Английская Википедия:Function composition

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Шаблон:Short description Шаблон:About Шаблон:Redirect-distinguish Шаблон:Use dmy dates Шаблон:Functions

In mathematics, function composition is an operation Шаблон:Math that takes two functions Шаблон:Math and Шаблон:Math, and produces a function Шаблон:Math such that Шаблон:Math. In this operation, the function Шаблон:Math is applied to the result of applying the function Шаблон:Math to Шаблон:Math. That is, the functions Шаблон:Math and Шаблон:Math are composed to yield a function that maps Шаблон:Math in domain Шаблон:Math to Шаблон:Math in codomain Шаблон:Math. Intuitively, if Шаблон:Math is a function of Шаблон:Math, and Шаблон:Math is a function of Шаблон:Math, then Шаблон:Math is a function of Шаблон:Math. The resulting composite function is denoted Шаблон:Math, defined by Шаблон:Math for all Шаблон:Math in Шаблон:Math.[nb 1]

The notation Шаблон:Math is read as "Шаблон:Math of Шаблон:Math ", "Шаблон:Math after Шаблон:Math ", "Шаблон:Math circle Шаблон:Math ", "Шаблон:Math round Шаблон:Math ", "Шаблон:Math about Шаблон:Math ", "Шаблон:Math composed with Шаблон:Math ", "Шаблон:Math following Шаблон:Math ", "Шаблон:Math then Шаблон:Math", or "Шаблон:Math on Шаблон:Math ", or "the composition of Шаблон:Math and Шаблон:Math ". Intuitively, composing functions is a chaining process in which the output of function Шаблон:Math feeds the input of function Шаблон:Math.

The composition of functions is a special case of the composition of relations, sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,[1] such as the property of associativity.

Composition of functions is different from multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not commutative.[2]

Examples

Файл:Example for a composition of two functions.svg
Concrete example for the composition of two functions.

Properties

The composition of functions is always associative—a property inherited from the composition of relations.[1] That is, if Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar are composable, then Шаблон:Math.[3] Since the parentheses do not change the result, they are generally omitted.

In a strict sense, the composition Шаблон:Math is only meaningful if the codomain of Шаблон:Mvar equals the domain of Шаблон:Mvar; in a wider sense, it is sufficient that the former be an improper subset of the latter.[nb 2] Moreover, it is often convenient to tacitly restrict the domain of Шаблон:Mvar, such that Шаблон:Mvar produces only values in the domain of Шаблон:Mvar. For example, the composition Шаблон:Math of the functions Шаблон:Math defined by Шаблон:Math and Шаблон:Math defined by <math>g(x) = \sqrt x</math> can be defined on the interval Шаблон:Math.

Файл:Absolute value composition.svg
Compositions of two real functions, the absolute value and a cubic function, in different orders, show a non-commutativity of composition.

The functions Шаблон:Mvar and Шаблон:Mvar are said to commute with each other if Шаблон:Math. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, Шаблон:Math only when Шаблон:Math. The picture shows another example.

The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that Шаблон:Math.[4]

Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno's formula.[3]

Composition monoids

Шаблон:Main Suppose one has two (or more) functions Шаблон:Math Шаблон:Math having the same domain and codomain; these are often called transformations. Then one can form chains of transformations composed together, such as Шаблон:Math. Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom) a composition monoid. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions Шаблон:Math is called the full transformation semigroup[5] or symmetric semigroup[6] on Шаблон:Mvar. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.[7])

Файл:SVG skew and rotation.svg
Composition of a Шаблон:W (red) and a clockwise rotation by 45° (green). On the left is the original object. Above is shear, then rotate. Below is rotate, then shear.

If the transformations are bijective (and thus invertible), then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions. A fundamental result in group theory, Cayley's theorem, essentially says that any group is in fact just a subgroup of a permutation group (up to isomorphism).[8]

The set of all bijective functions Шаблон:Math (called permutations) forms a group with respect to function composition. This is the symmetric group, also sometimes called the composition group.

In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroup.[9]

Functional powers

Шаблон:Main If Шаблон:Math, then Шаблон:Math may compose with itself; this is sometimes denoted as Шаблон:Math. That is: Шаблон:Block indent Шаблон:Block indent Шаблон:Block indent

More generally, for any natural number Шаблон:Math, the Шаблон:Mvarth functional power can be defined inductively by Шаблон:Math, a notation introduced by Hans Heinrich BürmannШаблон:Cn[10][11] and John Frederick William Herschel.[12][10][13][11] Repeated composition of such a function with itself is called iterated function.

