
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
and
of a set are equivalent with respect to an equivalence relation
the most common are "
" and "", which are used when
is implicit, and variations of "
", "", or "
" to specify
explicitly. Non-equivalence may be written "" or "
".
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 ...
on a set
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
and
in
*
(
Reflexivity)
*
if and only if
(
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
and
then
(
Transitivity)
together with the relation
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
under
denoted
is defined as
Examples
Simple example
On the set
, the relation
is an equivalence relation. The following sets are equivalence classes of this relation:
The set of all equivalence classes for
is
This set is a
of the set
with respect to
.
Equivalence relations
The following relations are all equivalence relations:
* "Is equal to" on the set of numbers. For example,
is equal to
* "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
, "is congruent to,
modulo " 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 ...
, "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
as" on the elements of
'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 ...
. For example,
and
have the same image under
, viz.
.
* "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
there exists some
[''If:'' Given let hold using seriality, then by symmetry, hence by transitivity. — ''Only if:'' Given choose then 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
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
is an equivalence relation on
and
is a property of elements of
such that whenever
is true if
is true, then the property
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
A frequent particular case occurs when
is a function from
to another set
if
implies
then
is said to be a for
a
or simply
This occurs, e.g. in the character theory of finite groups. The latter case with the function
can be expressed by a commutative triangle. See also
invariant. Some authors use "compatible with
" or just "respects
" instead of "invariant under
".
More generally, a function may map equivalent arguments (under an equivalence relation
) to equivalent values (under an equivalence relation
). Such a function is known as a morphism from
to
Equivalence class, quotient set, partition
Let
Some definitions:
Equivalence class
A subset ''Y'' of ''X'' such that
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
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
into a topological space; see
quotient space for the details.
Projection
The projection of
is the function
defined by