Quotient set
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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 S have a notion of equivalence (formalized as an
equivalence relation 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 i ...
), 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 bicondi ...
, 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 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 m ...
or a
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
) 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 (mathematics), 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 "factor ...
s,
homogeneous space 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 m ...
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 (linear algebra), quo ...
s, quotient monoids, and quotient categories.


Examples

* If X is the set of all cars, and \,\sim\, is the
equivalence relation 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 i ...
"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 language of ...
s, \Z, such that x \sim y if and only if their difference x - y is an
even number 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 ...
. 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, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
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 mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics) ...
of any
integral domain 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 ...
. * 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 geometry, geometric setting in which two real number, real quantities are required to determine the position (geometry), position of each point (mathematics), ...
, and L \sim M means that L and M are
parallel lines In geometry, parallel lines are coplanar straight line (geometry), lines that do not intersecting lines, intersect at any point. Parallel planes are plane (geometry), planes in the same three-dimensional space that never meet. ''Parallel curve ...
, 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 (mathematics), pencil of parallel l ...
.


Definition and notation

An
equivalence relation 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 i ...
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 Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
\,\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 /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 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 for ...
es. For example, "being
isomorphic 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 ...
" is an equivalence relation on groups, 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 In mathematics, a surjective function (also known as surjection, or onto function) is a Function (mathematics), function that every element can be mapped from element so that . In other words, every element of the function's codomain is the Im ...
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 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 from X / R to . Since its composition 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 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 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 ...
, 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 language of ...
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 Arithmetic () is an elementary part of 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 quantitie ...
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. Therefore, the set of all equivalence classes of X forms a 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 discrete 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 abstra ...
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 ...
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 ...
.


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 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 m ...
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 In mathematics, more specifically in group theory, the character of a group representation is a function (mathematics), function on the group (mathematics), group that associates to each group element the trace (linear algebra), trace of the corres ...
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(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 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 ...
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 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 mod ...
of sets equipped with an equivalence relation.


Quotient space in topology

In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a quotient space is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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 group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
,
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...
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 mathematical group (mathematics), 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 "factor ...
, where the quotient homomorphism is a
linear map 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 ...
. By extension, in abstract algebra, the term quotient space may be used for quotient modules,
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 (linear algebra), quo ...
s,
quotient group A quotient group or factor group is a mathematical group (mathematics), 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 "factor ...
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 (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical ...
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 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 ...
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, 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 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 m ...
, the quotient space of
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), 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 a ...
s * * * *


Notes


References

* * * *


Further reading

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


External links

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