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 division of X into non-overlapping "parts" or "blocks" that cover all of X.
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.)
The elements of 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} }
- { {1, 2}, {3} }
- { {1, 3}, {2} }
- { {1}, {2, 3} }
- { {1, 2, 3} }
- 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
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 some 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 Bn, 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