Characterization (mathematics) (original) (raw)
En langage mathématique, la caractérisation d'un objet X par une propriété P signifie que non seulement X possède la propriété P mais de plus X est le seul objet à posséder la propriété P. Il est également assez courant de rencontrer des affirmations telles que : « la propriété Q caractérise Y à isomorphisme près », qui indique que les objets vérifiant Q sont exactement les objets isomorphes à Y (à la place d'« isomorphisme » dans l'expression « à … près », une autre « relation d'équivalence entre objets » pourrait être spécifiée).
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| dbo:abstract | In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property P characterizes object X" is to say that not only does X have property P, but that X is the only thing that has property P (i.e., P is a defining property of X). Similarly, a set of properties P is said to characterize X, when these properties distinguish X from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P". It is also common to find statements such as "Property Q characterizes Y up to isomorphism". The first type of statement says in different words that the extension of P is a singleton set, while the second says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved). A reference on mathematical terminology notes that characteristic originates from the Greek term kharax, "a pointed stake": "From Greek kharax came kharakhter, an instrument used to mark or engrave an object. Once an object was marked, it became distinctive, so the character of something came to mean its distinctive nature. The Late Greek suffix -istikos converted the noun character into the adjective characteristic, which, in addition to maintaining its adjectival meaning, later became a noun as well." Just as in chemistry, the characteristic property of a material will serve to identify a sample, or in the study of materials, structures and properties will determine characterization, in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique to mathematics, but since the science is abstract, much of the activity can be described as "characterization". For instance, in Mathematical Reviews, as of 2018, more than 24,000 articles contain the word in the article title, and 93,600 somewhere in the review. In an arbitrary context of objects and features, characterizations have been expressed via the heterogeneous relation aRb, meaning that object a has feature b. For example, b may mean abstract or concrete. The objects can be considered the extensions of the world, while the features are expression of the intensions. A continuing program of characterization of various objects leads to their categorization. (en) En langage mathématique, la caractérisation d'un objet X par une propriété P signifie que non seulement X possède la propriété P mais de plus X est le seul objet à posséder la propriété P. Il est également assez courant de rencontrer des affirmations telles que : « la propriété Q caractérise Y à isomorphisme près », qui indique que les objets vérifiant Q sont exactement les objets isomorphes à Y (à la place d'« isomorphisme » dans l'expression « à … près », une autre « relation d'équivalence entre objets » pourrait être spécifiée). (fr) 数学において、「性質 P が対象 X を特徴づける (characterize)」とは、X が性質 P を持っているだけでなく、性質 P を持っているものが X のみである ことを意味する。「性質 Q は Y を同型の違いを除いて特徴づける」というような主張も一般的である。 (ja) Stwierdzenie, że „własność P charakteryzuje obiekt X” oznacza nie tylko, że X ma własność P, ale że X jest jedynym obiektem, który ma własność P. Często spotyka się także zdania takie jak „własność Q charakteryzuje obiekt Y co do izomorfizmu”. Stwierdzenie pierwszego rodzaju mówi innymi słowy, że P jest zbiór jednoelementowy; drugie zaś, że rozszerzeniem Q jest jedna klasa abstrakcji (w tym przypadku izomorfizmu – jednak o rodzaj relacji równoważności zależy od wyrażenia znajdującego się za słowami co do). (pl) |
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| rdfs:comment | En langage mathématique, la caractérisation d'un objet X par une propriété P signifie que non seulement X possède la propriété P mais de plus X est le seul objet à posséder la propriété P. Il est également assez courant de rencontrer des affirmations telles que : « la propriété Q caractérise Y à isomorphisme près », qui indique que les objets vérifiant Q sont exactement les objets isomorphes à Y (à la place d'« isomorphisme » dans l'expression « à … près », une autre « relation d'équivalence entre objets » pourrait être spécifiée). (fr) 数学において、「性質 P が対象 X を特徴づける (characterize)」とは、X が性質 P を持っているだけでなく、性質 P を持っているものが X のみである ことを意味する。「性質 Q は Y を同型の違いを除いて特徴づける」というような主張も一般的である。 (ja) Stwierdzenie, że „własność P charakteryzuje obiekt X” oznacza nie tylko, że X ma własność P, ale że X jest jedynym obiektem, który ma własność P. Często spotyka się także zdania takie jak „własność Q charakteryzuje obiekt Y co do izomorfizmu”. Stwierdzenie pierwszego rodzaju mówi innymi słowy, że P jest zbiór jednoelementowy; drugie zaś, że rozszerzeniem Q jest jedna klasa abstrakcji (w tym przypadku izomorfizmu – jednak o rodzaj relacji równoważności zależy od wyrażenia znajdującego się za słowami co do). (pl) In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property P characterizes object X" is to say that not only does X have property P, but that X is the only thing that has property P (i.e., P is a defining property of X). Similarly, a set of properties P is said to characterize X, when these properties distinguish X from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of X in terms of P include "P is necessary and sufficient for X", and "X holds if and only if P". (en) |
| rdfs:label | Characterization (mathematics) (en) Caractérisation (mathématiques) (fr) 特徴づけ (数学) (ja) Charakteryzacja (matematyka) (pl) |
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