D\u8FC7\uFF0C\u4F7F\u7528\u9012\u5F52\u5B9A\u4E49\u7684\u51FD\u6570\u6216\u96C6\u5408\uFF0C\u5B83\u4EEC\u7684\u6027\u8D28\u53EF\u4EE5\u7528\u6570\u5B66\u5F52\u7EB3\u6CD5\uFF0C\u901A\u8FC7\u9012\u5F52\u5B9A\u4E49\u7684\u5185\u5BB9\u6765\u8BC1\u660E\u3002"@zh . . . . . . "Una definici\u00F3n recursiva (o definici\u00F3n inductiva) en l\u00F3gica matem\u00E1tica y ciencias de la computaci\u00F3n se utiliza para definir los elementos de un conjunto en t\u00E9rminos de otros elementos del conjunto (Aczel 1978:740ff). La definici\u00F3n recursiva de una funci\u00F3n define los valores de las funciones para algunas entradas en t\u00E9rminos de los valores de la misma funci\u00F3n para otras entradas. Por ejemplo, la funci\u00F3n factorial n! est\u00E1 definida por las reglas 0! = 1. (n+1)! = (n+1)\u00B7n!."@es . "Definizione ricorsiva"@it . "Na l\u00F3gica matem\u00E1tica e em ci\u00EAncia da computa\u00E7\u00E3o, uma defini\u00E7\u00E3o recursiva (ou defini\u00E7\u00E3o indutiva) \u00E9 usada para definir um objeto em termos de si pr\u00F3prio (Aczel 1977). Uma defini\u00E7\u00E3o recursiva de uma fun\u00E7\u00E3o define valores das fun\u00E7\u00F5es para algumas entradas em termos dos valores da mesma fun\u00E7\u00E3o para outras entradas. Por exemplo, a fun\u00E7\u00E3o fatorial n! \u00E9 definida pelas regras 0! = 1.(n+1)! = (n+1)\u00B7n!. Esta defini\u00E7\u00E3o \u00E9 valida porque, para todo n, a recurs\u00E3o sempre vai alcan\u00E7ar o caso base de 0. Assim, a defini\u00E7\u00E3o \u00E9 bem-fundada. A defini\u00E7\u00E3o pode tamb\u00E9m ser vista como um procedimento que descreve como construir a fun\u00E7\u00E3o n!, a partir de n = 0 e prosseguindo em diante com n = 1, n = 2, n = 3 etc.. Uma defini\u00E7\u00E3o indutiva de um conjunto descreve os elementos de um conjunto em termos de outros elementos no conjunto. Por exemplo, uma defini\u00E7\u00E3o do conjunto dos n\u00FAmeros naturais \u00E9: 1. \n* 0 pertence a . 2. \n* Se um elemento n pertence a ent\u00E3o n+1 pertence a . 3. \n* \u00E9 o menor conjunto que satisfaz (1) e (2). H\u00E1 v\u00E1rios conjuntos que satisfazem (1) e (2) - por exemplo, o conjunto {0, 0.649, 1, 1.649, 2, 2.649, 3, 3.649, ...} satisfaz a defini\u00E7\u00E3o. No entanto, a condi\u00E7\u00E3o (3) especifica o conjunto dos n\u00FAmeros naturais, removendo os conjuntos com n\u00FAmeros externos. Propriedades de fun\u00E7\u00F5es e conjuntos definidos recursivamente muitas vezes podem ser provadas por um princ\u00EDpio de indu\u00E7\u00E3o que segue a defini\u00E7\u00E3o recursiva. Por exemplo, a defini\u00E7\u00E3o dos n\u00FAmeros naturais aqui apresentada implica o princ\u00EDpio da indu\u00E7\u00E3o matem\u00E1tica para os n\u00FAmeros naturais: se o n\u00FAmero natural 0 possui uma propriedade, e n+1 tem a propriedade sempre que n tamb\u00E9m a possui, ent\u00E3o a propriedade \u00E9 inerente a todos os n\u00FAmeros naturais (Aczel 1978:742)."@pt . . . . . . . . . . "En l\u00F2gica matem\u00E0tica i computaci\u00F3, una definici\u00F3 recursiva (o definici\u00F3 inductiva) s'usa per definir un objecte en termes de si mateix (Aczel 1978:740ff). Una definici\u00F3 recursiva d'una funci\u00F3 defineix els valors de la funci\u00F3 per algunes entrades en termes de valors de la mateixa funci\u00F3 donades diferents entrades. Per exemple, la funci\u00F3 factorial n! es defineix per les regles: 0! = 1.(n+1)! = (n+1)\u00B7n!. Una definici\u00F3 inductiva d'un conjunt descriu els elements d'un conjunt en termes d'altres elements del conjunt. Per exemple, una definici\u00F3 del conjunt N dels nombres naturals \u00E9s:"@ca . .