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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 ...
A bar chart can represent a function whose domain is a finite set, the natural numbers, or the integers. In this case, an element x of the domain is represented by an interval of the x -axis, and the corresponding value of the function, f ( x ) , is represented by a rectangle whose base is the interval corresponding to x and whose height is f ...
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 codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph. [1] The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it.
The domain-to-range ratio (DRR) is a ratio which describes how the number of outputs corresponds to the number of inputs of a given logical function or software component. The domain-to-range ratio is a mathematical ratio of cardinality between the set of the function's possible inputs (the domain) and the set of possible outputs (the range).
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Layers have a domain and codomain ,: defined in the obvious way. This forms a directed multigraph , also known as a quiver , with the types as vertices and the layers as edges. A string diagram d {\displaystyle d} is encoded as a path in this multigraph , i.e. it is given by:
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires .
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