Definition:Partition (Set Theory)

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

A partition is sometimes called a decomposition.

This same definition is sometimes also encountered in combinatorics.

The elements $S_1, S_2, \ldots \in \Bbb S$ are known as the components of the partition.

If $\Bbb S = \left\{{S_1, S_2, \ldots, S_n}\right\}$ forms a partition of $S$, the notations:

$S = S_1 \mid S_2 \mid \cdots \mid S_n$

or:

$\Bbb S = \left\{{S_1 \mid S_2 \mid \cdots \mid S_n}\right\}$

are sometimes seen.

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

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