TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, when the elements of some
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
$S$ have a notion of equivalence (formalized as an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
), then one may naturally split the set $S$ into equivalence classes. These equivalence classes are constructed so that elements $a$ and $b$ belong to the same equivalence class
if, and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicon ...
, they are equivalent. Formally, given a set $S$ and an equivalence relation $\,\sim\,$ on $S,$ the of an element $a$ in $S,$ denoted by is the set $\$ of elements which are equivalent to $a.$ It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of $S.$ This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of $S$ by $\,\sim\,,$ and is denoted by $S / \sim.$ When the set $S$ has some structure (such as a group operation or a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closin ...
) and the equivalence relation $\,\sim\,$ 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 group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
s,
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
s,
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algeb ...
s, quotient monoids, and quotient categories.

# Examples

* If $X$ is the set of all cars, and $\,\sim\,$ is the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
"has the same color as", then one particular equivalence class would consist of all green cars, and $X / \sim$ could be naturally identified with the set of all car colors. * Let $X$ be the set of all rectangles in a plane, and $\,\sim\,$ the equivalence relation "has the same area as", then for each positive real number $A,$ there will be an equivalence class of all the rectangles that have area $A.$ * Consider the modulo 2 equivalence relation on the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, $\Z,$ such that $x \sim y$ if and only if their difference $x - y$ is an
even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
. 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, and
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
$a / b,$ 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 In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fiel ...
of any
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
. * If $X$ consists of all the lines in, say, the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
, and $L \sim M$ means that $L$ and $M$ 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 geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
.

# Definition and notation

An
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
on a set $X$ is a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
$\,\sim\,$ on $X$ satisfying the three properties: * $a \sim a$ for all $a \in X$ ( reflexivity), * $a \sim b$ implies $b \sim a$ for all $a, b \in X$ (
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
), * if $a \sim b$ and $b \sim c$ then $a \sim c$ for all $a, b, c \in X$ ( transitivity). The equivalence class of an element $a$ is often denoted
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
", although some equivalence classes are not sets but
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
es. For example, "being
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
" is an equivalence relation on
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, and the equivalence classes, called isomorphism classes, are not sets. The set of all equivalence classes in $X$ with respect to an equivalence relation $R$ is denoted as $X / R,$ and is called $X$ modulo $R$ (or the of $X$ by $R$). The surjective map
injection Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...
from $X / R$ to . Since its
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
with the canonical surjection is the identity of $X / R,$ such an injection is called a
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section si ...
, when using the terminology of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
. 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 In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his b ...
, for every
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
greater than , the congruence modulo is an equivalence relation on the integers, for which two integers and are equivalent—in this case, one says ''congruent'' —if divides $a-b;$ this is denoted $a\equiv b \pmod m.$ Each class contains a unique non-negative integer smaller than $m,$ and these integers are the canonical representatives. The use of representatives for representing classes allows avoiding to consider 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 $a \bmod m,$ and produces the remainder of the
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller tha ...
of by .

# Properties

Every element $x$ of $X$ is a member of the equivalence class Every two equivalence classes

# Graphical representation

An
undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' ve ...
may be associated to any
symmetric relation A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X( ...
on a set $X,$ where the vertices are the elements of $X,$ and two vertices $s$ and $t$ are joined if and only if $s \sim t.$ Among these graphs are the graphs of equivalence relations; they are characterized as the graphs such that the connected components are
cliques A clique ( AusE, CanE, or ), in the social sciences, is a group of individuals who interact with one another and share similar interests. Interacting with cliques is part of normative social development regardless of gender, ethnicity, or popular ...
.

# Invariants

If $\,\sim\,$ is an equivalence relation on $X,$ and $P\left(x\right)$ is a property of elements of $X$ such that whenever $x \sim y,$ $P\left(x\right)$ is true if $P\left(y\right)$ is true, then the property $P$ is said to be an invariant of $\,\sim\,,$ or
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
under the relation $\,\sim.$ A frequent particular case occurs when $f$ is a function from $X$ to another set $Y$; if $f\left\left(x_1\right\right) = f\left\left(x_2\right\right)$ whenever $x_1 \sim x_2,$ then $f$ is said to be $\,\sim\,,$ or simply $\,\sim.$ This occurs, for example, in the character theory of finite groups. Some authors use "compatible with $\,\sim\,$" or just "respects $\,\sim\,$" instead of "invariant under $\,\sim\,$". Any function $f : X \to Y$ is ''class invariant under'' $\,\sim\,,$ according to which $x_1 \sim x_2$ if and only if $f\left\left(x_1\right\right) = f\left\left(x_2\right\right).$ The equivalence class of $x$ is the set of all elements in $X$ which get mapped to $f\left(x\right),$ that is, the class
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of $f.$ More generally, a function may map equivalent arguments (under an equivalence relation $\sim_X$ on $X$) to equivalent values (under an equivalence relation $\sim_Y$ on $Y$). Such a function is a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
of sets equipped with an equivalence relation.

# Quotient space in topology

In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closin ...
, a quotient space is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ...
,
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
s 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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, a quotient space is a vector space formed by taking a
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
, where the quotient homomorphism is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
. By extension, in abstract algebra, the term quotient space may be used for
quotient module In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups ...
s,
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algeb ...
s,
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
s, 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 In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorph ...
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
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and righ ...
s 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 $X,$ either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on $X,$ 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.

* Equivalence partitioning, a method for devising test sets in
software testing Software testing is the act of examining the artifacts and the behavior of the software under test by validation and verification. Software testing can also provide an objective, independent view of the software to allow the business to apprecia ...
based on dividing the possible program inputs into equivalence classes according to the behavior of the program on those inputs *
Homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
, the quotient space of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s * * * *

* * * *