Inmathematics, when the elements of someset${\displaystyle S}$ have a notion of equivalence (formalized as anequivalence relation), then one may naturally split the set${\displaystyle S}$ intoequivalence classes. These equivalence classes are constructed so that elements${\displaystyle a}$ and${\displaystyle b}$ belong to the sameequivalence classif, and only if, they are equivalent.

Formally, given a set${\displaystyle S}$ and an equivalence relation${\displaystyle \sim }$ on${\displaystyle S,}$ theequivalence class of an element${\displaystyle a}$ in${\displaystyle S}$ is denoted${\displaystyle [a]}$ or, equivalently,${\displaystyle [a{]}_{\sim }}$ to emphasize its equivalence relation${\displaystyle \sim }$, and is defined as the set of all elements in${\displaystyle S}$ with which${\displaystyle a}$ is${\displaystyle \sim }$-related. The definition of equivalence relations implies that the equivalence classes form apartition of${\displaystyle S,}$ meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called thequotient set or thequotient space of${\displaystyle S}$ by${\displaystyle \sim ,}$ and is denoted by${\displaystyle S/\sim .}$

When the set${\displaystyle S}$ has some structure (such as agroup operation or atopology) and the equivalence relation${\displaystyle \sim ,}$ is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples includequotient spaces in linear algebra,quotient spaces in topology,quotient groups,homogeneous spaces,quotient rings,quotient monoids, andquotient categories.

Anequivalence relation on a set${\displaystyle X}$  is abinary relation${\displaystyle \sim }$  on${\displaystyle X}$  satisfying the three properties:^[1]

• ${\displaystyle a\sim a}$  for all${\displaystyle a\in X}$  (reflexivity),
• ${\displaystyle a\sim b}$  implies${\displaystyle b\sim a}$  for all${\displaystyle a,b\in X}$  (symmetry),
• if${\displaystyle a\sim b}$  and${\displaystyle b\sim c}$  then${\displaystyle a\sim c}$  for all${\displaystyle a,b,c\in X}$  (transitivity).

The equivalence class of an element${\displaystyle a}$  is defined as^[2]

${\displaystyle [a]=\{x\in X:a\sim x\}.}$ 

The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although some equivalence classes are not sets butproper classes. For example, "beingisomorphic" is an equivalence relation ongroups, and the equivalence classes, calledisomorphism classes, are not sets.

The set of all equivalence classes in${\displaystyle X}$  with respect to an equivalence relation${\displaystyle R}$  is denoted as${\displaystyle X/R,}$  and is called${\displaystyle X}$ modulo${\displaystyle R}$  (or thequotient set of${\displaystyle X}$  by${\displaystyle R}$ ).^[3] Thesurjective map${\displaystyle x\mapsto [x]}$  from${\displaystyle X}$  onto${\displaystyle X/R,}$  which maps each element to its equivalence class, is called thecanonical surjection, or thecanonical projection.

Every element of an equivalence class characterizes the class, and may be used torepresent it. When such an element is chosen, it is called arepresentative of the class. The choice of a representative in each class defines aninjection from${\displaystyle X/R}$  toX. Since itscomposition with the canonical surjection is the identity of${\displaystyle X/R,}$  such an injection is called asection, when using the terminology ofcategory theory.

Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are calledcanonical representatives. For example, inmodular arithmetic, for everyintegerm greater than1, thecongruence modulom is an equivalence relation on the integers, for which two integersa andb are equivalent—in this case, one sayscongruent—ifm divides${\displaystyle a-b;}$  this is denoted$a\equiv b(\mathrm{mod}m).$  Each class contains a unique non-negative integer smaller than${\displaystyle m,}$  and these integers are the canonical representatives.

The use of representatives for representing classes allows avoiding considering explicitly classes as sets. In this case, the canonical surjection that maps an element to its class is replaced by the function that maps an element to the representative of its class. In the preceding example, this function is denoted${\displaystyle amodm,}$  and produces the remainder of theEuclidean division ofa bym.

