equivalence relation
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an equivalence relation is a
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es. Two elements of the given set are equivalent to each other
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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".


Definitions

A
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
\,\sim\, on a set X is said to be an equivalence relation, if 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 () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
). * If a \sim b and b \sim c then a \sim c ( transitivity). X together with the relation \,\sim\, is called a setoid. The
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of a under \,\sim, denoted is defined as = \.


Alternative definition using relational algebra

In
relational algebra In database theory, relational algebra is a theory that uses algebraic structures for modeling data and defining queries on it with well founded semantics (computer science), semantics. The theory was introduced by Edgar F. Codd. The main applica ...
, if R\subseteq X\times Y and S\subseteq Y\times Z are relations, then the composite relation SR\subseteq X\times Z is defined so that x \, SR \, z if and only if there is a y\in Y such that x \, R \, y and y \, S \, z.Sometimes the composition SR\subseteq X\times Z is instead written as R;S, or as RS; in both cases, R is the first relation that is applied. See the article on
Composition of relations In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
for more information.
This definition is a generalisation of the definition of functional composition. The defining properties of an equivalence relation R on a set X can then be reformulated as follows: * \operatorname \subseteq R. ( reflexivity). (Here, \operatorname denotes the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on X.) * R=R^ (
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
). * RR\subseteq R ( transitivity).


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 partition of the set X. It is also called the
quotient set In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of X by 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. * "Is similar to" on the set of all
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s. * "Is congruent to" on the set of all
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s. * Given a function f:X \to Y, "has the same
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
under f as" on the elements of f's domain X. For example, 0 and \pi have the same image under \sin, viz. 0. In particular: ** "Has the same absolute value as" on the set of real numbers ** "Has the same cosine as" on the set of all angles. ** Given a natural number n, "is congruent to,
modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
n" on the
integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. ** "Have the same length and direction" ( equipollence) on the set of directed line segments. ** "Has the same birthday as" on the set of all people.


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 greater than 1 with" between
natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
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 ''R'' (defined so that ''aRb'' is never true) on a set ''X'' is vacuously 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 In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
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 expressions, 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 is irreflexive, transitive, and asymmetric. * A partial equivalence relation is transitive and symmetric. Such a relation is reflexive
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is total, that is, if for all a, there exists some b \text a \sim b.''If:'' Given a, let a \sim b hold using totality, 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 total relation. * A ternary equivalence relation is a ternary analogue to the usual (binary) equivalence relation. * A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite. * A
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. The name is meant to suggest that preorders are ''almost'' partial orders, ...
is reflexive and transitive. * A
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 ...
is an equivalence relation whose domain X is also the underlying set for an
algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
, 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, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
s). * Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
), since it is equivalent to the
law of excluded middle In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and t ...
. * Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.


Well-definedness under an equivalence relation

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 well-defined 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(x_1\right) = f\left(x_2\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.


Related important definitions

Let a, b \in X, and \sim be an equivalence relation. Some key definitions and terminology follow:


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 \sim. 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 \sim, denoted X / \mathord := \, is the ''quotient set'' of X by \sim. If X 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 ...
, 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 \pi(x) = /math> which maps elements of X into their respective equivalence classes by \,\sim. : Theorem on projections: Let the function f : X \to B be such that if a \sim b then f(a) = f(b). Then there is a unique function g : X / \sim \to B such that f = g \pi. If f is a surjection and a \sim b \text f(a) = f(b), then g is a
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
.


Equivalence kernel

The equivalence kernel of a function f is the equivalence relation ~ defined by x \sim y \text f(x) = f(y). The equivalence kernel of an injection is the identity relation.


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 In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
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 ''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, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
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 lattice.


Generating equivalence relations

* Given any set X, an equivalence relation over the set \to X/math> of all functions X \to X can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
, corresponding to cycles of length one in a
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
. * An equivalence relation \,\sim\, on X is the equivalence kernel of its
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
projection \pi : X \to X / \sim. Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the
codomain In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...
. 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, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( re ...
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
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\times
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
and let ~ be the equivalence relation on ''X'' defined by (a, 0) \sim (a, 1) for all a \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> and (0, b) \sim (1, b) for all b \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
Then the quotient space X / \sim can be naturally identified (
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
) with a
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
: 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 relations.
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 pair of elements has a unique supremum (also called a least upper bou ...
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 abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
and, to a lesser extent, on the theory of lattices, categories, and
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...
s.


Group theory

Just as order relations are grounded in ordered sets, sets closed under pairwise
supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under
bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
s that preserve partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known as
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
s. Hence
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
s (also known as transformation groups) and the related notion of
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
shed light on the mathematical structure of equivalence relations. Let '~' denote an equivalence relation over some nonempty set ''A'', called the
universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
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 g \in G, g(x) \in 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 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 ''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 and inverse are elements of a set of bijections, ''A'' → ''A''. Moving to groups in general, let ''H'' be a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of some group ''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 of ''H'' on ''G''—are the right
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
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, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...
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 and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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 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, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
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: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
.Borceux, F. and Janelidze, G., 2001. ''Galois theories'', Cambridge University Press,


Lattices

The equivalence relations on any set ''X'', when ordered by set inclusion, form a complete lattice, called Con ''X'' by convention. The canonical map ker : ''X''^''X'' → Con ''X'', relates the
monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
''X''^''X'' of all functions on ''X'' and Con ''X''. ker is
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
but not
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
. Less formally, the equivalence relation ker on ''X'', takes each function ''f'' : ''X'' → ''X'' to its kernel 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 In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
. An implication of
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
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, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. The name is meant to suggest that preorders are ''almost'' partial orders, ...
; * ''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. Properties definable in
first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
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 The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
; * The number of elements in each equivalence class is the natural number ''n''.


See also

* * * * * * * *


Notes


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 theory. * Higgins, P.J., 1971.
Categories and groupoids.
' Van Nostrand. Downloadable since 2005 as a TAC Reprint. * 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.


External links

* * Alexander Bogomolny, Bogomolny, A.,
Equivalence Relationship
cut-the-knot. Accessed 1 September 2009
Equivalence relation
at PlanetMath * {{DEFAULTSORT:Equivalence Relation Equivalence (mathematics) Reflexive relations Symmetric relations Transitive relations