Definition:Set Partition/Definition 1

Definition
Let $S$ be a set.

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: $\forall T \in \Bbb S: T \ne \varnothing$.

Also see

 * Equivalence of Definitions of Partition of Sets