Definition:Mapping/Diagrammatic Presentations
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Diagrammatic Presentations of Mappings
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:
\(\ds \map f a\) | \(=\) | \(\ds \map f b = \map f c = p\) | ||||||||||||
\(\ds \map f i\) | \(=\) | \(\ds r\) | ||||||||||||
\(\ds \map f j\) | \(=\) | \(\ds \map f k = s\) |
The preimages of each of the elements of $T$ under $f$ are:
\(\ds \map {f^{-1} } p\) | \(=\) | \(\ds \set {a, b, c}\) | ||||||||||||
\(\ds \map {f^{-1} } q\) | \(=\) | \(\ds \O\) | ||||||||||||
\(\ds \map {f^{-1} } r\) | \(=\) | \(\ds \set i\) | ||||||||||||
\(\ds \map {f^{-1} } s\) | \(=\) | \(\ds \set {j, k}\) |
Mapping on Infinite Set
The following diagram illustrates the mapping:
- $f: S \to T$
where $S$ and $T$ are areas of the the plane, each containing an infinite number of points.
Note that by Image is Subset of Codomain:
- $\Img f \subseteq \Cdm f$
There are no other such constraints upon the domain, image and codomain.