# Domain (mathematics)

﻿
Domain (mathematics)

In mathematics, the domain of a given functionis the set of "input" values for which the function is defined. [Paley, H. "Abstract Algebra", Holt, Rinehart and Winston, 1966 (p. 16).] For instance, the domain of cosine would be all real numbers, while the domain of the square root would only be numbers greater than or equal to 0 (ignoring complex numbers in both cases). In a representation of a function in a "xy" Cartesian coordinate system, the domain is represented on the "x" axis (or abscissa).

Formal definition

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 $\left\{ f\left(x\right) : x in X \right\}$. [ Smith, William K. "Inverse Functions", MacMillan, 1966 (p. 8).] 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, $mathbb\left\{R\right\}$, cannot be its domain.In cases like this, the function is either defined on 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 $mathbb\left\{R\right\}$.

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.

In partial differential equations, a domain is an open connected subset of the euclidean space Rn, where the problem is posed, i.e., where the unknown function(s) are defined.

ee also

* Range (mathematics)
* Codomain
* Surjective function
* Injective function
* Bijection
* Domain decomposition
* Lipschitz domain

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Domain of discourse — In the formal sciences, the domain of discourse, also called the universe of discourse (or simply universe), is the set of entities over which certain variables of interest in some formal treatment may range. The domain of discourse is usually… …   Wikipedia

• Domain theory — is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science,… …   Wikipedia

• Domain decomposition methods — Domain dec …   Wikipedia

• Domain — may refer to: General Territory (administrative division), a non sovereign geographic area which has come under the authority of another government Public domain, a body of works and knowledge without proprietary interest Eminent domain, the… …   Wikipedia

• Domain knowledge — is that valid knowledge used to refer to an area of human endeavour, an autonomous computer activity, or other specialized discipline. Specialists and experts use and develop their own domain knowledge. If the concept domain knowledge or domain… …   Wikipedia

• Domain of a function — Venn diagram showing f, a function from domain X to codomain Y. The smaller oval inside Y is the image of f, sometimes called the range of f. In mathematics, the domain of definition or simply the domain of a function is the set of input or… …   Wikipedia

• mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

• Mathematics of radio engineering — A complex valued function. The mathematics of radio engineering is a pleasant and very useful subject. This article is an attempt to provide a reasonably comprehensive summary of this almost limitless topic. While the ideas have historically… …   Wikipedia

• Domain-specific language — Programming paradigms Agent oriented Automata based Component based Flow based Pipelined Concatenative Concurrent computing …   Wikipedia

• Domain (ring theory) — In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. (Sometimes such a ring is… …   Wikipedia