Function mathematics - Graph of a function

The graph of a function f is the set of all ordered pairs (x, f(x)), for all x in the domain X. If X and Y are the set of real numbers (or subsets thereof), then this definition coincides with the familiar sense of "graph" as a picture or plot of the function, with the ordered pairs being the Cartesian coordinates of the plot's points There are theorems formulated or proved most eas ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem. If X and Y are real lines, then this definition coincides with the familiar sense of "graph" as a picture of the function, with the ordered pairs plotted as Cartesian coordinates. Note that a binary relation on the two sets X and Y could be identified with an ordere ...

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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

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A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x). Because two definitions of binary relation are in use, there are actually two definitions of function, in ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler, in the 18th century. But the intuitive idea of a function as any rule or procedure that assigns an output to each given input proved to be naive. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. The concept of a function was not put on a rigorous basis u ...

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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

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If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f(x). More commonly, a function is defined by a formula, or more generally an algorithm — that is, a recipe that tells how to compute the value of f(x) given any x in the domain. See the squaring function sqr above. More generally, a function can also be defined by any mathematical condition relating the argument to the corresponding val ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows: (f + g)(x) = f(x) + g(x) (f × g)(x) = 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

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Functions are used in every quantitative science, to model relationships between all kinds of physical quantities — especially when one quantity is completely determined by another quantity. Thus, for example, one may use a function to describe how the temperature of water affects its density. Functions are also used in computer science to model data structures and the effects of algorithms. However, the word is also used in computing in the very different sense of pro ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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In the formal definition, a function represents a relationship between its domain and its codomain, rather than just a rule for taking an input to an output. A generalization of the notion of function is morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection. ...

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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

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As a mathematical term, "function" was coined by Gottfried Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope at 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 output or the change in the output as it depends on the input or the ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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If f is a function from X to Y, the set X is called the domain of f, and Y is called its codomain. Each element of the domain is called an argument of the function. For each argument x, the corresponding unique y in the codomain is called the function value at x, or the image of x by (or under) the function. The value of a function f at an argument x is traditionally written f(xSee also:

Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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One often extends the concept (and notation) of image of an argument to sets of arguments. Namely, if A is any subset of the domain X, the image of A under f is the subset of Y defined f(A) = {f(x) | x is in A} So, for example, the image of {-3,2,3} under the squaring function sqr is sqr({-3,2, 3}) = {4, 9}. This extension is consistent as long as no subset of the domain is also an element of the domain. A ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Function mathematics - Injective surjective bijective. Three important properties that a function may have are: injective (or one-to-one, or an injection) if it associates different arguments to different values; i.e., if f(a) = f(b) implies a = b, for any arguments a and b; surjective (or onto, or a surjection) if its range is equal to its codomain; in other words, if for every y in the ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Informally, a restriction of a function f is the result of trimming its graph to a smaller domain. More precisely, if f is a function from a X to Y, and S is any subset of X, the restriction of f to S is the function f|S from S to Y such that f|S(s) = f(s) for all s in S. The restriction f|S can also be expressed as the composition f incS,X, where incSSee also:

Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows: (f + g)(x) = f(x) + g(x) (f × g)(x) = f(x) × < ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The notion of function is generalizes to the notion of morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection. Ordinary functions are sometimes referred to as morphisms in a concrete category. ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x in X, there exists some y in Y such that x is related to y. f is single-valued: for each x in X, there is at most one y in Y such that x is related to y. In some contexts, a relation that satisfies condition (1), but not necessarily (2) ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codomain sets, and G is the graph of f. However, some mathematicians define a relation as being simply the set of pairs G, without explicitly giving the domain and co-domain. There are advantages and disadvantages to each definition, but either of them is satisfactory for most uses of functions in mathematics. The explicit domain and ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is ...

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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

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The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is justified by the fact that |YX| = |Y||X|. See the article on cardinal numbers for more details. It is traditional to write f: X → Y to mean f ∈ [X → Y]; that is, "f is a function from X to Y". This statement is sometimes read "f ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Set of all functions