Definition:Set Partition

From ProofWiki
Jump to navigation Jump to search

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
$(2): \quad \displaystyle S = \bigcup_{i \mathop \in I} S_i$, 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 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

  • Results about set partitions can be found here.