
The set {1, 2, 3} has the following partitions
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.
The set of all partitions of a set is a partially ordered set; one may say that one partition is "finer" than another if it splits the set into smaller blocks. This partially ordered set is a lattice.
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.
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