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 $f(a)=\perp.$

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