Equivalence class

An equivalence class is the set of all elements equivalent to a given element under an equivalence relation. The classes partition the underlying set into disjoint groups.

Equivalence relation: A binary relation that is reflexive, symmetric, and transitive. It supplies the rule that determines which elements belong to each class.
Symmetric relation: A relation in which, whenever one element is related to another, the reverse relation also holds. Symmetry makes membership in a class independent of which equivalent element is chosen.
Congruence modulo n: The relation on integers in which two integers differ by a multiple of a fixed positive integer n. Each class gathers integers with the same remainder upon division by n.
Quotient group: A group formed from the cosets of a normal subgroup, with an induced group operation. Its elements are equivalence classes that preserve group structure.
Binary relation: A relation that specifies which ordered pairs of elements are related. An equivalence relation is a binary relation with three additional properties.
Transitive relation: A relation in which related pairs compose: if a relates to b and b to c, then a relates to c. Transitivity prevents one class from splitting into inconsistent chains of equivalent elements.
Rational number: A number expressible as a ratio of two integers with nonzero denominator. A rational number can be constructed as a class of integer-pair representations of the same ratio.
Quotient space: A topological space formed by identifying points of a space according to an equivalence relation. Its points are classes, and its topology is defined from the original space.
Reflexive relation: A relation in which every element is related to itself. Reflexivity ensures each element belongs to its own equivalence class.
Congruence relation: An equivalence relation on algebraic objects that is compatible with their operations. Compatibility lets equivalence classes inherit operations, as in quotient structures.

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