Definition:Mapping/Diagrammatic Presentations/Finite

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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:

\(\ds S\) \(=\) \(\ds \set {a, b, c, i, j, k}\)
\(\ds T\) \(=\) \(\ds \set {p, q, r, s}\)

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:

\(\ds \map f a\) \(=\) \(\ds \map f b = \map f c = p\)
\(\ds \map f i\) \(=\) \(\ds r\)
\(\ds \map f j\) \(=\) \(\ds \map f k = s\)
$S$ 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:

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

Note that $f$ is neither injective nor surjective:

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