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The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...
The rough concept is older. In the 19th and early 20th century, the terms domain and region were often used informally (sometimes interchangeably) without explicit definition. [9] However, the term "domain" was occasionally used to identify closely related but slightly different concepts.
is a function from domain X to codomain Y. The yellow oval inside Y is the image of . Sometimes "range" refers to the image and sometimes to the codomain. In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or; the image of the function.
A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X, one often says that the partial function is a total function. In several areas of mathematics the term "function" refers to partial functions rather than to ...
More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain , codomain and graph ( ) = {(,) ():}. Similarly, one can define a right-restriction or range restriction R B . {\displaystyle R\triangleright B.}
Extension of a function, defined on a larger domain; Extension of a polyhedron, in geometry; Extension of a line segment (finite) into an infinite line (e.g., extended base) Exterior algebra, Grassmann's theory of extension, in geometry; Field extension, in Galois theory; Group extension, in abstract algebra and homological algebra
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree.