Category:Definitions/Set Partitions

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This category contains definitions related to partitions in the context of Set Theory.
Related results can be found in Category:Set Partitions.

Let $S$ be a set.

Definition 1

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

$(1): \quad$ $\Bbb S$ is pairwise disjoint: $\forall S_1, S_2 \in \Bbb S: S_1 \cap S_2 = \O$ when $S_1 \ne S_2$
$(2): \quad$ The union of $\Bbb S$ forms the whole set $S$: $\ds \bigcup \Bbb S = S$
$(3): \quad$ None of the elements of $\Bbb S$ is empty: $\forall T \in \Bbb S: T \ne \O$.

Definition 2

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