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]; it consists of precisely those elements of X which are equivalent to a.

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 ~. This operation can be thought of (very informally indeed) as the act of "dividing" the input set by the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division.

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.


(a,b) ~ (c,d) if and only if ad = bc.
  • The homotopy class of a continuous map f is the equivalence class of all maps homotopic to f.


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].

See also: -- rational numbers -- multiplicatively closed set -- homotopy theory -- up to