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## Definition: Partial and Total Maps (Functions)

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 $f(a)=\perp.$

### CC BY-SA 4.0

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

### Bibliography (further reading)

 Kane, Jonathan: “Writing Proofs in Analysis”, Springer, 2016

 Knauer Ulrich: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001

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