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 ...
, an equivalence relation 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 ...
that is reflexive,
symmetric Symmetry (from Greek συμμετρία ''symmetria'' "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 pre ...
and
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
. The relation '' is equal to'' is the canonical example of an equivalence relation. Each equivalence relation provides a of the underlying set into disjoint
equivalence 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). ...
es. Two elements of the given set are equivalent to each other
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
they belong to the same equivalence class.

# Notation

Various notations are used in the literature to denote that two elements $a$ and $b$ of a set are equivalent with respect to an equivalence relation $R;$ the most common are "$a \sim b$" and "", which are used when $R$ is implicit, and variations of "$a \sim_R b$", "", or "" to specify $R$ explicitly. Non-equivalence may be written "" or "$a \not\equiv b$".

# Definition

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 a set $X$ is said to be an equivalence relation,
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it is reflexive, symmetric and transitive. That is, for all $a, b,$ and $c$ in $X:$ * $a \sim a.$ ( Reflexivity) * $a \sim b$ if and only if $b \sim a.$ (
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 $a \sim c.$ ( Transitivity) $X$ together with the relation $\,\sim\,$ is called a
setoid 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 ...
. The
equivalence 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). ...
of $a$ under $\,\sim,$ denoted is defined as

# Examples

## Simple example

On the set $X = \$, the relation $R = \$ is an equivalence relation. The following sets are equivalence classes of this relation: The set of all equivalence classes for $R$ is $\.$ This set is a of the set $X$ with respect to $R$.

## Equivalence relations

The following relations are all equivalence relations: * "Is equal to" on the set of numbers. For example, $\tfrac$ is equal to $\tfrac.$ * "Has the same birthday as" on the set of all people. * "Is similar to" on the set of all
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...
s. * "Is
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
to" on the set of all
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...
s. * Given a natural number $n$, "is congruent to, modulo $n$" on the
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

. * Given a
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 ...
$f:X \to Y$, "has the same
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
under $f$ as" on the elements of $f$'s
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
$X$. For example, $0$ and $\pi$ have the same image under $\sin$, viz. $0$. * "Has the same absolute value as" on the set of real numbers * "Has the same cosine as" on the set of all angles.

## Relations that are not equivalences

* The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7. * The relation "has a
common factor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is d ...
greater than 1 with" between
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1. * The
empty relation Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...
''R'' (defined so that ''aRb'' is never true) on a set ''X'' is
vacuously 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 ...
symmetric and transitive; however, it is not reflexive (unless ''X'' itself is empty). * The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. However, if the approximation is defined asymptotically, for example by saying that two functions ''f'' and ''g'' are approximately equal near some point if the limit of ''f − g'' is 0 at that point, then this defines an equivalence relation.

# Connections to other relations

*A
partial order upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion Inclusion or Include may refer to: Sociology * Social inclusion, affirmative action to change the circumstances and habits that leads to s ...
is a relation that is reflexive, , and transitive. * Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In
algebraic expressionIn 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 ha ...
s, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. *A
strict partial order In mathematical 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 ...
is irreflexive, transitive, and asymmetric. *A partial equivalence relation is transitive and symmetric. Such a relation is reflexive
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it is
serial Serial may refer to: Arts, entertainment, and media The presentation of works in sequential segments * Serial (literature), serialised fiction in print * Serial (publishing), periodical publications and newspapers * Serial (radio and television), ...
, that is, if for all $a,$ there exists some $b \text a \sim b.$''If:'' Given $a,$ let $a \sim b$ hold using seriality, then $b \sim a$ by symmetry, hence $a \sim a$ by transitivity. — ''Only if:'' Given $a,$ choose $b = a,$ then $a \sim b$ by reflexivity. Therefore, an equivalence relation may be alternatively defined as a symmetric, transitive, and serial relation. *A ternary equivalence relation is a ternary analogue to the usual (binary) equivalence relation. *A reflexive and symmetric relation is a
dependency relation In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algori ...
(if finite), and a
tolerance relationIn mathematics, a tolerance relation is a binary relation, relation that is reflexive relation, reflexive and symmetric relation, symmetric, but not necessarily transitive relation, transitive; a set ''X'' that possesses a tolerance relation can be d ...
if infinite. *A
preorder 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 ...
is reflexive and transitive. *A
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) ...
is an equivalence relation whose domain $X$ is also the underlying set for an
algebraic 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 calculus, change (mathematical analysis, analysis). It ...
, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases, congruence relations have an alternative representation as substructures of the structure on which they are defined (e.g., the congruence relations on groups correspond to the
normal subgroup 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). *Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in
classical mathematicsIn the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories ...
(as opposed to
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In classical mathematics, one can prove the existence of a mathematical object without "finding ...
), since it is equivalent to the
law of excluded middle In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
. *Each relation that is both reflexive and left (or right)
Euclidean Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...
is also an equivalence relation.

