Function (mathematics)

The red curve is the graph of a function f in the Cartesian plane. The horizontal axis represents the input x and the vertical axis the output f(x), so that the graph consists of all ordered pairs (x, f(x)).

In mathematics, a function associates an input with exactly one output. The output of a function f with input x is denoted f(x) (read f of x). For example if f(x) = 2x, then the function f associates any input with the number twice as large. If x = 5 then f(x) = 10.

Function inputs are often called arguments and outputs are often called values. They can be real numbers or they can be elements of any given set. For example, a function could associate the letter A with the number 1, the letter B with the number 2, and so on. The set of all inputs for a function is called its domain, and the set of all outputs its range or image.

There are many ways to describe or represent a function. One way is with a formula or algorithm that computes the output for a given input. Another is with a picture or graph of the function's input–output ordered pairs. Some functions are given by a table of outputs for selected inputs. A function can also be described through its relations to other functions, for instance as an inverse function or as a solution of a differential equation. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, and division of functions. Another important operation defined on functions is function composition, where the output from one function becomes the input of another function.

Formal definitions of functions give the set of inputs (the domain), the set of paired inputs and outputs, and another set known as the codomain in which the outputs are constrained to fall. Collections of functions with the same domain and codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis. Functions and their various analogues or generalisations, such as the functors of category theory, are the central objects of investigation^[1] in most fields of modern mathematics.

[edit] Overview

Because functions are so widely used, many traditions have grown up around their use. The symbol for the input to a function is often called the independent variable or argument and is often represented by the letter x or, if the input is a particular time, by the letter t. The symbol for the output is called the dependent variable or value and is often represented by the letter y. The function itself is most often called f, and thus the notation y = f(x) indicates that a function named f has an input named x and an output named y.

A function ƒ takes an input, x, and returns an output ƒ(x). One metaphor describes the function as a machine or black box that converts the input into the output.

The set of all permitted inputs to a given function is called the domain of the function. The set of all resulting outputs is called the image or range of the function. The image is often a subset of a set of permissable outputs, called the codomain of the function. Thus, for example, the function f(x) = x² could take as its domain the set of all real numbers, as its image the set of all non-negative real numbers, and as its codomain the set of all real numbers. In that case, we would describe f as a real-valued function of a real variable. Sometimes, especially in computer science, the term range refers to the codomain rather than the image, so care needs to be taken when using the word.

It is usual practice in mathematics to introduce functions with temporary names like ƒ. For example, ƒ(x) = 2x+1, implies ƒ(3) = 7; when a name for the function is not needed, the form y = 2x+1 may be used. If a function is often used, it may be given a more permanent name as, for example,

Functions need not act on numbers: the domain and codomain of a function may be arbitrary sets. One example of a function that acts on non-numeric inputs takes English words as inputs and returns the first letter of the input word as output. Furthermore, functions need not be described by any expression, rule or algorithm: indeed, in some cases it may be impossible to define such a rule. For example, the association between inputs and outputs in a choice function often lacks any fixed rule, although each input element is still associated to one and only one output.

A function of two or more variables is considered in formal mathematics as having a domain consisting of ordered pairs or tuples of the argument values. For example Sum(x,y) = x+y operating on integers is the function Sum with a domain consisting of pairs of integers. Sum then has a domain consisting of elements like (3,4), a codomain of integers, and an association between the two that can be described by a set of ordered pairs like ((3,4), 7). Evaluating Sum(3,4) then gives the value 7 associated with the pair (3,4).

A family of objects indexed by a set is equivalent to a function. For example, the sequence 1, 1/2, 1/3, ..., 1/n, ... can be written as the ordered sequence <1/n> where n is a natural number, or as a function f(n) = 1/n from the set of natural numbers into the set of rational numbers.

[edit] Definition

This diagram represents a function with domain {1,2,3}, codomain {A,B,C,D} and set of ordered pairs {(1,D),(2,C),(3,C)}. The image is {C,D}.

This does not represent a function since 2 is the first element in more than one ordered pair, in particular, (2,B) and (2,C) are both elements of the set of ordered pairs.

