# Definition:Set Partition

## Definition

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 $S$: $\displaystyle \bigcup \Bbb S = S$
- $(3): \quad$ 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 $\Bbb S$.

## Component

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

## Finite Expansion

Let $S$ be a set.

Let $\Bbb S = \set {S_1, S_2, \ldots, S_n}$ form a partition of $S$.

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

The notations:

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

or:

- $\Bbb S = \set {S_1 \mid S_2 \mid \cdots \mid S_n}$

are sometimes seen.

## Partitioning

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 known as

A partition is sometimes called a **decomposition**.

This same definition is also encountered in the field of combinatorics.

## Also defined as

Some sources do not impose the condition that all sets in $\Bbb S$ are non-empty.

This is most probably more likely to be an accidental omission rather than a deliberate attempt to allow $\O$ to be an element of a **partition**.

The point is minor; proofs of **partitionhood** usually include a demonstration that all elements of such a **partition** are indeed non-empty.

## Examples

### Integers by Sign

Let $\Z$ denote the set of integers.

Let $\Z_{> 0}$ denote the set of strictly positive integers.

Let $\Z_{< 0}$ denote the set of strictly negative integers.

Let $\Z_0$ denote the singleton $\set 0$

Then $P = \set {\Z_{> 0}, \Z_{< 0}, \Z_0}$ forms a partition of $\Z$.

### Partition into Singletons

Let $S$ be a set.

Consider the family of subsets $\family {\set x}_{x \mathop \in S}$ indexed by $S$ itself.

Then $\family {\set x}_{x \mathop \in S}$ is a partitioning of $S$ into singletons.

Its associated partition is:

- $\set {\set x: x \in S}$

## Also see

- Definition:Partition (Topology): a slightly more specialized definition.

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