function

A function is a triplet $(f,A,B)$ where:

1. $A$ is a set (called the domain of the function).

2. $B$ is a set (called the codomain of the function).

3. $f$ is a binary relation between $A$ and $B$.

4. For every $a\in A$, there exists $b\in B$ such that $(a,b)\in f$.

5. If $a\in A$, $b_1,b_2\in B$, and $(a,b_1)\in f$ and $(a,b_2)\in f$, then $b_1=b_2$.

The triplet $(f,A,B)$ is usually written with the specialized notation $f:A\to B$. This notation visually conveys the fact that $f$ maps elements of $A$ into elements of $B$.

Other standard notations for functions are as follows:

• For $a\in A$, one denotes by $f(a)$ the unique element $b\in B$ such that $(a,b)\in f$.

• The image of $(f,A,B)$, denoted $f(A)$, is the set

  $$\{b\in B\mid f(a)=b\text{ for some }a\in A\}$$

  consisting of all elements of $B$ which equal $f(a)$ for some element $a\in A$. Note that, by abuse of notation, the set $f(A)$ is almost always called the image of $f$, rather than the image of $(f,A,B)$.

• In cases where the function $f$ is clear from context, the notation $a\mapsto b$ is equivalent to the statement $f(a)=b$.

• Given two functions $f:A\to B$ and $g:B\to C$, there exists a unique function $g\circ f:A\to C$ satisfying the equation $g\circ f(a)=g(f(a))$. The function $g\circ f$ is called the composition of $f$ and $g$, and a function constructed in this manner is called a composite function. Composition is associative, meaning that $h\circ (g\circ f)=(h\circ g)\circ f$ provided that either expression is defined.

• When a function $f:A\to A$ has its domain equal to its codomain, one often writes $f^n$ for the $n$-fold composition

  $$\underset{n\text{ times}}{\underbrace{f\circ f\circ \mathrm{\cdots }\circ f}}$$

  where $n$ is any natural number. Occasionally this can be confused with ordinary exponentiation (for example the function $x\mapsto (\sin x)(\sin x)$ is conventionally written as ${\sin }^2$); in such cases one usually writes $f^{[n]}$ to denote the $n$-fold composition.

There is no universal agreement as to the definition of the range of a function. Some authors define the range of a function to be equal to the codomain, and others define the range of a function to be equal to the image.

Remark. In set theory, a function is defined as a relation $f$, such that whenever $(a,b),(a,c)\in f$, then $b=c$. Notice that the sets $A,B$ are not specified in advance, unlike the defintion given in the beginning of the article. The domain and range of the function $f$ is the domain and range of $f$ as a relation. Using this definition of a function, we may recapture the defintion at the top of the entry by saying that a function $f$ maps from a set $A$ into a set $B$, if the domain of $f$ is $A$, and the range of $f$ is a subset of $B$.

Title	function
Canonical name	Function
Date of creation	2013-03-22 11:48:58
Last modified on	2013-03-22 11:48:58
Owner	djao (24)
Last modified by	djao (24)
Numerical id	23
Author	djao (24)
Entry type	Definition
Classification	msc 03E20
Classification	msc 44A20
Classification	msc 33E20
Classification	msc 30D15
Synonym	map
Related topic	Mapping
Related topic	InjectiveFunction
Related topic	Surjective
Related topic	Bijection
Related topic	Relation
Defines	domain
Defines	codomain
Defines	composition
Defines	image
Defines	range
Defines	composite function