Inverse Mapping/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 $Y$:

\(\text {(1)}: \quad\) \(\ds \map {f_1} a\) \(=\) \(\ds a\)
\(\ds \map {f_1} b\) \(=\) \(\ds b\)


\(\text {(2)}: \quad\) \(\ds \map {f_2} a\) \(=\) \(\ds a\)
\(\ds \map {f_2} b\) \(=\) \(\ds a\)


\(\text {(3)}: \quad\) \(\ds \map {f_3} a\) \(=\) \(\ds b\)
\(\ds \map {f_3} b\) \(=\) \(\ds b\)


\(\text {(4)}: \quad\) \(\ds \map {f_4} a\) \(=\) \(\ds b\)
\(\ds \map {f_4} b\) \(=\) \(\ds a\)


We have that:

$f_1$ is the inverse mapping of itself
$f_4$ is the inverse mapping of itself
the inverse of neither $f_2$ nor $f_3$ are mappings.


Sources