# Total Number of Set Partitions/Examples/2

## Example of Total Number of Set Partitions

Let $S$ be a set whose cardinality is $2$.

Then the number of partitions of $S$ is $2$.

## Proof

Let $p \paren n$ denote the cardinality of the set of partitions of a set whose cardinality is $n$.

From Total Number of Set Partitions, $p \paren n$ is the $n$th Bell number $B_n$.

Thus:

 $\displaystyle p \paren 2$ $=$ $\displaystyle B_2$ $\displaystyle$ $=$ $\displaystyle {2 \brace 0} + {2 \brace 1} + {2 \brace 2}$ Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind $\displaystyle$ $=$ $\displaystyle 0 + 1 + 1$ Definition of Stirling Numbers of the Second Kind $\displaystyle$ $=$ $\displaystyle 2$

$\blacksquare$

### Illustration

Let a set $S$ of cardinality $2$ be exemplified by $S = \set {a, b}$.

Then the partitions of $S$ are:

$\set {a, b}$
$\set {a \mid b}$