Let \(A\) and \(B\) be two (not necessarily different) sets and let \(f \subseteq A \times B\) be a binary relation

- a (
**total**)**function**(or a**map**), if it is left-total and right-unique. This is equivalant to saying that for every element \(x\in A\) there is exactly one element \(y\in B\) with \((x,y)\in f\). To express this, we write $f(x)=y$ for functions instead of writing $(x,y)\in f$ as we wrote for general relations. - a (
**partial**)**function**(or a**map**), if it is right-unique, but not left-total.

The following terms are strongly related to functions:

- $A$ is called the
**domain**of $f$. - $B$ is called the
**codomain**of $f$. - The element \(f(x)=y\) for some $x\in A$ and $b\in B$ is called the
**value of $f$ at the point $x$**. - The set \(f[A]:=\{y\in B:f(x)=y\;\text{ for all }x\in A\}\) is called the
**range:**(or**image**) of $f$. - For some $y\in B$, the set $f^{-1}(y):=\{x\in A:f(x)=y\}$ is called the
**fiber of $y$ under $f$**. - The set $f^{-1}[B]:=\{x\in A:f(x)=y\text{ for all }y\in B\}$ is called the
**inverse image**of $f$. - If a partial function $f$ is
**undefined**for an $a\in A$, i.e. $\not\exists b\in B:f(a)=b,$ then we write $f(a)=\perp.$

| | | | | created: 2014-05-02 22:30:29 | modified: 2020-08-02 11:13:14 | by: *bookofproofs* | references: [577], [979], [6823], [6907]

[6907] **Brenner, Prof. Dr. rer. nat., Holger**: “Various courses at the University of Osnabrück”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014

[6823] **Kane, Jonathan**: “Writing Proofs in Analysis”, Springer, 2016

[577] **Knauer Ulrich**: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001

[979] **Reinhardt F., Soeder H.**: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10