Composition of Mappings/Examples/Arbitrary Finite Set with Itself
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Example of Compositions of Mappings
Let $X = Y = \set {a, b}$.
Consider the mappings from $X$ to $Y$:
| \(\text {(1)}: \quad\) | \(\ds \map {f_1} a\) | \(=\) | \(\ds a\) | |||||||||||
| \(\ds \map {f_1} b\) | \(=\) | \(\ds b\) |
| \(\text {(2)}: \quad\) | \(\ds \map {f_2} a\) | \(=\) | \(\ds a\) | |||||||||||
| \(\ds \map {f_2} b\) | \(=\) | \(\ds a\) |
| \(\text {(3)}: \quad\) | \(\ds \map {f_3} a\) | \(=\) | \(\ds b\) | |||||||||||
| \(\ds \map {f_3} b\) | \(=\) | \(\ds b\) |
| \(\text {(4)}: \quad\) | \(\ds \map {f_4} a\) | \(=\) | \(\ds b\) | |||||||||||
| \(\ds \map {f_4} b\) | \(=\) | \(\ds a\) |
The Cayley table illustrating the compositions of these $4$ mappings is as follows:
- $\begin{array}{c|cccc} \circ & f_1 & f_2 & f_3 & f_4 \\ \hline f_1 & f_1 & f_2 & f_3 & f_4 \\ f_2 & f_2 & f_2 & f_2 & f_2 \\ f_3 & f_3 & f_3 & f_3 & f_3 \\ f_4 & f_4 & f_3 & f_2 & f_1 \\ \end{array}$
We have that $f_1$ is the identity mapping and is also the identity element in the algebraic structure under discussion.
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.9$