In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation into equivalence classes belong to the same equivalence class if, and only if, they are equivalent.

the -related. The definition of equivalence relations implies that the equivalence classes form a partition meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space

has some structure (such as a group operation or a topology is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

Definition and notation

An equivalence relation is a binary relation satisfying the three properties:

• (reflexivity),

• (symmetry),

• (transitivity).

is defined as

The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although some equivalence classes are not sets but proper classes. For example, "being isomorphic" is an equivalence relation on groups, and the equivalence classes, called isomorphism classes, are not sets.

modulo (or the ). The surjective map which maps each element to its equivalence class, is called the , or the canonical projection.

Every element of an equivalence class characterizes the class, and may be used to represent it. When such an element is chosen, it is called a representative of the class. The choice of a representative in each class defines an injection to . Since its composition such an injection is called a section, when using the terminology of category theory.

Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are called . For example, in modular arithmetic, for every integer greater than , the congruence modulo m is an equivalence relation on the integers, for which two integers and are equivalent—in this case, one says congruent—if and these integers are the canonical representatives.

and produces the remainder of the Euclidean division of by .

Properties

with an equivalence relation by reflexivity, or disjoint forms a partition belongs to one and only one equivalence class.

belong to the same set of the partition.

the following statements are equivalent:

• ,

• , and

Examples

• For example, the rectangles with side lengths 2 by 4 and 1 by 8 both have the area 8. Although they differ, in shape, they are equivalent under the relation "has the same area" so they are in the same equivalence class.

Consider the modulo 2 equivalence relation on the set of integers is an even number

• Congruence modulo 3 on the set of integers can be examined to demonstrate this. Under this equivalence relation,

[0] = {..., -9, -6, -3, 0, 3, 6, 9,...}

[1] = {..., -8, -5, -2, 1, 4, 7, 10,...}

[2] = {..., -7, -4, -1, 2, 5, 8, 11,...}

belongs to only one of the classes where elements 0, 1 and 2 are the canonical representatives of their classes.

be the set of ordered pairs can be identified with the rational number and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers. The same construction can be generalized to the field of fractions of any integral domain.

• pairing represents the rational number .

consists of all the lines in the Euclidean plane are parallel lines, then the set of lines that are parallel to each other form an equivalence class, as long as a line is considered parallel to itself. In this situation, each equivalence class determines a point at infinity.

• In the Cartesian plane have the same slope—and are parallel—so they belong to the same equivalence class.

is an equivalence relation in which every equivalence class is a set consisting of a single element.

• as no two distinct elements are considered equivalent under the equality relation.

Graphical representation

An undirected graph may be associated to any symmetric relation Among these graphs are the graphs of equivalence relations. These graphs, called cluster graphs, are characterized as the graphs such that the connected components are cliques.

Invariants

is said to be an invariant or well-defined

is said to be or simply This occurs, for example, in the character theory".

Any function is class invariant under is the inverse image This equivalence relation is known as the kernel

). Such a function is a morphism of sets equipped with an equivalence relation.

Quotient space in topology

In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes.

In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map. By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a group action.

The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroup on the group by left translations, or respectively the left cosets as orbits under right translation.

A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously.

or to the orbits of a group action. Both the sense of a structure preserved by an equivalence relation, and the study of invariants under group actions, lead to the definition of invariants of equivalence relations given above.

See also

• Equivalence partitioning, a method for devising software test sets based on program coverage of possible inputs

• Homogeneous space, the quotient space of Lie groups

Notes

References

Further reading