# 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:

\(\displaystyle \map f a\) | \(=\) | \(\displaystyle p\) | |||||||||||

\(\displaystyle \map f b\) | \(=\) | \(\displaystyle r\) | |||||||||||

\(\displaystyle \map f c\) | \(=\) | \(\displaystyle s\) | |||||||||||

\(\displaystyle \map f d\) | \(=\) | \(\displaystyle q\) |

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

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Example $47$