Note: If Шаблон:Mvar takes its values in a ring (in particular for real or complex-valued Шаблон:Math), there is a risk of confusion, as Шаблон:Math could also stand for the Шаблон:Mvar-fold product of Шаблон:Mvar, e.g. Шаблон:Math.[11] For trigonometric functions, usually the latter is meant, at least for positive exponents.[11] For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: Шаблон:Math. However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., Шаблон:Math.

In some cases, when, for a given function Шаблон:Mvar, the equation Шаблон:Math has a unique solution Шаблон:Mvar, that function can be defined as the functional square root of Шаблон:Mvar, then written as Шаблон:Math.

More generally, when Шаблон:Math has a unique solution for some natural number Шаблон:Math, then Шаблон:Math can be defined as Шаблон:Math.

Under additional restrictions, this idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow, specified through solutions of Schröder's equation. Iterated functions and flows occur naturally in the study of fractals and dynamical systems.

To avoid ambiguity, some mathematiciansШаблон:Cn choose to use Шаблон:Math to denote the compositional meaning, writing Шаблон:Math for the Шаблон:Mvar-th iterate of the function Шаблон:Math, as in, for example, Шаблон:Math meaning Шаблон:Math. For the same purpose, Шаблон:Math was used by Benjamin Peirce[14][11] whereas Alfred Pringsheim and Jules Molk suggested Шаблон:Math instead.[15][11][nb 3]

Alternative notations

Many mathematicians, particularly in group theory, omit the composition symbol, writing Шаблон:Math for Шаблон:Math.[16]

In the mid-20th century, some mathematicians decided that writing "Шаблон:Math" to mean "first apply Шаблон:Mvar, then apply Шаблон:Mvar" was too confusing and decided to change notations. They write "Шаблон:Math" for "Шаблон:Math" and "Шаблон:Math" for "Шаблон:Math".[17] This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, when Шаблон:Mvar is a row vector and Шаблон:Mvar and Шаблон:Mvar denote matrices and the composition is by matrix multiplication. This alternative notation is called postfix notation. The order is important because function composition is not necessarily commutative (e.g. matrix multiplication). Successive transformations applying and composing to the right agrees with the left-to-right reading sequence.

Mathematicians who use postfix notation may write "Шаблон:Math", meaning first apply Шаблон:Mvar and then apply Шаблон:Mvar, in keeping with the order the symbols occur in postfix notation, thus making the notation "Шаблон:Math" ambiguous. Computer scientists may write "Шаблон:Math" for this,[18] thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the Z notation the ⨾ character is used for left relation composition.[19] Since all functions are binary relations, it is correct to use the [fat] semicolon for function composition as well (see the article on composition of relations for further details on this notation).

Composition operator

Шаблон:Main Given a function Шаблон:Math, the composition operator Шаблон:Math is defined as that operator which maps functions to functions as <math display="block">C_g f = f \circ g.</math>

Composition operators are studied in the field of operator theory.

In programming languages

Шаблон:Main Function composition appears in one form or another in numerous programming languages.

Multivariate functions

Partial composition is possible for multivariate functions. The function resulting when some argument Шаблон:Math of the function Шаблон:Mvar is replaced by the function Шаблон:Mvar is called a composition of Шаблон:Mvar and Шаблон:Mvar in some computer engineering contexts, and is denoted Шаблон:Math <math display="block">f|_{x_i = g} = f (x_1, \ldots, x_{i-1}, g(x_1, x_2, \ldots, x_n), x_{i+1}, \ldots, x_n).</math>

When Шаблон:Mvar is a simple constant Шаблон:Mvar, composition degenerates into a (partial) valuation, whose result is also known as restriction or co-factor.[20]

<math display="block">f|_{x_i = b} = f (x_1, \ldots, x_{i-1}, b, x_{i+1}, \ldots, x_n).</math>

