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 $Y$:

 $(1):\quad$ $\displaystyle \map {f_1} a$ $=$ $\displaystyle a$ $\displaystyle \map {f_1} b$ $=$ $\displaystyle b$

 $(2):\quad$ $\displaystyle \map {f_2} a$ $=$ $\displaystyle a$ $\displaystyle \map {f_2} b$ $=$ $\displaystyle a$

 $(3):\quad$ $\displaystyle \map {f_3} a$ $=$ $\displaystyle b$ $\displaystyle \map {f_3} b$ $=$ $\displaystyle b$

 $(4):\quad$ $\displaystyle \map {f_4} a$ $=$ $\displaystyle b$ $\displaystyle \map {f_4} b$ $=$ $\displaystyle a$

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