Definition:Partitioning

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Definition

Let $S$ be a set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ such that:

$(1): \quad \forall i \in I: S_i \ne \O$, that is, none of $S_i$ is empty
$(2): \quad \displaystyle S = \bigcup_{i \mathop \in I} S_i$, that is, $S_i$ is the union of $\family {S_i}_{i \mathop \in I}$
$(3): \quad \forall i, j \in I: i \ne j \implies S_i \cap S_j = \O$, that is, the elements of $\family {S_i}_{i \mathop \in I}$ are pairwise disjoint.

Then $\family {S_i}_{i \in I}$ is a partitioning of $S$.


The image of this partitioning is the set $\set {S_i: i \in I}$ and is called a partition of $S$.

Note the difference between:

the partitioning, which is an indexing function (that is a mapping)

and

the partition, which is the effect (that is, the image) of that mapping.


Also see

  • Results about set partitions can be found here.


Sources