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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if 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 \,\sim\, on a set X is said to be an equivalence relation, if and only 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 (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
). * If a \sim b and b \sim c then a \sim c ( transitivity). X together with the relation \,\sim\, is called a setoid. The equivalence class of a under \,\sim, denoted is defined as = \.


Alternative definition using relational algebra

In relational algebra, 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 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantitie ...
on X.) * R=R^ (
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
). * 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 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 with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
s. * "Is congruent to" on the set of all
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
s. * Given a natural number n, "is congruent to, modulo n" on the integers. * Given a function f:X \to Y, "has the same
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
under f as" on the elements of f's domain 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 greater than 1 with" between natural numbers 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 partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
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 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 is reflexive and transitive. *A congruence relation is an equivalence relation whose domain X is also the underlying set for an
algebraic structure In mathematics, an algebraic structure consists of a nonempty Set (mathematics), set ''A'' (called the underlying set, carrier set or domain), a collection of operation (mathematics), operations on ''A'' (typically binary operations such as addit ...
, 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 subgroups). *Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle. *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.


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, 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.


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 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 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, corresponding to cycles of length one in a
permutation In mathematics, a permutation of a Set (mathematics), set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers ...
. * An equivalence relation \,\sim\, on X is the equivalence kernel of its surjective 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. 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, 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 are ...
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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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\times , 1 and let ~ be the equivalence relation on ''X'' defined by (a, 0) \sim (a, 1) for all a \in , 1/math> and (0, b) \sim (1, b) for all b \in , 1 Then the quotient space X / \sim can be naturally identified ( homeomorphism) with a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) 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 ...
: 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 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 groupoids.


Group theory

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under bijections 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 (mathematics), set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers ...
s. Hence permutation groups (also known as transformation groups) 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 pos ...
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, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmology, cosmological description of the development of ...
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 In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X'' under Function composition, composition of morphisms. For example, if ''X'' is a Dimension (vector space), finite-dime ...
''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, 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 ''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 cosets 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 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 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 actions, 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, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
.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 ''X''^''X'' of all functions on ''X'' and Con ''X''. ker is surjective but not injective. 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
. An implication of model theory 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; * ''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 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 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 John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, Academi ...
.


External links

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