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

## 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.