One precise definition of a function is an ordered triple of sets, written (X, Y, F), where X is the domain, Y is the codomain, and F is a set of ordered pairs (a, b).^[2] In each of the ordered pairs, the first element a is from the domain, the second element b is from the codomain, and a necessary condition is that every element in the domain is the first element in exactly one ordered pair. The set of all b is known as the image of the function, and need not be the whole of the codomain. Most authors use the term range to mean the image, while some use range to mean the codomain.

The notation f:X→Y indicates that f is a function with domain X and codomain Y, and the function f is said to map or associate elements of X to elements of Y.

Instead of talking about a 'function' and specifying the domain and codomain in its definition, it is normally more convenient to talk about functions with a specific domain and codomain, functions of a given type can then be defined by the set of ordered pairs F. For example one 'function from the reals to the reals' would be given by the set of pairs (x, 2x) where x is a real.

If the domain and codomain are both the set of real numbers, as is commonly the case, we say f is a real valued function of a real variable, and the study of such functions is called real variables. If the domain and codomain are both the set of complex numbers, then we say f is a complex valued function of a complex variable. The study of these functions is called complex variables. In most situations, the domain and codomain are understood from context, and only the relationship between the input and output is given, but if f(x) = √x, then in real variables the domain is limited to non-negative numbers, while in complex variables the domain is all complex numbers.

In set theory especially, a function f is often defined as the set of ordered pairs. The domain is simply the set of element which appear as the first element of a pair and there is no explicit codomain separate from the image.

A function can also be called a map or a mapping. Some authors, however, use the terms function and map to refer to different types of functions. Other specific types of functions include functionals and operators.

A specific input in a function is called an argument of the function. For each argument value x, the corresponding unique y in the codomain is called the function value at x, output of ƒ for an argument x, or the image of x under ƒ. The image of x may be written as ƒ(x) or as y.

The graph of a function is its set of ordered pairs F. This is an abstraction of the idea of a graph as a picture showing the function plotted on a pair of coordinate axes; for example, (3, 9), the point above 3 on the horizontal axis and to the right of 9 on the vertical axis, lies on the graph of y = x².

The domain X may be void, but if X = ∅ then F = ∅. The codomain Y may be also void, but if Y = ∅ then X = ∅ and F = ∅. Such void functions are not usual, but the theory assures their existence.

The set of all functions f:X→Y is sometimes denoted by Y^X. If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. For example, the function f such that f(x) = 1/x does not define a value for x = 0, and so is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph. In other parts of mathematics, non-single-valued relations are similarly conflated with functions: these are called multivalued functions, with the corresponding term single-valued function for ordinary functions.

Many operations in set theory, such as the power set, have the class of all sets as their domain, and therefore, although they are informally described as functions, they do not fit the set-theoretical definition outlined above, because a class is not necessarily a set.

A function is a special case of a more general mathematical concept, the relation, for which the restriction that each element of the domain appear as the first element in one and only one ordered pair is removed. In other words, an element of the domain may not be the first element of any ordered pair, or may be the first element of two or more ordered pairs. A relation is single-valued when if an element of the domain is the first element of one ordered pair, it is not the first element of any other ordered pair. A relation is left-total or simply total if every element of the domain is the first element of some ordered pair. Thus a function is a total, single-valued relation.

[edit] Notation

Formal description of a function typically involves the function's name, its domain, its codomain, and a rule of correspondence. Thus we frequently see a two-part notation, an example being

where the first part is read:

• ƒ is a function from N to R (one often writes informally Let ƒ: X → Y to mean Let ƒ be a function from X to Y), or
• ƒ is a function on N into R, or
• ƒ is an R-valued function of an N-valued variable,

and the second part is read:

• maps to

Here the function named ƒ has the natural numbers as domain, the real numbers as codomain, and maps n to itself divided by π. Less formally, this long form might be abbreviated

where f(n) is read as f as function of n or f of n. There is some loss of information: we no longer are explicitly given the domain N and codomain R.

It is common to omit the parentheses around the argument when there is little chance of confusion, thus: sin x; this is known as prefix notation. Writing the function after its argument, as in x ƒ, is known as postfix notation; for example, the factorial function is customarily written n!, even though its generalization, the gamma function, is written Γ(n). Parentheses are still used to resolve ambiguities and denote precedence, though in some formal settings the consistent use of either prefix or postfix notation eliminates the need for any parentheses.

