Domain (mathematics)

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In mathematics, a domain of a k-place relation L ⊆ X₁ × … × X_k is one of the sets X_j, 1 ≤ j ≤ k.

In the special case where k = 2 and L ⊆ X₁ × X₂ is a function L : X₁ → X₂, it is conventional to refer to X₁ as the domain of the function and to refer to X₂ as the codomain of the function.

1 Domain of a function
2 Domain of a partial function
3 Category theory
4 Complex analysis
5 See also

[edit] Domain of a function

Given a function f:X→Y, the set X of input values is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values.

A well-defined function must map every element of the 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 R of real numbers cannot be its domain. In cases like this, the function is usually either defined on R\{0}, or the "gap" is plugged by specifically 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 we can choose its domain to be 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.

[edit] 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. Some (particularly category theorists), however, consider the domain of a partial function f:X→Y to be X, irrespective of whether f(x) exists for all x in X.

[edit] Category theory

In category theory, instead of functions, one deals with morphisms, which are simply arrows from one object to another. The domain of any morphism is then simply the object where the arrow starts. In this context, many set theoretic ideas about domains have to 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.