For a set${\displaystyle X}$  with anequivalence relation ~, every element${\displaystyle x}$  of${\displaystyle X}$  is a member of the equivalence class${\displaystyle [x]}$  byreflexivity (${\displaystyle a\sim a}$  for all${\displaystyle a\in X}$ ). Every two equivalence classes${\displaystyle [x]}$  and${\displaystyle [y]}$  are either equal if${\displaystyle x\sim y}$ , ordisjoint otherwise. (The proof is shown below.) Therefore, the set of all equivalence classes of${\displaystyle X}$  forms apartition of${\displaystyle X}$ : every element${\displaystyle x}$  of${\displaystyle X}$  belongs to one and only one equivalence class${\displaystyle [x]}$ , which may be the equivalence classes for other elements of${\displaystyle X}$ .^[4] (I.e., all elements in${\displaystyle X}$  are grouped into non-empty sets, that are here equivalence classes of${\displaystyle X}$ .)

Conversely, for a set${\displaystyle X}$ , every partition comes from an equivalence relation in this way, and different relations give different partitions. Thus${\displaystyle x\sim y}$  if and only if${\displaystyle x}$  and${\displaystyle y}$  belong to the same set of the partition.^[5]

It follows from the properties in the previous section that if${\displaystyle \sim }$  is an equivalence relation on a set${\displaystyle X,}$  and${\displaystyle x}$  and${\displaystyle y}$  are two elements of${\displaystyle X,}$  the following statements are equivalent:

• ${\displaystyle x\sim y}$ 
• ${\displaystyle [x]=[y]}$ 
• ${\displaystyle [x]\cap [y]\ne \mathrm{\varnothing }.}$ 

• Proof of "${\displaystyle x\sim y}$  if and only if${\displaystyle [x]=[y]}$ ".
  • Proof of "If${\displaystyle x\sim y}$  then${\displaystyle [x]=[y]}$ ".
    1. For${\displaystyle c\in [x]}$ ,${\displaystyle x\sim c}$ . By symmetry${\displaystyle y\sim x}$  from${\displaystyle x\sim y}$ , and by transitivity${\displaystyle y\sim c}$  or${\displaystyle c\in [y]}$ , Thus,${\displaystyle [x]\subseteq [y]}$ .
    2. For${\displaystyle c^{\prime }\in [y]}$ ,${\displaystyle y\sim c^{\prime }}$ . By transitivity${\displaystyle x\sim c^{\prime }}$  or${\displaystyle c^{\prime }\in [x]}$ , Thus,${\displaystyle [y]\subseteq [x]}$ .
    3. Thus${\displaystyle [x]=[y]}$ .
  • Proof of "If${\displaystyle [x]=[y]}$  then${\displaystyle x\sim y}$ ".
    • For${\displaystyle c\in [x]}$ ,${\displaystyle x\sim c}$ , and${\displaystyle y\sim c}$  by${\displaystyle [x]=[y]}$ . By symmetry and transitivity,${\displaystyle x\sim y}$ .

• Proof of "If${\displaystyle [x]\cap [y]\ne \mathrm{\varnothing }}$  then${\displaystyle [x]=[y]}$ ".
  • If${\displaystyle [x]\cap [y]\ne \mathrm{\varnothing }}$ , then there is${\displaystyle c}$  such that${\displaystyle x\sim c}$  and${\displaystyle y\sim c}$ . By symmetry and transitivity${\displaystyle x\sim y}$ , and by the above theorem,${\displaystyle [x]=[y]}$ .

• Let${\displaystyle X}$  be the set of all rectangles in a plane, and${\displaystyle \sim }$  the equivalence relation "has the same area as", then for each positive real number${\displaystyle A,}$  there will be an equivalence class of all the rectangles that have area${\displaystyle A.}$ ^[6]
• Consider themodulo 2 equivalence relation on the set ofintegers,${\displaystyle \mathbb{Z},}$  such that${\displaystyle x\sim y}$  if and only if their difference${\displaystyle x-y}$  is aneven number. This relation gives rise to exactly two equivalence classes: one class consists of all even numbers, and the other class consists of all odd numbers. Using square brackets around one member of the class to denote an equivalence class under this relation,${\displaystyle [7],[9],}$  and${\displaystyle [1]}$  all represent the same element of${\displaystyle \mathbb{Z}/\sim .}$ ^[2]
• Let${\displaystyle X}$  be the set ofordered pairs of integers${\displaystyle (a,b)}$  with non-zero${\displaystyle b,}$  and define an equivalence relation${\displaystyle \sim }$  on${\displaystyle X}$  such that${\displaystyle (a,b)\sim (c,d)}$  if and only if${\displaystyle ad=bc,}$  then the equivalence class of the pair${\displaystyle (a,b)}$  can be identified with therational number${\displaystyle a/b,}$  and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.^[7] The same construction can be generalized to thefield of fractions of anyintegral domain.
• If${\displaystyle X}$  consists of all the lines in, say, theEuclidean plane, and${\displaystyle L\sim M}$  means that${\displaystyle L}$  and${\displaystyle M}$  areparallel lines, then the set of lines that are parallel to each other form an equivalence class, as long as aline is considered parallel to itself. In this situation, each equivalence class determines apoint at infinity.

