Function in mathematics

A function ( mapping , operator , transformation ) is a mathematical concept that reflects the relationship between the elements of a set. In other words, a function is a rule according to which each element of one set (called the definition domain) is associated with some element of another set (called a range of values).

The mathematical concept of a function expresses an intuitive idea of ​​how one quantity completely determines the value of another quantity. So the value of the variable uniquely determines the value of an expression , and the value of the month uniquely determines the value of the month following it. Similarly, some pre-designed algorithm for variable input data produces certain output data.

Often, the term “function” refers to a numeric function; that is, a function that puts some numbers in line with others. These functions are conveniently represented in the figures as graphs.

Story

The term "function" (in a somewhat narrower sense) was first used by Leibniz (1692). In turn, Johann Bernoulli in a letter to the same Leibniz used this term in a sense that is closer to the modern one [1] .

Initially, the concept of function was indistinguishable from the concept of analytical representation. Subsequently, the definition of a function, given by Euler (1751), then - in Lacroix (1806) - was already practically in its present form. Finally, the general definition of a function (in modern form, but for numerical functions) was given by Lobachevsky (1834) and Dirichlet (1837) [2] .

By the end of the 19th century, the concept of function had grown beyond the framework of numerical systems. The vector functions were the first to do this; Frege soon introduced logical functions (1879), and after the appearance of the theory of the Dedekind sets (1887) and Peano (1911), they formulated a modern universal definition.

Definitions

The function that associates each of the four figures with its color.

The most rigorous definition of a function is the set-theoretic definition (based on the concept of a binary relation). Often, instead of defining a function, its intuitive description is given; that is, the concept of a function is translated into ordinary language using the words “law”, “rule” or “conformity”.

Intuitive description

Function ( mapping , operation , operator ) is a law or a rule according to which each [3] element from the set the only element mapped from the set [4] .

At the same time they say that the function set on set or what displays at .

If the item matched item then say item is in functional dependence from item . In this case, the variable is called a function argument independent variable set is called the scope of the task or scope of the function, and the element corresponding to a specific element - private value of the function at the point . Lots of all possible particular values ​​of the function called its domain of values or region of change .

Set-theoretic definition

Function there are many ordered pairs ), which satisfies the following condition: for any [3] there is only one element such that .

Thus, a function is an ordered triple (or tuple) of objects. where

• lots of called the scope of the definition ;
• lots of called the field of values ;
• set of ordered pairs or, equivalently, a function graph.

Designations

If the function is set which is defined on the set and takes values ​​in the set that is, the function displays a lot at then

• this fact is briefly recorded as or .
• function domain (lots of ) is denoted by , or ;
• function range (lots of ) is denoted by ( ), or ( ).
• The presence of a functional relationship between the element and element most commonly referred to as

  ,

  or

  ;

• less commonly used without brackets , or ,
• and where it is necessary to emphasize duality, symbols with brackets are used: or ;
• also there is an operator designation which can be found in general algebra.
• in the lambda calculus of Church.

Multiple argument functions

Graph of two variables function:

The definition of a function is easy to generalize to the case of a function of many arguments.

If many is a cartesian product of sets then mapping it turns out - local display, with the elements of an ordered set are called arguments (given - local function), each of which runs its own set:

Where .

In this case means that .

Ways to set functions

Analytical method

A function as a mathematical object is a binary relation satisfying certain conditions . The function can be specified directly as a set of ordered pairs, for example: there is a function . However, this method is completely unsuitable for functions on infinite sets (which are the usual real functions: power, linear, exponential, logarithmic, etc.).

To set the function use the expression: . Wherein, there is a variable running through the function definition domain, and - range of values. This entry indicates the presence of a functional relationship between the elements of the sets. x and y can run through any sets of objects of any nature. These can be numbers, vectors, matrices, apples, colors of the rainbow. Let us explain by example:

Let there be many apple, plane, pear, chair and many man, locomotive, square . Set the function f as follows: (apple, man), (plane, locomotive), (pear, square), (chair, man) . If we introduce the variable x, the running set and the variable y running through the set The specified function can be defined analytically, like: .

Similarly, you can set numeric functions. For example: where x runs through the set of real numbers, sets some function f. It is important to understand that the expression itself not a function. The function, as an object, is a set (ordered pairs). And this expression, as an object, is the equality of two variables. It sets the function, but is not.

However, in many branches of mathematics, it is possible to denote by f (x) both the function itself and the analytical expression defining it. This syntax is extremely convenient and justified.

