# 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.

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.6$