HOME

TheInfoList




In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, when the elements of some set S have a notion of equivalence (formalized as an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
) defined on them, 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 Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argumen ...
, 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
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 form the Rubik's Cube group. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...
or a
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
) 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 math Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...
s,
homogeneous space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s,
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
s,
quotient monoid In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to pr ...
s, and quotient categories.


Examples

* If X is the set of all cars, and \,\sim\, is the
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
"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 (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, \Z, such that x \sim y if and only if their difference x - y is an
even number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. 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 /math> all represent the same element of \Z / \sim. * Let X be the set of
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

ordered pair
s of integers (a, b) with non-zero b, and define an equivalence relation \,\sim\, on X such that (a, b) \sim (c, d) if and only if a d = b c, then the equivalence class of the pair (a, b) can be identified with the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
of any
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. * If X consists of all the lines in, say, the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, and L \sim M means that L and M are
parallel lines In geometry, parallel lines are line (geometry), lines in a plane (geometry), plane which do not meet; that is, two straight lines in a plane that do not intersecting lines, intersect at any point are said to be parallel. Colloquially, curves tha ...

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 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
.


Definition and notation

An
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on a set X is a
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
\,\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 Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
), * 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 /math> or , and is defined as the set \ of elements that are related to a by \,\sim. 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 class Proper may refer to: Mathematics * Proper map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...
es. For example, "being
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
" is an equivalence relation on
groups A group is a number of people 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 identi ...
, and the equivalence classes, called
isomorphism class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
es, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
x \mapsto /math> from X onto X / R, which maps each element to its equivalence class, is called the , or the canonical projection. Every element of an equivalent 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 Injection or injected may refer to: Science and technology * Injection (medicine) An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...

injection
from X / R to . Since its
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
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 sign ...
, when using the terminology of
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. 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 #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
, for every
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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 n, 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 division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
of by .


Properties

Every element x of X is a member of the equivalence class Every two equivalence classes /math> and /math> are either equal or
disjoint
disjoint
. Therefore, the set of all equivalence classes of X forms a
partition
partition
of X: every element of X belongs to one and only one equivalence class. Conversely, every partition of X comes from an equivalence relation in this way, according to which x \sim y if and only if x and y belong to the same set of the partition. It follows from the properties of an equivalence relation that x \sim y if and only if = In other words, if \,\sim\, is an equivalence relation on a set X, and x and y are two elements of X, then these statements are equivalent: * x \sim y * = /math> * \cap \ne \emptyset.


Graphical representation

An
undirected graph In mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions cal ...

undirected graph
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 (mathematics), set ''X'' is symmetric if: : ...
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 Cane or caning may refer to: *Walking stick or walking cane, a device used primarily to aid walking *Assistive cane, a walking stick used as a mobility aid for better balance *White cane, a mobility or safety device used b ...
.


Invariants

If \,\sim\, is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x \sim y, P(x) is true if P(y) is true, then the property P is said to be an invariant of \,\sim\,, or
well-defined In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
under the relation \,\sim. A frequent particular case occurs when f is a function from X to another set Y; if f\left(x_1\right) = f\left(x_2\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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
f : X \to Y itself defines an equivalence relation on X according to which x_1 \sim x_2 if and only if f\left(x_1\right) = f\left(x_2\right). The equivalence class of x is the set of all elements in X which get mapped to f(x), that is, the class /math> is the
inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

inverse image
of f(x). This equivalence relation is known as the
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 learnin ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

morphism
of sets equipped with an equivalence relation.


Quotient space in topology

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, a quotient space is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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 algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
,
congruence relation In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
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 matrix (mat ...
, a quotient space is a vector space formed by taking a
quotient group A quotient group or factor group is a math Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...
, where the quotient homomorphism is a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
. By extension, in abstract algebra, the term quotient space may be used for
quotient moduleIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
s,
quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
s,
quotient group A quotient group or factor group is a math Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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.


See also

*
Equivalence partitioning Equivalence partitioning or equivalence class partitioning (ECP) is a software testing technique that divides the input data of a software unit into partitions of equivalent data from which test cases can be derived. In principle, test cases are de ...
, a method for devising test sets in
software testing #REDIRECT Software testing#REDIRECT Software testing Software testing is an investigation conducted to provide stakeholders with information about the quality of the software product or service under test. Software testing can also provide an ob ...
based on dividing the possible program inputs into equivalence classes according to the behavior of the program on those inputs *
Homogeneous space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, the quotient space of
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s * * *


Notes


References

* * * *


Further reading

* * * * * * * * * * * * * *


External links

* {{Authority control Equivalence (mathematics) Algebra Binary relations Set theory