Definition:Set Partition
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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
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.