Equivalence class

Given a set X and an equivalence relation ~ over X, an equivalence class is a subset of X of the form

{ x in X | x ~ a }

where a is an element in X. This equivalence class is usually denoted as [a].

Because of the properties of an equivalence relation it holds that a is in [a] and that any two equivalence classes are either equal or disjoint. It follows that the set of all equivalence classes of X forms a partition of X: every element of X belongs to one and only one equivalence class. Conversely every partition of X also defines an equivalence relation over X.

It also follows from the properties of an equivalence relation that

a ~ b if and only if [a] = [b].

The notion of equivalence classes is useful for constructing sets out of already constructed ones. The set of all equivalence classes in X given an equivalence relation ~ is usually denoted as X / ~ and called the quotient set of X by ~. In cases where X has some additional structure preserved under ~, the quotient becomes an object of the same type in a natural fashion; the map that sends a to [a] is then a homomorphism.

Examples:

If X is the set of all cars, and ~ is the equivalence relation of "having the same color", then one particular equivalence class consists of all green cars. X / ~ could be naturally identified with the set of all car colors.
The rational numbers can be constructed as the set of equivalence classes of pairs of integers (a,b) where the equivalence relation is defined by

(a,b) ~ (c,d) if and only if ad = bc.

The real numbers can be constructed as the set of equivalence classes on the set of Cauchy sequences of rational numbers, where the equivalence relation is defined by

(x_n)_{n=1...infinity} ~ (y_n)_{n=1...infinity} if and only if (x_n - y_n) -> 0 as n -> infinity

Every group can be partitioned into equivalence classes called conjugacy classes.
The homotopy class of a continuous map f is the equivalence class of all maps homotopic to f.

Examples:

See also: