# Total Number of Set Partitions/Examples/2

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## 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:

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

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $16 \ \text{(i)}$