Definition:Set Partition

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: In some places, e.g. Quotient Set Determined by Relation Induced by Partition is That Partition, $\Bbb S$ is instead written as $\PP$. While not wrong, it would be an idea to settle on a single notation You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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 $S$: $\ds \bigcup \Bbb S = S$

$(3): \quad$ 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 $\Bbb S$.

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 $\ds \bigcup_{k \mathop = 1}^n S_k = S$ 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

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

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

• Equivalence of Definitions of Set Partition

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

• Results about set partitions can be found here.