# Definition:Mapping/Diagrammatic Presentations/Finite

## Diagrammatic Presentation of Mapping on Finite Set

The following diagram illustrates the mapping:

$f: S \to T$

where $S$ and $T$ are the finite sets:

 $\displaystyle S$ $=$ $\displaystyle \set {a, b, c, i, j, k}$ $\displaystyle T$ $=$ $\displaystyle \set {p, q, r, s}$

and $f$ is defined as:

$f = \set {\tuple {a, p}, \tuple {b, p}, \tuple {c, p}, \tuple {i, r}, \tuple {j, s}, \tuple {k, s} }$

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

 $\displaystyle \map f a$ $=$ $\displaystyle \map f b = \map f c = p$ $\displaystyle \map f i$ $=$ $\displaystyle r$ $\displaystyle \map f j$ $=$ $\displaystyle \map f k = s$ $S$ is the domain of $f$.
$T$ is the codomain of $f$.
$\set {p, r, s}$ is the image of $f$.

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

 $\displaystyle \map {f^{-1} } p$ $=$ $\displaystyle \set {a, b, c}$ $\displaystyle \map {f^{-1} } q$ $=$ $\displaystyle \O$ $\displaystyle \map {f^{-1} } r$ $=$ $\displaystyle \set i$ $\displaystyle \map {f^{-1} } s$ $=$ $\displaystyle \set {j, k}$

Note that $f$ is neither injective nor surjective:

$\map {f^{-1} } p$ is not a singleton: $\map f a = \map f b = \map f c$
$\map {f^{-1} } q = \O$