Editing Equivalence relation (section)

{{Short description|Mathematical concept for comparing objects}}
{{About|the mathematical concept|the patent doctrine|Doctrine of equivalents}}
{{Redirect|Equivalency||Equivalence (disambiguation){{!}}Equivalence}}
{{stack|{{Binary relations}}}}
[[File:Set partitions 5; matrices.svg|right|thumb|The [[Bell number|52]] equivalence relations on a 5-element set depicted as <math>5 \times 5</math> [[Logical matrix|logical matrices]] (colored fields, including those in light gray, stand for ones; white fields for zeros). The row and column indices of nonwhite cells are the related elements, while the different colors, other than light gray, indicate the equivalence classes (each light gray cell is its own equivalence class).]]

In [[mathematics]], an '''equivalence relation''' is a [[binary relation]] that is [[Reflexive relation|reflexive]], [[Symmetric relation|symmetric]], and [[Transitive relation|transitive]]. The [[Equipollence (geometry)|equipollence]] relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number <math>a</math> is equal to itself (reflexive). If <math>a = b</math>, then <math>b = a</math> (symmetric). If <math>a = b</math> and <math>b = c</math>, then <math>a = c</math> (transitive).

Each equivalence relation provides a [[Partition of a set|partition]] of the underlying set into disjoint [[equivalence class]]es. Two elements of the given set are equivalent to each other [[if and only if]] they belong to the same equivalence class.