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In mathematics, a domain is most often defined as the set of values, D for which a function is defined.^[1] A function that has a domain N is said to be a function over N, where N is an arbitrary set.

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

The range of f is the set of all output values of f; this is the set . The range of f is a subset of the codomain Y. 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, , 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 .

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.

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.

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.

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

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

1. ^ Paley, H: "Abstract Algebra", page 16. Holt Rinehart and Winston, Inc., 1966