When the elements of some set have a notion of equivalence, then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class iff they are equivalent.

Formally, given a set and an equivalence relation on , the equivalence class of an element in , denoted by , is the set

of elements which are equivalent to . The equivalence classes form a partition of and is sometimes called the quotient set or the quotient space of by , and is denoted by .

Example:

If is the set of all cars, and is the equivalence relation of βhas the same color asβ, then one particular equivalence class would consist of all green cars, and could be naturally identified with the set of all car colors.