Cardinality of Set of All Mappings/Examples/2 Elements to 2 Elements

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Example of Cardinality of Set of All Mappings

Let $X = \set {a, b}$.

Let $Y = \set {u, v}$.

Then the mappings from $X$ to $Y$ are:

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

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

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

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

$f_1$ and $f_4$ are bijections.

$f_2$ and $f_3$ are neither surjections nor injections.