          # Equivalence relation

## This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents. "Equivalency" redirects here. For other uses, see Equivalence. The 52 equivalence relations on a 5-element set depicted as 5×5 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, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c: a = a (reflexive property), if a = b then b = a (symmetric property), and if a = b and b = c then a = c (transitive property). As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Contents 1 Notation 2 Definition 3 Examples 3.1 Simple example 3.2 Equivalence relations 3.3 Relations that are not equivalences 4 Connections to other relations 5 Well-definedness under an equivalence relation 6 Equivalence class, quotient set, partition 6.1 Equivalence class 6.2 Quotient set 6.3 Projection 6.4 Equivalence kernel 6.5 Partition 6.5.1 Counting partitions 7 Fundamental theorem of equivalence relations 8 Comparing equivalence relations 9 Generating equivalence relations 10 Algebraic structure 10.1 Group theory 10.2 Categories and groupoids 10.3 Lattices 11 Equivalence relations and mathematical logic 12 Euclidean relations 13 See also 14 Notes 15 References 16 External links  