# Definition:Set Partition/Finite Expansion

## Definition

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 $\displaystyle \bigcup_{k \mathop = 1}^n S_k = S$ is also called a finite expansion of $S$.

The notations:

$S = S_1 \mid S_2 \mid \cdots \mid S_n$

or:

$\Bbb S = \set {S_1 \mid S_2 \mid \cdots \mid S_n}$

are sometimes seen.