Double counting (proof technique) (original) (raw)
Doppeltes Abzählen ist ein Beweisverfahren aus dem Gebiet der abzählenden Kombinatorik, das aber auch auf anderen Gebieten Anwendung findet. Das Prinzip besteht darin, die Mächtigkeit einer Menge auf zwei verschiedene Arten zu ermitteln. Die beiden Ergebnisse müssen dann gleich sein, so dass man eine Gleichung erhält.
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| dbo:abstract | Doppeltes Abzählen ist ein Beweisverfahren aus dem Gebiet der abzählenden Kombinatorik, das aber auch auf anderen Gebieten Anwendung findet. Das Prinzip besteht darin, die Mächtigkeit einer Menge auf zwei verschiedene Arten zu ermitteln. Die beiden Ergebnisse müssen dann gleich sein, so dass man eine Gleichung erhält. (de) In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which call "one of the most important tools in combinatorics", one describes a finite set from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other. (en) En mathématiques combinatoires, une preuve par double dénombrement, ou double comptage, ou encore double décompte, est une technique de preuve combinatoire servant à démontrer que deux expressions sont égales en prouvant qu'il y a deux façons de compter le nombre d'éléments d'un même ensemble. Van Lint et Wilson décrivent cette technique comme « un des outils les plus importants en combinatoire ». (fr) Una dimostrazione mediante doppio conteggio è un genere di dimostrazione utilizzata in combinatoria che ha come scopo una uguaglianza di due espressioni enumerative che forniscono la cardinalità di un insieme finito X e consiste in due diversi modi di contare gli elementi di tale insieme. I due modi di contare riguardano due diverse prospettive per l'organizzazione dell'insieme X, oppure due diversi procedimenti per costruirlo, oppure due diversi percorsi per (cioè per passare su tutti e suoi elementi una sola volta). (it) 在數學中，算兩次是一個常用的證明技巧，常在證明恆等式時被提到。其思想是，對一個具體的量用方法甲來計算，得到的答案是A，而用方法乙則得到B，那麼等式A = B成立。此思想雖然明顯，但在實際使用時由於方法甲與方法乙通常有明顯的差異，因此能把兩個表面上相去甚遠的式子聯繫起來。算兩次產生過很多。 (zh) |
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| rdfs:comment | Doppeltes Abzählen ist ein Beweisverfahren aus dem Gebiet der abzählenden Kombinatorik, das aber auch auf anderen Gebieten Anwendung findet. Das Prinzip besteht darin, die Mächtigkeit einer Menge auf zwei verschiedene Arten zu ermitteln. Die beiden Ergebnisse müssen dann gleich sein, so dass man eine Gleichung erhält. (de) In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which call "one of the most important tools in combinatorics", one describes a finite set from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other. (en) En mathématiques combinatoires, une preuve par double dénombrement, ou double comptage, ou encore double décompte, est une technique de preuve combinatoire servant à démontrer que deux expressions sont égales en prouvant qu'il y a deux façons de compter le nombre d'éléments d'un même ensemble. Van Lint et Wilson décrivent cette technique comme « un des outils les plus importants en combinatoire ». (fr) Una dimostrazione mediante doppio conteggio è un genere di dimostrazione utilizzata in combinatoria che ha come scopo una uguaglianza di due espressioni enumerative che forniscono la cardinalità di un insieme finito X e consiste in due diversi modi di contare gli elementi di tale insieme. I due modi di contare riguardano due diverse prospettive per l'organizzazione dell'insieme X, oppure due diversi procedimenti per costruirlo, oppure due diversi percorsi per (cioè per passare su tutti e suoi elementi una sola volta). (it) 在數學中，算兩次是一個常用的證明技巧，常在證明恆等式時被提到。其思想是，對一個具體的量用方法甲來計算，得到的答案是A，而用方法乙則得到B，那麼等式A = B成立。此思想雖然明顯，但在實際使用時由於方法甲與方法乙通常有明顯的差異，因此能把兩個表面上相去甚遠的式子聯繫起來。算兩次產生過很多。 (zh) |
| rdfs:label | Doppeltes Abzählen (de) Double counting (proof technique) (en) Preuve par double dénombrement (fr) Dimostrazione mediante doppio conteggio (it) 算兩次 (zh) |
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