# Definition:Bijection/Graphical Depiction

## 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:

 $\displaystyle \map f a$ $=$ $\displaystyle p$ $\displaystyle \map f b$ $=$ $\displaystyle r$ $\displaystyle \map f c$ $=$ $\displaystyle s$ $\displaystyle \map f d$ $=$ $\displaystyle q$
$S$ 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 $x \in \Cdm f$