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. For instance, stands for the function , stands for the integral function , and so on.

[edit] Injective and surjective functions

Three important kinds of function are the injections (or one-to-one functions), which have the property that if ƒ(a) = ƒ(b) then a must equal b; the surjections (or onto functions), which have the property that for every y in the codomain there is an x in the domain such that ƒ(x) = y; and the bijections, which are both one-to-one and onto. This nomenclature was introduced by the Bourbaki group.

When the definition of a function by its graph only is used, since the codomain is not defined, the surjection must be accompanied with a statement about the set the function maps onto. For example, we might say ƒ maps onto the set of all real numbers.

[edit] Functions with multiple inputs and outputs

The concept of function can be extended to an object that takes a combination of two (or more) argument values to a single result. This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets.

For example, consider the function that associates two integers to their product: ƒ(x, y) = x·y. This function can be defined formally as having domain Z×Z , the set of all integer pairs; codomain Z; and, for graph, the set of all pairs ((x,y), x·y). Note that the first component of any such pair is itself a pair (of integers), while the second component is a single integer.

The function value of the pair (x,y) is ƒ((x,y)). However, it is customary to drop one set of parentheses and consider ƒ(x,y) a function of two variables, x and y. Functions of two variables may be plotted on the three-dimensional Cartesian as ordered triples of the form (x,y,f(x,y)).

The concept can still further be extended by considering a function that also produces output that is expressed as several variables. For example, consider the integer divide function, with domain Z×N and codomain Z×N. The resultant (quotient, remainder) pair is a single value in the codomain seen as a Cartesian product.

[edit] Currying

An alternative approach to handling functions with multiple arguments is to transform them into a chain of functions that each takes a single argument. For instance, one can interpret Add(3,5) to mean first produce a function that adds 3 to its argument, and then apply the 'Add 3' function to 5. This transformation is called currying: Add 3 is curry(Add) applied to 3. There is a bijection between the function spaces C^A×B and (C^B)^A.

When working with curried functions it is customary to use prefix notation with function application considered left-associative, since juxtaposition of multiple arguments—as in (ƒ x y)—naturally maps to evaluation of a curried function. Conversely, the → and ⟼ symbols are considered to be right-associative, so that curried functions may be defined by a notation such as ƒ: Z → Z → Z = x ⟼ y ⟼ x·y

[edit] Binary operations

The familiar binary operations of arithmetic, addition and multiplication, can be viewed as functions from R×R to R. This view is generalized in abstract algebra, where n-ary functions are used to model the operations of arbitrary algebraic structures. For example, an abstract group is defined as a set X and a function ƒ from X×X to X that satisfies certain properties.

Traditionally, addition and multiplication are written in the infix notation: x+y and x×y instead of +(x, y) and ×(x, y).

[edit] Function composition

A composite function g(f(x)) can be visualized as the combination of two machines. The first takes input x and outputs f(x). The second takes f(x) and outputs g(f(x)).

The function composition of two or more functions takes the output of one or more functions as the input of others. The functions ƒ: X → Y and g: Y → Z can be composed by first applying ƒ to an argument x to obtain y = ƒ(x) and then applying g to y to obtain z = g(y). The composite function formed in this way from general ƒ and g may be written

This notation follows the form such that

The function on the right acts first and the function on the left acts second, reversing English reading order. We remember the order by reading the notation as g of ƒ. The order is important, because rarely do we get the same result both ways. For example, suppose ƒ(x) = x² and g(x) = x+1. Then g(ƒ(x)) = x²+1, while ƒ(g(x)) = (x+1)², which is x²+2x+1, a different function.

In a similar way, the function given above by the formula y = 5x−20x³+16x⁵ can be obtained by composing several functions, namely the addition, negation, and multiplication of real numbers.

An alternative to the colon notation, convenient when functions are being composed, writes the function name above the arrow. For example, if ƒ is followed by g, where g produces the complex number e^ix, we may write

A more elaborate form of this is the commutative diagram.

