Equivalence relation

A binary relation that is reflexive, symmetric, and transitive. It partitions a set into disjoint equivalence classes.

Binary relation: A relation between elements of two sets, represented as a set of ordered pairs. Equivalence relations are binary relations whose pairs satisfy three additional properties.
Equivalence class: The set of elements equivalent to a specified element under an equivalence relation. Each class gathers exactly the elements related to its representative.
Congruence modulo n: An integer relation in which two integers are congruent when their difference is divisible by a positive integer n. Remainder classes modulo n are a basic example of equivalence classes.
Preorder: A reflexive and transitive relation that need not be symmetric. A preorder lacks the symmetry required for equivalence classes.
Reflexive relation: A relation on a set in which every element is related to itself. Reflexivity requires every element to belong to its own equivalence class.
Partition of a set: A collection of nonempty, pairwise disjoint subsets whose union is the original set. The equivalence classes form a partition, and every partition determines an equivalence relation.
Rational number: A number expressible as a ratio of integers with a nonzero denominator. Rational numbers can be constructed from equivalent pairs of integers.
Partial order: A reflexive, antisymmetric, and transitive relation. A partial order replaces symmetry with antisymmetry, allowing distinct elements to be comparable without being equivalent.
Symmetric relation: A relation in which reversing any related pair produces another related pair. Symmetry makes equivalence run in both directions between related elements.
Quotient set: The set whose elements are the equivalence classes of an equivalence relation. Replacing each class with one element produces the quotient set.

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