# 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
- $(2): \quad \ds S = \bigcup_{i \mathop \in I} S_i$, that is, $S$ 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 $\family {S_i}_{i \mathop \in I}$ 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

## Also see

- Results about
**set partitions**can be found here.

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products: Exercise $3$