[edit] Identity function

The unique function over a set X that maps each element to itself is called the identity function for X, and typically denoted by id_X. Each set has its own identity function, so the subscript cannot be omitted unless the set can be inferred from context. Under composition, an identity function is neutral: if ƒ is any function from X to Y, then

[edit] Restrictions and extensions

Informally, a restriction of a function ƒ is the result of trimming its domain.

More precisely, if ƒ is a function from a X to Y, and S is any subset of X, the restriction of ƒ to S is the function ƒ|_S from S to Y such that ƒ|_S(s) = ƒ(s) for all s in S.

If g is a restriction of ƒ, then it is said that ƒ is an extension of g.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|_{X \ W}. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. If f|_{X ∩ W} and g|_{X ∩ W} are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

[edit] Inverse function

If ƒ is a function from X to Y then an inverse function for ƒ, denoted by ƒ⁻¹, is a function in the opposite direction, from Y to X, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible. The inverse function exists if and only if ƒ is a bijection.

As a simple example, if ƒ converts a temperature in degrees Celsius C to degrees Fahrenheit F, the function converting degrees Fahrenheit to degrees Celsius would be a suitable ƒ⁻¹.

The notation for composition is similar to multiplication; in fact, sometimes it is denoted using juxtaposition, gƒ, without an intervening circle. With this analogy, identity functions are like the multiplicative identity, 1, and inverse functions are like reciprocals (hence the notation).

For functions that are injections or surjections, generalized inverse functions can be defined, called left and right inverses respectively. Left inverses map to the identity when composed to the left; right inverses when composed to the right.

[edit] Image of a set

The concept of the image can be extended from the image of a point to the image of a set. If A is any subset of the domain, then ƒ(A) is the subset of im ƒ consisting of all images of elements of A. We say the ƒ(A) is the image of A under f.

Use of ƒ(A) to denote the image of a subset A⊆X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is ƒ[A] for the set { ƒ(x): x ∈ A }.

Notice that the image of ƒ is the image ƒ(X) of its domain, and that the image of ƒ is a subset of its codomain.

[edit] Inverse image

The inverse image (or preimage, or more precisely, complete inverse image) of a subset B of the codomain Y under a function ƒ is the subset of the domain X defined by

So, for example, the preimage of {4, 9} under the squaring function is the set {−3,−2,2,3}.

In general, the preimage of a singleton set (a set with exactly one element) may contain any number of elements. For example, if ƒ(x) = 7, then the preimage of {5} is the empty set but the preimage of {7} is the entire domain. Thus the preimage of an element in the codomain is a subset of the domain. The usual convention about the preimage of an element is that ƒ⁻¹(b) means ƒ⁻¹({b}), i.e

In the same way as for the image, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write ƒ⁻¹[B] and ƒ⁻¹[b] for the preimage of a set and a singleton.

The preimage of a singleton set is sometimes called a fiber. The term kernel can refer to a number of related concepts.

[edit] Specifying a function

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

There are many other ways of defining functions. Examples include piecewise definitions, induction or recursion, algebraic or analytic closure, limits, analytic continuation, infinite series, and as solutions to integral and differential equations. The lambda calculus provides a powerful and flexible syntax for defining and combining functions of several variables.

[edit] Computability

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

[edit] Function spaces

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by Y^X.

The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |Y^X| = |Y|^|X|. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. Other examples are the multiplication sign X×Y used for the Cartesian product, where |X×Y| = |X|·|Y|; the factorial sign X!, used for the set of permutations where |X!| = |X|!; and the binomial coefficient sign , used for the set of n-element subsets where

If ƒ: X → Y, it may reasonably be concluded that ƒ ∈ [X → Y].

