# Partition of a set

A partition of

*U*into 6 blocks:

a Venn diagram representation.

**partition**of a set

*X*is a division of

*X*into non-overlapping

**"parts"**or

**"blocks"**that cover all of

*X*.

Table of contents |

2 Examples 3 Partitions and equivalence relations 4 Partial ordering of the lattice of partitions 5 Noncrossing partitions 6 The number of partitions |

## Definition

A partition of a set *X* is a set of nonempty subsets of *X* such that every element *x* in *X* is in exactly one of these subsets.

Equivalently, a set *P* of subsets of *X*, is a partition of *X* if

- No element of
*P*is empty. - The union of the elements of
*P*is equal to*X*. (We say the elements of*P**cover**X*.) - The intersection of any two elements of
*P*is empty. (We say the elements of*P*are pairwise disjoint.)

*P*are sometimes called the

**blocks**of the partition.

## Examples

- Every singleton set {
*x*} has exactly one partition, namely { {*x*} }. - The empty set has exactly one partition, namely itself. (Axioms 1 and 3 are vacuously satisfied.)
- Forgetting momentarily about certain exotic cases, the set of all humans can be partitioned into two blocks: the males and the females.
- The set { 1, 2, 3 } has these five partitions.
- { {1}, {2}, {3} }, sometimes denoted by 1/2/3.
- { {1, 2}, {3} }, sometimes denoted by 12/3.
- { {1, 3}, {2} }, sometimes denoted by 13/2.
- { {1}, {2, 3} }, sometimes denoted by 1/23.
- { {1, 2, 3} }, sometimes denoted by 123.

- Note that
- { {}, {1,3}, {2} } is not a partition (of any set) because it contains the empty set.
- { {1,2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one distinct subset.
- { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partition of {1, 2}.

## Partitions and equivalence relations

If an equivalence relation is given on the set *X*, then the set of all equivalence classes forms a partition of *X*. Conversely, if a partition *P* is given on *X*, we can define an equivalence relation on *X* by writing *x* ~ *y* iff there exists a member of *P* which contains both *x* and *y*. The notions of "equivalence relation" and "partition" are thus essentially equivalent.

## Partial ordering of the lattice of partitions

The relation of "being-finer-than" is a partial order on the set of all partitions of the set *X*, and indeed even a complete lattice. In case *n* = 4, the partial order of the set of all 15 partitions is depicted in this Hasse diagram:

## Noncrossing partitions

The lattice of noncrossing partitions of a finite set has recently taken on importance because of its role in free probability theory. These form a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.

## The number of partitions

The Bell number *B*_{n}, named in honor of Eric Temple Bell, is the number of different partitions of a set with *n* elements. The first several Bell numbers are *B*_{0} = 1,
*B*_{1} = 1, *B*_{2} = 2, *B*_{3} = 5, *B*_{4} = 15, *B*_{5} = 52, *B*_{6} = 203.

The Stirling number *S*(*n*, *k*) of the second kind
is the number of partitions of a set of size *n* into *k* blocks.

The number of partitions of a set of size *n* corresponding to the integer partition

*n*, is the Faà di Bruno coefficient;

*n*is the

*n*th Catalan number, given by