# Well-definedness under an equivalence relation

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
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 ...
or a under the relation $\,\sim.$ A frequent particular case occurs when $f$ is a function from $X$ to another set $Y;$ if $x_1 \sim x_2$ implies $f\left\left(x_1\right\right) = f\left\left(x_2\right\right)$ then $f$ is said to be a for $\,\sim,$ a $\,\sim,$ or simply $\,\sim.$ This occurs, e.g. in the character theory of finite groups. The latter case with the function $f$ can be expressed by a commutative triangle. See also invariant. Some authors use "compatible with $\,\sim$" or just "respects $\,\sim$" instead of "invariant under $\,\sim$". More generally, a function may map equivalent arguments (under an equivalence relation $\,\sim_A$) to equivalent values (under an equivalence relation $\,\sim_B$). Such a function is known as a morphism from $\,\sim_A$ to $\,\sim_B.$

# Equivalence class, quotient set, partition

Let $a, b \in X.$ Some definitions:

## Equivalence class

A subset ''Y'' of ''X'' such that $a \sim b$ holds for all ''a'' and ''b'' in ''Y'', and never for ''a'' in ''Y'' and ''b'' outside ''Y'', is called an equivalence class of ''X'' by ~. Let denote the equivalence class to which ''a'' belongs. All elements of ''X'' equivalent to each other are also elements of the same equivalence class.

## Quotient set

The set of all equivalence classes of ''X'' by ~, denoted $X / \mathord := \,$ is the quotient set of ''X'' by ~. If ''X'' 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 ...
, there is a natural way of transforming $X / \sim$ into a topological space; see quotient space for the details.

## Projection

The projection of $\,\sim\,$ is the function $\pi : X \to X/\mathord$ defined by
surjection 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 ...

and $a \sim b \text f\left(a\right) = f\left(b\right),$ then $g$ is a
bijection 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 ...

.

## Equivalence kernel

The equivalence kernel of a function $f$ is the equivalence relation ~ defined by $x \sim y \text f\left(x\right) = f\left(y\right).$ The equivalence kernel of 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 ...

is the
identity relation Identity may refer to: Social sciences * Identity (social science) Identity is the qualities, beliefs, personality, looks and/or expressions that make a person (self-identity as emphasized in psychology) or group (collective identity as p ...
.

## Partition

A partition of ''X'' is a set ''P'' of nonempty subsets of ''X'', such that every element of ''X'' is an element of a single element of ''P''. Each element of ''P'' is a ''cell'' of the partition. Moreover, the elements of ''P'' are
pairwise disjoint Two disjoint sets. 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 anal ...
and their union is ''X''.

### Counting partitions

Let ''X'' be a finite set with ''n'' elements. Since every equivalence relation over ''X'' corresponds to a partition of ''X'', and vice versa, the number of equivalence relations on ''X'' equals the number of distinct partitions of ''X'', which is the ''n''th
Bell number In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, ...
''Bn'': :$B_n = \frac \sum_^\infty \frac \quad$ ( Dobinski's formula).

# Fundamental theorem of equivalence relations

A key result links equivalence relations and partitions: * An equivalence relation ~ on a set ''X'' partitions ''X''. * Conversely, corresponding to any partition of ''X'', there exists an equivalence relation ~ on ''X''. In both cases, the cells of the partition of ''X'' are the equivalence classes of ''X'' by ~. Since each element of ''X'' belongs to a unique cell of any partition of ''X'', and since each cell of the partition is identical to an equivalence class of ''X'' by ~, each element of ''X'' belongs to a unique equivalence class of ''X'' by ~. Thus there is a natural
bijection 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 ...

between the set of all equivalence relations on ''X'' and the set of all partitions of ''X''.

