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:
\(\ds p \paren 2\) | \(=\) | \(\ds B_2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {2 \brace 0} + {2 \brace 1} + {2 \brace 2}\) | Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + 1 + 1\) | Definition of Stirling Numbers of the Second Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds 2\) |
$\blacksquare$
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 $16 \ \text{(i)}$