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