Definition:Bijection/Graphical Depiction

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Diagrammatic Presentation of Bijection on Finite Set

The following diagram illustrates the bijection:

$f: S \to T$

and its inverse, where $S$ and $T$ are the finite sets:

\(\displaystyle S\) \(=\) \(\displaystyle \set {a, b, c, d}\)
\(\displaystyle T\) \(=\) \(\displaystyle \set {p, q, r, s}\)

and $f$ is defined as:

$f = \set {\tuple {a, p}, \tuple {b, r}, \tuple {c, s}, \tuple {d, q} }$


Thus the images of each of the elements of $S$ under $f$ are:

\(\displaystyle \map f a\) \(=\) \(\displaystyle p\)
\(\displaystyle \map f b\) \(=\) \(\displaystyle r\)
\(\displaystyle \map f c\) \(=\) \(\displaystyle s\)
\(\displaystyle \map f d\) \(=\) \(\displaystyle q\)
Bijection-and-Inverse.png
$S$ is the domain of $f$.
$T$ is the codomain of $f$.
$\set {p, q, r, s}$ is the image of $f$.


The preimages of each of the elements of $T$ under $f$ are:

\(\displaystyle \map {f^{-1} } p\) \(=\) \(\displaystyle \set a\)
\(\displaystyle \map {f^{-1} } q\) \(=\) \(\displaystyle \set d\)
\(\displaystyle \map {f^{-1} } r\) \(=\) \(\displaystyle \set c\)
\(\displaystyle \map {f^{-1} } s\) \(=\) \(\displaystyle \set c\)


$f$ is surjective and injective:

$\map {f^{-1} } x$ a singleton for all $x \in \Cdm f$


Sources