Definition:Set Partition

Definition
Let $S$ be a set.

A partition of $S$ is a set of subsets $\mathbb S$ of $S$ such that:


 * 1) All sets in $\mathbb S$ are pairwise disjoint: $\forall S_1, S_2 \in \mathbb S: S_1 \cap S_2 = \varnothing$;
 * 2) The union of all the sets forms the whole set $S$: $\displaystyle \bigcup \mathbb S = S$;
 * 3) None of the sets in $\mathbb S$ is null: $\forall T \in \mathbb S: T \ne \varnothing$.

A partition is sometimes called a decomposition.

This same definition is sometimes also encountered in combinatorics.

Finite Expansion
If $S_1, S_2, \ldots, S_n$ form a partition of $S$, the notations:
 * $S = S_1 | S_2 | \cdots | S_n$

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

are sometimes seen.

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

Note
The definition of a partition in the field of topology is slightly more specialized.