Graphic way

Numeric functions can also be specified using a graph. Let be - real function of n variables.

Consider some (n + 1) -dimensional linear space over the field of real numbers (since the function is real). Choose in this space any basis ( ). To each point of the function we associate a vector: . Thus, we will have a set of vectors of linear space corresponding to points of this function according to the indicated rule. The points of the corresponding affine space will form some surface.

If the Euclidean space of free geometric vectors (of directed segments) is taken as the linear space, and the number of arguments of the function f does not exceed 2, the specified set of points can be visualized as a drawing (graphic). If, moreover, the original basis is taken orthonormal, we obtain the “school” definition of the graph of a function.

For functions of 3 arguments and more, such a representation is not applicable due to the lack of geometrical intuition of multidimensional spaces in a person.

However, even for such functions, you can come up with a vivid semigeometric representation (for example, for each value of the fourth coordinate of a point, to associate a certain color on the graph).

Related Definitions

Narrowing and continuing function

Main article: Narrowing and continuing a function

Let the mapping be given and .

Display which takes on same values ​​as function , is called the judgment (or otherwise limitation ) of the function on the set .

Function narrowing on the set denoted by .

If the function such that it is a contraction for some function then function in turn, is called the continuation of the function on the set .

Image and preimage (when displayed)

Element which is mapped to an element called the image of the element (point) (when displaying ).

If we take the whole subset function definition areas then we can consider the set of images of all elements of the set , namely a subset of the range of values ​​(functions ) view

,

which is called the image set (when displaying ). This set is sometimes referred to as or .

On the contrary, taking some subset function domain , we can consider the totality of those elements of the domain ), whose images fall into the set , namely - many species

,

which is called the ( complete ) prototype of the set (when displaying ).

In the particular case when many consists of one element let's say , lots of has a simpler designation .

Identity mapping

Mappings that have a domain of definition and a domain of values ​​are called mappings of a given set into itself or transformations .

In particular, the conversion which matches each point sets her or herself, which is the same

for each ,

called the identity .

This mapping has a special designation: or simpler (if it is clear from the context, what set is meant). Such a designation is due to the English. the word identity ("identity, identity").

Another designation of the identity transformation is . Such a mapping is a unary operation defined on the set . Therefore, often, the identity transformation is called the unit transformation.

Mapping composition

Main article: Feature Composition

Let be and - two specified mappings such that the range of values ​​of the first map is a subset of the range of definition of the second map. Then for everyone uniquely identified element such that but for this very uniquely identified element such that . That is, for everyone uniquely identified element such that . In other words, the mapping is defined such that

for all .

This mapping is called a mapping composition . and and is denoted by

• or or ,
• or (in that order!) that is most commonly used.

Reverse mapping

Main article: Reverse function

If the mapping is one-to-one or bijective (see below), then the mapping is defined , which one

• scope (set ) coincides with the display value area ;
• the range of values ​​(set ) is the same as the display definition ;
• then and only if .

This mapping is called the reverse of the mapping. .

A mapping for which the inverse is defined is called reversible .

In terms of the composition of a function, the property of reversibility consists in the simultaneous fulfillment of two conditions: and .

Properties

Let function be given where and - data sets, and . Each such function may have some properties, which are described below.

Image and preimage while rendering

Image capture

Set and are subsets of the domain of definition. Taking an image (or, equivalently, using an operator ) has the following properties:

• ;
• ;
• .

Further

• image of the union is equal to the union of images: ;
• intersection image is a subset of intersection of images .

The last two properties, generally speaking, admit a generalization to any number of sets greater than two (as it is formulated here).

Taking a type

Set and - subsets of the set .

By analogy with taking an image, taking a type (transition to a type) also has the following two obvious properties:

• the preimage of the union is equal to the union of the preimages: ;
• the preimage of the intersection is equal to the intersection of the preimages .

These properties also allow generalization to any number of sets greater than two (as it is formulated here).

В случае, если отображение обратимо (см. ниже), прообраз каждой точки области значений одноточечный, поэтому для обратимых отображений выполняется следующее усиленное свойство для пересечений:

• образ пересечения равен пересечению образов: .

Поведение функций

Сюръективность

Основная статья: Сюръекция

Function называется сюръективной (или, коротко, сюръекция ), если каждому элементу множества прибытия может быть сопоставлен хотя бы один элемент области определения. Другими словами, функция сюръективна , если образ множества при отображении совпадает с множеством : .