# Comparing equivalence relations

If $\sim$ and $\approx$ are two equivalence relations on the same set $S$, and $a \sim b$ implies $a \approx b$ for all $a, b \in S,$ then $\approx$ is said to be a coarser relation than $\sim$, and $\sim$ is a finer relation than $\approx$. Equivalently, * $\sim$ is finer than $\approx$ if every equivalence class of $\sim$ is a subset of an equivalence class of $\approx$, and thus every equivalence class of $\approx$ is a union of equivalence classes of $\sim$. * $\sim$ is finer than $\approx$ if the partition created by $\sim$ is a refinement of the partition created by $\approx$. The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. The relation "$\sim$ is finer than $\approx$" on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a
geometric latticeIn the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, ...
.

# Generating equivalence relations

* Given any set $X,$ an equivalence relation over the set
fixpoint In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function (mathematics), function is an element of the function's Domain of a function, domain that is mapped to itself by the function. That is to ...
s have the same
cardinality 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 ...
, corresponding to cycles of length one in a
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

. * An equivalence relation $\,\sim\,$ on $X$ is the equivalence kernel of its
surjective 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 ...
$\pi : X \to X / \sim.$ Conversely, any
surjection 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 ...

between sets determines a partition on its domain, the set of
preimage 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 singletons in the
codomain 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 ...

. Thus an equivalence relation over $X,$ a partition of $X,$ and a projection whose domain is $X,$ are three equivalent ways of specifying the same thing. * The intersection of any collection of equivalence relations over ''X'' (binary relations viewed as a
subset 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 ...

of $X \times X$) is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given any binary relation ''R'' on ''X'', the equivalence relation is the intersection of all equivalence relations containing ''R'' (also known as the smallest equivalence relation containing ''R''). Concretely, ''R'' generates the equivalence relation ::$a \sim b$ if there exists a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
$n$ and elements $x_0, \ldots, x_n \in X$ such that $a = x_0$, $b = x_n$, and $x_ \mathrel x_i$ or $x_i \mathrel x_$, for $i = 1, \ldots, n.$ :The equivalence relation generated in this manner can be trivial. For instance, the equivalence relation generated by any
total order 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 ...
on ''X'' has exactly one equivalence class, ''X'' itself. * Equivalence relations can construct new spaces by "gluing things together." Let ''X'' be the unit
Cartesian square 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 ...

, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
\times
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
and let ~ be the equivalence relation on ''X'' defined by $\left(a, 0\right) \sim \left(a, 1\right)$ for all $a \in$
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> and $\left(0, b\right) \sim \left(1, b\right)$ for all $b \in$
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
Then the quotient space $X / \sim$ can be naturally identified (
homeomorphism In the mathematical 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 quantiti ...
) with a
torus In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ...

: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.

# Algebraic structure

Much of mathematics is grounded in the study of equivalences, and
order relation Order theory is a branch of 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 (mathematic ...
s.
Lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper boun ...
captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily on
group theory 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 ...
and, to a lesser extent, on the theory of lattices,
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
, and
groupoid 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.

## Group theory

Just as
order relation Order theory is a branch of 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 (mathematic ...
s are grounded in ordered sets, sets closed under pairwise
supremum 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 ...

and
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...
, equivalence relations are grounded in , which are sets closed under
bijection 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 ...

s that preserve partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known as
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s. Hence
permutation 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 (mathematical analysis, analysis). It ...
s (also known as
transformation groups 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 ...
) and the related notion of
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...
shed light on the mathematical structure of equivalence relations. Let '~' denote an equivalence relation over some nonempty set ''A'', called the
universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development ...
or underlying set. Let ''G'' denote the set of bijective functions over ''A'' that preserve the partition structure of ''A'', meaning that for all $x \in A$ and Then the following three connected theorems hold: * ~ partitions ''A'' into equivalence classes. (This is the , mentioned above); * Given a partition of ''A'', ''G'' is a transformation group under composition, whose orbits are the
cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Closed spaces * Monastic cell, a small room, hut, or cave in which a monk or religious recluse lives * Prison cell, a room used to hold peopl ...
of the partition; * Given a transformation group ''G'' over ''A'', there exists an equivalence relation ~ over ''A'', whose equivalence classes are the orbits of ''G''. In sum, given an equivalence relation ~ over ''A'', there exists a
transformation group In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X''. For example, if ''X'' is a Dimension (vector space), finite-dimensional vector space, then the automorphism group of ' ...
''G'' over ''A'' whose orbits are the equivalence classes of ''A'' under ~. This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe ''A''. Meanwhile, the arguments of the transformation group operations
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 ...
and
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
are elements of a set of
bijections 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 ...
, ''A'' → ''A''. Moving to groups in general, let ''H'' be a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...
of some
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
''G''. Let ~ be an equivalence relation on ''G'', such that $a \sim b \text a b^ \in H.$ The equivalence classes of ~—also called the orbits of the
action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
of ''H'' on ''G''—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 ''H'' in ''G''. Interchanging ''a'' and ''b'' yields the left cosets. Related thinking can be found in Rosen (2008: chpt. 10).

