PlanetMath: function

(more info)

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function

(Definition)

A function is a triplet where:

is a set (called the domain of the function).
is a set (called the codomain of the function).
is a binary relation between and .
For every $a \in A$ , there exists $b \in B$ such that $(a,b) \in f$ .
If $a \in A$ , $b_1,b_2 \in B$ , and $(a,b_1) \in f$ and $(a,b_2) \in f$ , then .

The triplet

is usually written with the specialized notation $f\colon A \to B$ . This notation visually conveys the fact that

maps elements of

into elements of

Other standard notations for functions are as follows:

For $a \in A$ , one denotes by the unique element $b \in B$ such that $(a,b) \in f$ .
The image of , denoted , is the set
$\displaystyle \{b \in B \mid f(a) = b$ for some $\displaystyle a \in A\}$
consisting of all elements of which equal for some element $a \in A$ . Note that, by abuse of notation, the set is almost always called the image of , rather than the image of .
In cases where the function is clear from context, the notation $a \mapsto b$ is equivalent to the statement .
Given two functions $f\colon A \to B$ and $g\colon B \to C$ , there exists a unique function $g \circ f\colon A \to C$ satisfying the equation $g \circ f(a) = g(f(a))$ . The function $g \circ f$ is called the composition of and . Composition is associative, meaning that $h \circ (g \circ f) = (h \circ g) \circ f$ provided that either expression is defined.
When a function $f\colon A \to A$ has its domain equal to its codomain, one often writes for the -fold composition
$\displaystyle \underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}}$
where 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 -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 , such that whenever $(a,c),(b,c)\in f$ , then . Notice that the sets are not specified in advance, unlike the defintion given in the beginning of the article. The domain and range of the function is the domain and range of 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 maps from a set into a set , if the domain of is , and the range of is a subset of .