In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function. Given Шаблон:Mvar, a Шаблон:Mvar-ary function, and Шаблон:Mvar Шаблон:Mvar-ary functions Шаблон:Math, the composition of Шаблон:Mvar with Шаблон:Math, is the Шаблон:Mvar-ary function <math display="block">h(x_1,\ldots,x_m) = f(g_1(x_1,\ldots,x_m),\ldots,g_n(x_1,\ldots,x_m)).</math>

This is sometimes called the generalized composite or superposition of f with Шаблон:Math.[21] The partial composition in only one argument mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen projection functions. Here Шаблон:Math can be seen as a single vector/tuple-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.[22]

A set of finitary operations on some base set X is called a clone if it contains all projections and is closed under generalized composition. A clone generally contains operations of various arities.[21] The notion of commutation also finds an interesting generalization in the multivariate case; a function f of arity n is said to commute with a function g of arity m if f is a homomorphism preserving g, and vice versa i.e.:[21] <math display="block">f(g(a_{11},\ldots,a_{1m}),\ldots,g(a_{n1},\ldots,a_{nm})) = g(f(a_{11},\ldots,a_{n1}),\ldots,f(a_{1m},\ldots,a_{nm})).</math>

A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called medial or entropic.[21]

Generalizations

Composition can be generalized to arbitrary binary relations. If Шаблон:Math and Шаблон:Math are two binary relations, then their composition Шаблон:Math is the relation defined as Шаблон:Math. Considering a function as a special case of a binary relation (namely functional relations), function composition satisfies the definition for relation composition. A small circle Шаблон:Math has been used for the infix notation of composition of relations, as well as functions. When used to represent composition of functions <math>(g \circ f)(x) \ = \ g(f(x))</math> however, the text sequence is reversed to illustrate the different operation sequences accordingly.

The composition is defined in the same way for partial functions and Cayley's theorem has its analogue called the Wagner–Preston theorem.[23]

The category of sets with functions as morphisms is the prototypical category. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.[24] The structures given by composition are axiomatized and generalized in category theory with the concept of morphism as the category-theoretical replacement of functions. The reversed order of composition in the formula Шаблон:Math applies for composition of relations using converse relations, and thus in group theory. These structures form dagger categories.

The standard "foundation" for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions.


. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.

- Saunders Mac Lane, Mathematics: Form and Function[25]

Typography

The composition symbol Шаблон:Math is encoded as Шаблон:Unichar; see the Degree symbol article for similar-appearing Unicode characters. In TeX, it is written \circ.

See also

Notes

Шаблон:Reflist

References

Шаблон:Reflist

External links

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  1. ↑ 1,0 1,1 Ошибка цитирования Неверный тег <ref>; для сносок Velleman_2006 не указан текст
  2. ↑ Шаблон:Cite web
  3. ↑ 3,0 3,1 Шаблон:Cite web
  4. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Rodgers_2000 не указан текст
  5. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Hollings_2014 не указан текст
  6. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Grillet_1995 не указан текст
  7. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Dömösi-Nehaniv_2005 не указан текст
  8. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Carter_2009 не указан текст
  9. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Ganyushkin-Mazorchuk_2008 не указан текст
  10. ↑ 10,0 10,1 10,2 Ошибка цитирования Неверный тег <ref>; для сносок Herschel_1820 не указан текст
  11. ↑ 11,0 11,1 11,2 11,3 11,4 11,5 11,6 Ошибка цитирования Неверный тег <ref>; для сносок Cajori_1929 не указан текст
  12. ↑ 12,0 12,1 Ошибка цитирования Неверный тег <ref>; для сносок Herschel_1813 не указан текст
  13. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Peano_1903 не указан текст
  14. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Peirce_1852 не указан текст
  15. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Pringsheim-Molk_1907 не указан текст
  16. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Ivanov_2009 не указан текст
  17. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Gallier_2011 не указан текст
  18. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Barr-Wells_1990 не указан текст
  19. ↑ Ошибка цитирования Неверный тег <ref>; для сносок ISOIEC13568 не указан текст
  20. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Bryant_1986 не указан текст
  21. ↑ 21,0 21,1 21,2 21,3 Ошибка цитирования Неверный тег <ref>; для сносок Bergman_2011 не указан текст
  22. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Tourlakis_2012 не указан текст
  23. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Lipcomb_1997 не указан текст
  24. ↑ Ошибка цитирования Неверный тег <ref>; для сносок Hilton-Wu_1989 не указан текст
  25. ↑ Шаблон:Cite web
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