Такое отображение называется ещё отображением на .

If the surjectivity condition is violated, then such a mapping is called a mapping in .

Injectivity

Main article: Injection (mathematics)

Function is called injective (or, briefly, injection ) if different elements of the set are associated with different elements of the set .More formally, the function is injective , if for any two elements such that it is necessarily executed .

In other words, a surjection is when “every image has a type”, and an injection is when “different - into different”. That is, the injection does not happen so that two or more different elements are displayed in the same element .And with a surjection, it does not happen that some element has no type.

Bijectivity

Main article: Bijection

If the function is both surjective and injective , then such a function is called bijective or one-to-one .

Increase and decrease

Main article: Monotonic function

Let function be given Then

• function is called increasing on , if a

• the function is called increasing on , if a

• function is called decreasing by , if a

• function is strictly decreasing on , if a

The (strictly) increasing or decreasing function is called (strictly) monotone.

Periodicity

Main article: Periodic function

Function called periodic with period if true

.

If this equality is not fulfilled for any , then the function is called aperiodic .

Parity

Main article: Odd and even functions

• Function called odd if equality holds

• Function is called even if equality is true

Extremum functions

Main article: Extremum

Let function be given and - internal point of the domain Then

• is called a point of absolute (global) maximum, if

  

• called the point of absolute minimum if

  

Examples

В зависимости от того, какова природа области определения и области значений, различают случаи, когда эти области — это:

• абстрактные множества — множества без какой-либо дополнительной структуры;
• множества, которые наделены некоторой структурой.

В первом случае рассматриваются отображения в самом общем виде и решаются наиболее общие вопросы. Таким общим вопросом, например, является вопрос о сравнении множеств по мощности: если между двумя множествами существует взаимно однозначное отображение (биекция), то два данных множества называют эквивалентными или равномощными . Это позволяет провести классификацию множеств в виде единой шкалы, начальный фрагмент выглядит следующим образом:

• конечные множества — здесь мощность множества совпадает с количеством элементов;
• countable sets are sets equivalent to the set of natural numbers;
• sets of the power of the continuum (for example, a segment of a real straight line or the real straight line itself).

Accordingly, it makes sense to consider the following mapping examples:

• finite functions - mappings of finite sets;
• sequences - mapping of a countable set to an arbitrary set;
• continual functions are mappings of uncountable sets into finite, countable or uncountable sets.

Во втором случае, основной объект рассмотрения — заданная на множестве структура и то, что происходит с этой структурой при отображении: если существует взаимно однозначное отображение одной структуры в другую, что при отображении сохраняются свойства заданной структуры, то говорят, что между двумя структурами установлен изоморфизм. Таким образом, изоморфные структуры, заданные в различных множествах, невозможно различить, поэтому в математике принято говорить, что данная структура рассматривается «с точностью до изоморфизма».

Существует великое разнообразие структур, которые могут быть заданы на множествах. Сюда относится:

• структура порядка — частичный или линейный порядок.
• алгебраическая структура — группоид, полугруппа, группа, кольцо, тело, область целостности или поле.
• структура метрического пространства — здесь задаётся функция расстояния;
• структура евклидового пространства — здесь задаётся скалярное произведение;
• структура топологического пространства — здесь задаётся совокупность т. н. «открытых множеств»;
• структура измеримого пространства — здесь задаётся сигма-алгебра подмножеств исходного множества (например, посредством задания меры с данной сигма-алгеброй в качестве области определения)

Природа множеств определяет и свойства соответствующих функций, поскольку эти свойства формулируются в терминах структур, заданных на множествах. Например, свойство непрерывности требует задания топологической структуры .

Вариации и обобщения

Основная статья: Обобщённая функция

Частично определённые функции

A partially defined function from set to set is a function with a domain. .

Some authors understand the function as a partially defined function. This has its advantages, for example recording is possible. where in this case .

Multivalued functions

Main article: Multivalued function

By virtue of the function definition, the specified value of the argument corresponds to exactly one value of the function. Despite this, one can often hear about so-called "multi-valued" functions. In fact, this is nothing more than a convenient designation of a function, the range of values ​​of which is itself a family of sets.

Let be where - family of subsets of a set . Then will be set for everyone .

A function is unique if each value of the argument corresponds to a single value of the function. A function is multivalued if at least one value of the argument corresponds to two or more values ​​of the function [5] .

see also

• Functional equation
• Algorithm
• The equation
• Boolean function