Composition of Mappings/Examples/Arbitrary Finite Set with Itself

Example of Compositions of Mappings
Let $X = Y = \set {a, b}$.

Consider the mappings from $X$ to $Y$:

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.