Concept

# Partition of a set

Summary
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. 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. Definition and notation A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: *The family P does not contain the empty set (that is \emptyset \notin P). *The union of the sets in P is equal to X (that is \textstyle\bigcup_{A\in P} A = X). The sets in P are said to exhaust or cover X. See also collectively exh
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