function

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

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 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 $f\colon A\to B$. This notation visually conveys the fact that $f$ maps elements of $A$ into elements of $B$.

Other standard notations for functions are as follows:

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