morphism

Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is a subset of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in G(f). X is called the domain of f, Y is called the codomain of F, and G(f) is called th ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Formal definition

X, the set of input values, is called the domain of f, sometimes denoted dom(f), and Y. The distinction between "codomain" and "range" is best understood by means of an example. Consider, once again, f(x) = x2. This function has no negative outputs, but does permit negative inputs. The domain is R, the set of all real numbers. It is often convenient to consider this as a function from R into R, and so it is possible to call R the codomain and to write f:R → R. The "range", however, is the set of all outputs, which in this ca ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Domains codomains and ranges

The image of an element x in X under f is the output f(x). The image of a subset A of X under f is the subset of Y formally defined by f[A] = {f(x) | x is in A} Usually (when subsets of the domain are not at the same time elements of the domain) one writes f(A) instead of f[A]. (An old-fashioned notation writes f'x instead of f(x ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Images and preimages

As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, a ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - History

A straightforward generalization is to allow functions depending on several arguments. For instance, g(x,y) = xy is a function which takes two inputs, x and y, and outputs their product, xy. The input (x,y) is often thought of as an ordered pair. Functions whose inputs consist of ordered pairs are called "binary" or "2-ary". In the sciences, we often encounter functions that are not given by (known) formulas. Consider ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Functions of more than one variable

An arbitrary topological space X can be considered a locally ringed space by taking OX to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of X (there may exist continuous functions over open subsets of X which are not the restriction of any continuous function over X). The stalk at a point x can be thought of as the set of all germs of continuous functions at x; this is a local ring with maximal ideal consisting of ...

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Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - O_X modules

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Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let X be locally ringed space with structure sheaf OX; we want to define the tangent space Tx at the point x ∈ X. Take the local ring (stalk) Rx at the point x, with maximal ideal mx. Then kx := Rx/mx is a field and mx/mx2 is a vector space over that field (the cotangent space). The tangent space RxSee also:

Ringed space, Ringed space - Definition, Ringed space - Examples, Ringed space - Morphisms, Ringed space - Tangent spaces, Ringed space - O_X modules

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One may consider associative algebras over a commutative ring R: these are modules over R together with a R-bilinear map which yields an associative multiplication. In this case, a unital R-algebra A can equivalently be defined as a ring A with a ring homomorphism R→A. The n-by-n matrices with integer entries form an associative algebra over the integers and the polynomials with coefficients in the ring Z/nZ (see modular arithmetic) form ...

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Associative algebra, Associative algebra - Definition, Associative algebra - Examples, Associative algebra - Algebra homomorphisms, Associative algebra - Index-free notation, Associative algebra - Generalizations, Associative algebra - Coalgebras, Associative algebra - Representations, Associative algebra - Motivation for a Hopf algebra, Associative algebra - Motivation for a Lie algebra

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