Information about Domain (mathematics)
For the term in algebra, see Domain (ring theory). For the type of a partially ordered set, see Domain theory.
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.
Domain of a function
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
, 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,
.
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. 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.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.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
References
In abstract algebra, a domain is a ring with 0 ≠ 1 such that ab = 0 implies that either a = 0 or b = 0 (the zero-product property). That is, it is a nontrivial ring without left or right zero divisors.
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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.
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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In mathematics, the codomain of a function : → is the set .
The domain of is the set .
The range of is the set defined as .
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The domain of is the set .
The range of is the set defined as .
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In mathematics, the range of a function is the set of all "output" values produced by that function. Sometimes it is called the image, or more precisely, the image of the domain of the function.
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non-surjective function.]] In mathematics, a function f is said to be surjective if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y .
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion or containment.
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In mathematics, a partial function is a binary relation that associates each element of a set, sometimes called its domain, with at most one element of another (possibly the same) set, called its codomain.
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Recursion theory, also called computability theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability.
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In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
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In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and have been a unifying notion.
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In mathematics, a morphism is an abstraction of a structure-preserving mapping between two mathematical structures.
The most common example occurs when the process is a function or map which preserves the structure in some sense.
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The most common example occurs when the process is a function or map which preserves the structure in some sense.
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In category theory, there is a general definition of subobject extending the idea of subset and subgroup.
In detail, suppose we are given some category C and monomorphisms
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In detail, suppose we are given some category C and monomorphisms
- u: S → A and
- v:
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Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education.
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics.
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In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U.
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In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open spaces.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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In mathematics, the codomain of a function : → is the set .
The domain of is the set .
The range of is the set defined as .
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The domain of is the set .
The range of is the set defined as .
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In mathematics, the range of a function is the set of all "output" values produced by that function. Sometimes it is called the image, or more precisely, the image of the domain of the function.
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non-injective function.]] In mathematics, an injective function is a function which associates distinct arguments to distinct values. More precisely, a function f is said to be injective if it maps distinct x in the domain to distinct y
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In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.
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f(x) = y.
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