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- For the concept in language, see Definition In mathematics, the term well-defined is used to specify that a certain concept or object is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy the required properties. Sometimes however, it is economical to state a definition in terms of an arbitrary choice; one then has to check that the definition is independent of that choice. On other occasions, the required properties might not all be obvious; one then has to verify them. These issues commonly arise in the definition of functions. For instance, in group theory, the term well-defined is often used when dealing with cosets, where a function on a coset space is often defined by choosing a representative: it is then as important that we check that we get the same result regardless of which representative of the coset we choose as it is that we always get the same result when we perform arithmetical operations (e.g. , whenever we add 2 and 3, we always get the answer 5). f(x1)=f(x2) if x1~x2, then the definition makes sense, and f is well-defined on X/~. Although the distinction is often ignored, the function on X/~, having a different domain, should be viewed as a distinct map <math>\tilde{f}</math>. In this view, one says that <math>\tilde{f}</math> is well-defined if the diagram shown commutes. That is, that f factors through π, where π is the canonical projection map X → X/~, so that <math>f=\tilde{f}\circ\pi</math>. As an example, consider the equivalence relation between real numbers defined by θ1~θ2 if there is an integer n such that θ1-θ2 = 2πn, where π (not italicized) denotes Pi. The quotient set X/~ may then be identified with a circle, as an equivalence class [θ] represents an angle. (In fact this is the coset space R/2πZ of the additive subgroup 2πZ of R. ) Now if f:R→R is the cosine function, then <math>\tilde{f}=\cos\theta</math> is well-defined, whereas if f(&theta) = θ then <math>\tilde{f}=\theta</math> is not well-defined. Two other issues of well-definition arise when defining a function f from a set X to a set Y. First, f should actually be defined on all elements of X. For example, the function f(x) = 1/x is not well-defined as a function from the real numbers to itself, as f(0) is not defined. Secondly, f(x) should be an element of Y for all x∈X. For example, the function f(x) = x is not well-defined as a function from the real numbers to the positive real numbers, as f(0) is not positive. A set is well-defined if any given object either is an element of the set, or is not an element of the set.
- Man kann in der Mathematik ein Objekt nicht nur durch eine Definitionsgleichung (explizit), sondern auch durch eine charakteristische Eigenschaft (implizit) definieren. Während eine explizite Definition immer zulässig ist, ist eine implizite Definition nur unter der Bedingung zulässig, dass es tatsächlich genau ein Objekt mit der angegebenen Eigenschaft gibt. Diese Bedingung nennt man die Wohldefiniertheit der impliziten Definition. Den Beweis der Wohldefiniertheit kann man in zwei Teile zerlegen: die Existenz und die Eindeutigkeit des zu definierenden Begriffs. Implizite Definitionen tauchen oft unbemerkt auf, wenn man Abbildungen auf Faktormengen definiert. Das Ergebnis wird durch eine Abbildung auf einem Repräsentanten definiert. Wohldefiniertheit läuft hier auf die Unabhängigkeit des Ergebnisses von der Wahl des Repräsentanten hinaus.
- well-defined は、ある概念が数学的あるいは論理学的に特定の条件を公理に用いて定義・導入されるとき、その定義（における公理の組）が自己矛盾を孕まぬ状態にあることを言い表す修飾語句である。文脈により、「うまく定義されている」「矛盾なく定まった」「定義可能である」などと表現されることもある。 well-defined は「状態」を表す形容詞であるが、日本語の定訳はなく慣例的に形容詞と動詞の複合語に訳されるか、そのまま形容動詞的に「well-defined である」といった形で用いる。名詞形 well-definedness などもあり、これを well-defined 性と記すことはできるが日本語訳としてこなれたものは特には存在しない（文脈によっては「定義可能性」などで代用可能である）。
- Em matemáticas, o termo bem definido se utiliza para especificar que um conceito se define de forma lógica ou matemática usando um conjunto de axiomas básicos sem ambigüidade alguma. Usualmente as definições se enunciam sem ambigüidade, e não há dúvidas sobre sua definição. Ocasionalmente, contudo, se enuncia uma definição com base a uma escolha arbitrária por motivos de economia; então deve-se comprovar que a definição é independente da dita escolha.
- 在数学里，术语定义良好（定义良好的 well-defined，名词 well-definition）用于确认用一组基本公理以数学或逻辑的方式定义的某个概念或对象（一个函数，性质，关系，等等）是完全无歧义的，满足它必需满足的那些性质。通常定义是无歧义地表述，明白地满足它们所需的性质。但有时候，使用任意选择的方式来陈述定义是经济的，这时我们便要验证定义与选择无关。另一种情形，所需的性质可能不都是显然的，这时要验证它们。这些问题通常来自函数的定义。 譬如，在群论中，术语“定义良好”经常用于处理陪集时，陪集空间上的函数经常选取一个代表来定义：这时非常重要的是验证无论选取陪集的哪个代表，就像算术运算一样（比如，2加3总是5）我们总得到同样的结果。 f(x1)=f(x2) 只要 x1~x2，则定义有意义，从而 f 在 X/~ 上定义良好。函数在 X/~ 上有不同定义域，应该视为不同的映射 <math>\tilde{f}</math>，尽管这种差别通常被忽略。以这种观点来看，我们说 <math>\tilde{f}</math> 是定义良好的如果图表交换，即 f 穿过 π，使得<math>f=\tilde{f}\circ\pi</math>，这里 π 是典范投影映射 X → X/~。 作为一个例子，考虑实数如下定义的等价关系：θ1~θ2 如果存在整数 n 使得θ1-θ2 = 2πn，这里 π 为圆周率。商集 X/~ 可以和一个圆周等价，作为等价类 [θ] 表示一个角度（事实上这是 R 的加法子群 2πZ 的陪集空间 R/2πZ）。现在如果 f:R→R 是正弦函数，则 <math>\tilde{f}=\cos\theta</math> 是定义良好的；但是如果 f(&theta) = θ 则 <math>\tilde{f}=\theta</math> 不是定义良好的。 “定义良好”的另外两个问题发生在定义从一个集合 X 到集合 Y 的函数时。首先，f 需定义在 X 的所有元素上。譬如，函数 f(x)=1/x 不是从实数到自身定义良好的函数，因为 f(0) 没有定义。第二，对任何 x ∈ X 需有 f(x) 是 Y 中的元素。譬如，函数 f(x)=x 不是从实数到正实数定义良好的函数，因为 f(0) 不是正数。 一个集合是定义良好的如果任何给定的对象要么是、要么不是这个集合的对象。
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- For the concept in language, see Definition In mathematics, the term well-defined is used to specify that a certain concept or object is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy the required properties.
- Man kann in der Mathematik ein Objekt nicht nur durch eine Definitionsgleichung (explizit), sondern auch durch eine charakteristische Eigenschaft (implizit) definieren. Während eine explizite Definition immer zulässig ist, ist eine implizite Definition nur unter der Bedingung zulässig, dass es tatsächlich genau ein Objekt mit der angegebenen Eigenschaft gibt. Diese Bedingung nennt man die Wohldefiniertheit der impliziten Definition.
- Em matemáticas, o termo bem definido se utiliza para especificar que um conceito se define de forma lógica ou matemática usando um conjunto de axiomas básicos sem ambigüidade alguma. Usualmente as definições se enunciam sem ambigüidade, e não há dúvidas sobre sua definição. Ocasionalmente, contudo, se enuncia uma definição com base a uma escolha arbitrária por motivos de economia; então deve-se comprovar que a definição é independente da dita escolha.
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