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:
\(\ds \map f a\) | \(=\) | \(\ds \map f b = \map f c = p\) | ||||||||||||
\(\ds \map f i\) | \(=\) | \(\ds r\) | ||||||||||||
\(\ds \map f j\) | \(=\) | \(\ds \map f k = s\) |
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: $\map f a = \map f b = \map f c$
- $\map {f^{-1} } q = \O$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Example $47 \ \text{(i)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 20$: Introduction: Remarks $\text{(h)}$