Domain (mathematics)

All or part of this article may be confusing or unclear. Please help clarify the article. Suggestions may be on the talk page. (January 2008)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined.^[1] In a representation of a function in a xy Cartesian coordinate system, the domain is represented on the abscissa (the x axis).

Domain of a function

Given a function f:X→Y, the set X of input values is the domain of f; the set Y is the codomain of f.

The range of f is the set of all output values of f; this is the set Failed to parse (Missing texvc executable; please see math/README to configure.): \{ f(x) : x \in X \} . ^[2] The range of f can be the same set as the codomain or it can be a proper subset of it. It is in general smaller than the codomain unless f is a surjective function.

A well defined function must map every element of its domain to an element of its codomain. For example, the function f defined by

f(x) = 1/x

has no value for f(0). Thus, the set of real numbers, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} , cannot be its domain. In cases like this, the function is either defined on Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} \backslash \{0 \}

or the "gap is plugged" by explicitly defining f(0).

If we extend the definition of f to

f(x) = 1/x, for x ≠ 0

f(0) = 0,

then f is defined for all real numbers, and its domain is Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} .

Any function can be restricted to a subset of its domain. The restriction of g : A → B to S, where S ⊆ A, is written g |_S : S → B.

Domain of a partial function

There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the term "domain of f" for the set of all values x such that f(x) is defined. But some, particularly category theorists, consider the domain of a partial function f:X→Y to be X, irrespective of whether f(x) exists for every x in X.

Category theory

In category theory one deals with morphisms instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject for more.

Real and complex analysis

In real and complex analysis, a domain is an open connected subset of a real or complex vector space.

See also

• Range (mathematics)
• Codomain
• Surjective function
• Injective function
• Bijection

References

1. ^ Paley, H. Abstract Algebra, Holt, Rinehart and Winston, 1966 (p. 16).
2. ^ Smith, William K. Inverse Functions, MacMillan, 1966 (p. 8).

ca:Domini (matemàtiques) cs:Definiční obor da:Definitionsmængde de:Definitionsmenge et:Määramispiirkond es:Dominio de definición eo:Fonto-aro fr:Ensemble de définition hr:Domena (matematika) io:Ensemblo di defino is:Formengi it:Dominio (matematica) nl:Domein (wiskunde) pl:Dziedzina sk:Definičný obor sv:Definitionsmängd ta:ஆட்களம் (கணிதம்) vi:Tập xác định