Partition of a set

In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.

A set of stamps partitioned into bundles: No stamp is in two bundles, no bundle is empty, and every stamp is in a bundle.

The 52 partitions of a set with 5 elements. A colored region indicates a subset of X that forms a member of the enclosing partition. Uncolored dots indicate single-element subsets. The first shown partition contains five single-element subsets; the last partition contains one subset having five elements.
The traditional Japanese symbols for the 54 chapters of the Tale of Genji are based on the 52 ways of partitioning five elements (the two red symbols represent the same partition, and the green symbol is added for reaching 54).[1]

Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory.


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