\begin{definition}[Definition:Set Partition/Finite Expansion]
Let $S$ be a set.
Let $\Bbb S = \set {S_1, S_2, \ldots, S_n}$ form a partition of $S$.
Then the representation by such a partition $\ds \bigcup_{k \mathop = 1}^n S_k = S$ is also called a '''finite expansion''' of $S$.
The notations:
:$S = S_1 \mid S_2 \mid \cdots \mid S_n$
or:
:$\Bbb S = \set {S_1 \mid S_2 \mid \cdots \mid S_n}$
are sometimes seen.
Category:Definitions/Set Partitions
\end{definition}