[edit] Pointwise operations

Pointwise operations inherit properties from the corresponding operations on the codomain. For example if ƒ: X → R and g: X → R are functions with a common domain of X and common codomain of a ring R, then the sum function ƒ + g: X → R and the product function ƒ ⋅ g: X → R can be defined as follows:

[edit] Other properties

There are many other special classes of functions that are important to particular branches of mathematics, or particular applications. Here is a partial list:

[edit] History

[edit] Functions prior to Leibniz

Historically, some mathematicians can be regarded as having foreseen and come close to a modern formulation of the concept of function. Among them is Oresme (1323–1382) . . . In his theory, some general ideas about independent and dependent variable quantities seem to be present.^[3][4]

Ponte further notes that The emergence of a notion of function as an individualized mathematical entity can be traced to the beginnings of infinitesimal calculus.^[3]

[edit] The notion of function in analysis

As a mathematical term, function was coined by Gottfried Leibniz, in a 1673 letter, to describe a quantity related to a curve, such as a curve's slope at a specific point.^[5][6] The functions Leibniz considered are today called differentiable functions. 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 change in the input. Such functions are the basis of calculus.

Johann Bernoulli by 1718, had come to regard a function as any expression made up of a variable and some constants,^[7] and Leonhard Euler during the mid-18th century used the word to describe an expression or formula involving variables and constants e.g., x²+3x+2.^[8]

Alexis Claude Clairaut (in approximately 1734) and Euler introduced the familiar notation f(x) .^[8]

At first, the idea of a function was rather limited. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. By broadening the definition of functions, mathematicians were able to study strange mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called monsters as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are, in a precise sense, more common than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion.

During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis).

Dirichlet and Lobachevsky are traditionally credited with independently giving the modern formal definition of a function as a relation in which every first element has a unique second element. Eves asserts that the student of mathematics usually meets the Dirichlet definition of function in his introductory course in calculus,^[9] but Dirichlet's claim to this formalization is disputed by Imre Lakatos:

There is no such definition in Dirichlet's works at all. But there is ample evidence that he had no idea of this concept. In his [1837], for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the function has two values: ...

(Proofs and Refutations, 151, Cambridge University Press 1976.)

In the context of the Differential Calculus George Boole defined (circa 1849) the notion of a function as follows:

That quantity whose variation is uniform . . . is called the independent variable. That quantity whose variation is referred to the variation of the former is said to be a function of it. The Differential calculus enables us in every case to pass from the function to the limit. This it does by a certain Operation. But in the very Idea of an Operation is . . . the idea of an inverse operation. To effect that inverse operation in the present instance is the business of the Int[egral] Calculus.^[10]

[edit] The logician's function prior to 1850

Logicians of this time were primarily involved with analyzing syllogisms (the 2000 year-old Aristotelian forms and otherwise), or as Augustus De Morgan (1847) stated it: the examination of that part of reasoning which depends upon the manner in which inferences are formed, and the investigation of general maxims and rules for constructing arguments.^[11] At this time the notion of (logical) function is not explicit, but at least in the work of De Morgan and George Boole it is implied: we see abstraction of the argument forms, the introduction of variables, the introduction of a symbolic algebra with respect to these variables, and some of the notions of set theory.

De Morgan's 1847 FORMAL LOGIC OR, The Calculus of Inference, Necessary and Probable observes that [a] logical truth depends upon the structure of the statement, and not upon the particular matters spoken of; he wastes no time (preface page i) abstracting: In the form of the proposition, the copula is made as absract as the terms. He immediately (p. 1) casts what he calls the proposition (present-day propositional function or relation) into a form such as X is Y, where the symbols X, is, and Y represent, respectively, the subject, copula, and predicate. While the word function does not appear, the notion of abstraction is there, variables are there, the notion of inclusion in his symbolism “all of the Δ is in the О” (p. 9) is there, and lastly a new symbolism for logical analysis of the notion of relation (he uses the word with respect to this example X)Y (p. 75) ) is there:

A₁ X)Y To take an X it is necessary to take a Y [or To be an X it is necessary to be a Y]

A¹ Y)X To take an Y it is sufficient to take a X [or To be a Y it is sufficient to be an X], etc.

In his 1848 The Nature of Logic Boole asserts that logic . . . is in a more especial sense the science of reasoning by signs, and he briefly discusses the notions of belonging to and class: An individual may possess a great variety of attributes and thus belonging to a great variety of different classes .^[12] Like De Morgan he uses the notion of variable drawn from analysis; he gives an example of represent[ing] the class oxen by x and that of horses by y and the conjunction and by the sign + . . . we might represent the aggregate class oxen and horses by x + y.^[13]

[edit] The logicians' function 1850–1950

Eves observes that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory of sets, or classes, from a foundation in the logic of propositions and propositional functions.^[14] But by the late 19th century the logicians' research into the foundations of mathematics was undergoing a major split. The direction of the first group, the Logicists, can probably be summed up best by Bertrand Russell 1903:9 – to fulfil two objects, first, to show that all mathematics follows from symbolic logic, and secondly to discover, as far as possible, what are the principles of symbolic logic itself.

