Definition:Set Partition

Definition
Let $S$ be a set.

Definition 1
A partition of $S$ is a set of subsets $\Bbb S$ of $S$ such that:


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

Definition 2
A partition of $S$ is a set of subsets $\Bbb S$ of $S$ such that each element of $S$ lies in exactly one element of $\Bbb S$.

Also known as
A partition is sometimes called a decomposition.

This same definition is also encountered in the field of combinatorics.

Also see

 * Equivalence of Definitions of Partition of Sets


 * Definition:Partition (Topology): a slightly more specialized definition.