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: $$\forall T_1, T_2 \in \mathbb{S}: T_1 \cap T_2 = \varnothing$$;
 * 2) The union of all the sets forms the whole set $$S$$: $$\bigcup \mathbb{S} = S$$;
 * 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.

If $$S_1, S_2, \ldots, S_n$$ form a partition of $$S$$, the notation $$S = S_1 | S_2 | \cdots | S_n$$ or $$S = \left\{{S_1 | S_2 | \cdots | S_n}\right\}$$ are sometimes seen.

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

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