Anundirected graph may be associated to anysymmetric relation on a set${\displaystyle X,}$  where the vertices are the elements of${\displaystyle X,}$  and two vertices${\displaystyle s}$  and${\displaystyle t}$  are joined if and only if${\displaystyle s\sim t.}$  Among these graphs are the graphs of equivalence relations. These graphs, calledcluster graphs, are characterized as the graphs such that theconnected components arecliques.^[2]

If${\displaystyle \sim }$  is an equivalence relation on${\displaystyle X,}$  and${\displaystyle P(x)}$  is a property of elements of${\displaystyle X}$  such that whenever${\displaystyle x\sim y,}$ ${\displaystyle P(x)}$  is true if${\displaystyle P(y)}$  is true, then the property${\displaystyle P}$  is said to be aninvariant of${\displaystyle \sim ,}$  orwell-defined under the relation${\displaystyle \sim .}$ 

A frequent particular case occurs when${\displaystyle f}$  is a function from${\displaystyle X}$  to another set${\displaystyle Y}$ ; if${\displaystyle f\left(x_1\right)=f\left(x_2\right)}$  whenever${\displaystyle x_1\sim x_2,}$  then${\displaystyle f}$  is said to beclass invariant under${\displaystyle \sim ,}$  or simplyinvariant under${\displaystyle \sim .}$  This occurs, for example, in thecharacter theory of finite groups. Some authors use "compatible with${\displaystyle \sim }$ " or just "respects${\displaystyle \sim }$ " instead of "invariant under${\displaystyle \sim }$ ".

Anyfunction${\displaystyle f:X\to Y}$  isclass invariant under${\displaystyle \sim ,}$  according to which${\displaystyle x_1\sim x_2}$  if and only if${\displaystyle f\left(x_1\right)=f\left(x_2\right).}$  The equivalence class of${\displaystyle x}$  is the set of all elements in${\displaystyle X}$  which get mapped to${\displaystyle f(x),}$  that is, the class${\displaystyle [x]}$  is theinverse image of${\displaystyle f(x).}$  This equivalence relation is known as thekernel of${\displaystyle f.}$ 

More generally, a function may map equivalent arguments (under an equivalence relation${\displaystyle {\sim }_X}$  on${\displaystyle X}$ ) to equivalent values (under an equivalence relation${\displaystyle {\sim }_Y}$  on${\displaystyle Y}$ ). Such a function is amorphism of sets equipped with an equivalence relation.

Intopology, aquotient space is atopological 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.

Inabstract 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 aquotient algebra. Inlinear algebra, aquotient space is a vector space formed by taking aquotient group, where the quotient homomorphism is alinear map. By extension, in abstract algebra, the term quotient space may be used forquotient 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 agroup 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 rightcosets 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.

Although the term can be used for any equivalence relation's set of equivalence classes, possibly with further structure, the intent of using the term is generally to compare that type of equivalence relation on a set${\displaystyle X,}$  either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on${\displaystyle X,}$  or to the orbits of a group action. Both the sense of a structure preserved by an equivalence relation, and the study ofinvariants under group actions, lead to the definition ofinvariants of equivalence relations given above.

• Equivalence partitioning, a method for devising test sets insoftware testing based on dividing the possible program inputs into equivalence classes according to the behavior of the program on those inputs
• Homogeneous space, the quotient space ofLie groups
• Partial equivalence relation – Mathematical concept for comparing objects
• Quotient by an equivalence relation – Generalization of equivalence classes to scheme theory
• Setoid – Mathematical construction of a set with an equivalence relation
• Transversal (combinatorics) – Set that intersects every one of a family of sets

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