Partition of a set

A Partition of U into 6 blocks:
a Venn diagram representation.

In mathematics, a partition of a set X is a way to divide X into different "blocks" that cover all of X and do not overlap.

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

1. The union of the elements of P is equal to X; and
2. The intersection of any two elements of P is empty; and
3. No element of P is empty.

The elements of P are sometimes called the blocks of the partition.

Examples

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} },
• { {1, 2}, {3} },
• { {1, 3}, {2} },
• { {1}, {2, 3} } and
• { {1, 2, 3} }.

• { {}, {1,3}, {2} } is not a partition because it contains an empty subset.
• { {1,2}, {2, 3} } is not a partition because the element 2 is contained in more than one subset.
• { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3. 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

Given two partitions P and Q of a given set X, we say that P is finer than Q if it splits the set X into smaller blocks, i.e. if every element of P is a subset of an element of Q. In that case, one writes P ≤ Q.

With this relation of "being-finer-than", the set of all partitions of a set X is a partially ordered set and indeed even a complete lattice.

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 B₀=1, B₁=1, B₂=2, B₃=5, B₄=15, B₅=52, B₆=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

of n, is the Faà di Bruno coefficient

Referenced By