The second group of logicians, the set-theorists, emerged with Georg Cantor's set theory (1870–1890) but were driven forward partly as a result of Russell's discovery of a paradox that could be derived from Frege's conception of function, but also as a reaction against Russell's proposed solution.^[15] Zermelo's set-theoretic response was his 1908 Investigations in the foundations of set theory I – the first axiomatic set theory; here too the notion of propositional function plays a role.

[edit] George Boole's The Laws of Thought 1854; John Venn's Symbolic Logic 1881

In his An Investigation into the laws of thought Boole now defined a function in terms of a symbol x as follows:

8. Definition. – Any algebraic expression involving symbol x is termed a function of x, and may be represented by the abbreviated form f(x)^[16]

Boole then used algebraic expressions to define both algebraic and logical notions, e.g., 1−x is logical NOT(x), xy is the logical AND(x,y), x + y is the logical OR(x, y), x(x+y) is xx+xy, and the special law xx = x² = x.^[17]

In his 1881 Symbolic Logic Venn was using the words logical function and the contemporary symbolism ( x = f(y), y = f⁻¹(x), cf page xxi) plus the circle-diagrams historically associated with Venn to describe class relations,^[18] the notions 'quantifying' our predicate, propositions in respect of their extension, the relation of inclusion and exclusion of two classes to one another, and propositional function (all on p. 10), the bar over a variable to indicate not-x (page 43), etc. Indeed he equated unequivocally the notion of logical function with class [modern set]: ... on the view adopted in this book, f(x) never stands for anything but a logical class. It may be a compound class aggregated of many simple classes; it may be a class indicated by certain inverse logical operations, it may be composed of two groups of classes equal to one another, or what is the same thing, their difference declared equal to zero, that is, a logical equation. But however composed or derived, f(x) with us will never be anything else than a general expression for such logical classes of things as may fairly find a place in ordinary Logic.^[19]

[edit] Frege's Begriffsschrift 1879

Gottlob Frege's Begriffsschrift (1879) preceded Giuseppe Peano (1889), but Peano had no knowledge of Frege 1879 until after he had published his 1889.^[20] Both writers strongly influenced Bertrand Russell (1903). Russell in turn influenced much of 20th-century mathematics and logic through his Principia Mathematica (1913) jointly authored with Alfred North Whitehead.

At the outset Frege abandons the traditional concepts subject and predicate, replacing them with argument and function respectively, which he believes will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention.^[21]

Frege begins his discussion of function with an example: Begin with the expression^[22] Hydrogen is lighter than carbon dioxide. Now remove the sign for hydrogen (i.e., the word hydrogen) and replace it with the sign for oxygen (i.e., the word oxygen); this makes a second statement. Do this again (using either statement) and substitute the sign for nitrogen (i.e., the word nitrogen) and note that This changes the meaning in such a way that oxygen or nitrogen enters into the relations in which hydrogen stood before.^[23] There are three statements:

• Hydrogen is lighter than carbon dioxide.
• Oxygen is lighter than carbon dioxide.
• Nitrogen is lighter than carbon dioxide.

Now observe in all three a stable component, representing the totality of [the] relations;^[24] call this the function, i.e.,

... is lighter than carbon dioxide, is the function.

Frege calls the argument of the function [t]he sign [e.g., hydrogen, oxygen, or nitrogen], regarded as replaceable by others that denotes the object standing in these relations.^[25] He notes that we could have derived the function as Hydrogen is lighter than . . .. as well, with an argument position on the right; the exact observation is made by Peano (see more below). Finally, Frege allows for the case of two (or more arguments). For example, remove carbon dioxide to yield the invariant part (the function) as:

• ... is lighter than ...

The one-argument function Frege generalizes into the form Φ(