## Categories and groupoids

Let ''G'' be a set and let "~" denote an equivalence relation over ''G''. Then we can form a
groupoid 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 ...
representing this equivalence relation as follows. The objects are the elements of ''G'', and for any two elements ''x'' and ''y'' of ''G'', there exists a unique morphism from ''x'' to ''y''
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
$x \sim y.$ The advantages of regarding an equivalence relation as a special case of a groupoid include: *Whereas the notion of "free equivalence relation" does not exist, that of a free groupoid on a
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a Graph (discrete mathematics), graph that is made up of a set of Vertex (graph theory), vertices connected by directed Edge (graph theory), edges often called ...

does. Thus it is meaningful to speak of a "presentation of an equivalence relation," i.e., a presentation of the corresponding groupoid; * Bundles of groups,
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). ...
s, sets, and equivalence relations can be regarded as special cases of the notion of groupoid, a point of view that suggests a number of analogies; *In many contexts "quotienting," and hence the appropriate equivalence relations often called congruences, are important. This leads to the notion of an internal groupoid in a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
.Borceux, F. and Janelidze, G., 2001. ''Galois theories'', Cambridge University Press,

## Lattices

The equivalence relations on any set ''X'', when ordered by
set inclusion 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). ...

, form a
complete lattice 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 th ...
, called Con ''X'' by convention. The canonical
map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...
ker: ''X''^''X'' → Con ''X'', relates the
monoid 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 (mathemati ...
''X''^''X'' of all
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 ...
s on ''X'' and Con ''X''. ker is
surjective 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 ...
but not
injective 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 ...
. Less formally, the equivalence relation ker on ''X'', takes each function ''f'': ''X''→''X'' to its
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 ...
ker ''f''. Likewise, ker(ker) is an equivalence relation on ''X''^''X''.

# Equivalence relations and mathematical logic

Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω- categorical, but not categorical for any larger
cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
. An implication of
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples: * ''Reflexive and transitive'': The relation ≤ on N. Or any
preorder 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 ...
; * ''Symmetric and transitive'': The relation ''R'' on N, defined as ''aRb'' ↔ ''ab'' ≠ 0. Or any partial equivalence relation; * ''Reflexive and symmetric'': The relation ''R'' on Z, defined as ''aRb'' ↔ "''a'' − ''b'' is divisible by at least one of 2 or 3." Or any
dependency relation In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algori ...
. Properties definable in
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
that an equivalence relation may or may not possess include: *The number of equivalence classes is finite or infinite; *The number of equivalence classes equals the (finite) natural number ''n''; *All equivalence classes have infinite
cardinality 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 number of elements in each equivalence class is the natural number ''n''.

*
Conjugacy 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 ...
*
Equipollence (geometry) In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; ...
*
Quotient by 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). It h ...
*
Topological conjugacyIn 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 ha ...
*
Up to Two mathematical 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 ...

# References

*Brown, Ronald, 2006.
Topology and Groupoids.
' Booksurge LLC. . *Castellani, E., 2003, "Symmetry and equivalence" in Brading, Katherine, and E. Castellani, eds., ''Symmetries in Physics: Philosophical Reflections''. Cambridge Univ. Press: 422–433. * Robert Dilworth and Crawley, Peter, 1973. ''Algebraic Theory of Lattices''. Prentice Hall. Chpt. 12 discusses how equivalence relations arise in lattice (order), lattice theory. *Higgins, P.J., 1971.
Categories and groupoids.
' Van Nostrand. Downloadable since 2005 as a TAC Reprint. *John Lucas (philosopher), John Randolph Lucas, 1973. ''A Treatise on Time and Space''. London: Methuen. Section 31. *Rosen, Joseph (2008) ''Symmetry Rules: How Science and Nature are Founded on Symmetry''. Springer-Verlag. Mostly chapters. 9,10. *Raymond Wilder (1965) ''Introduction to the Foundations of Mathematics'' 2nd edition, Chapter 2-8: Axioms defining equivalence, pp 48–50, John Wiley & Sons.