Definition:Set Partition/Finite Expansion

Definition
Let $S$ be a set.

Let $\Bbb S = \left\{{S_1, S_2, \ldots, S_n}\right\}$ form a partition of $S$.

Then the representation by such a partition $\displaystyle \bigcup_{k \mathop = 1}^n S_k = S$ is also called a finite expansion of $S$.

The notations:
 * $S = S_1 \mid S_2 \mid \cdots \mid S_n$

or:
 * $\Bbb S = \left\{{S_1 \mid S_2 \mid \cdots \mid S_n}\right\}$

are sometimes seen.