In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being "equivalent" in some way. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of Let a, b, and c be arbitrary elements of some set X. Then "a ~ b" or "a ≡ b" denotes that a is equivalent to b.

An equivalence relation "~" is reflexive, symmetric, and transitive. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In other words, the following must hold for "~" to be an equivalence relation on X:

An equivalence relation partitions a set into several disjoint subsets, called equivalence classes. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In Mathematics, two sets are said to be disjoint if they have no element in common In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X All the elements in a given equivalence class are equivalent among themselves, and no element is equivalent with any element from a different class.

• Reflexivity: a ~ a
• Symmetry: if a ~ b then b ~ a
• Transitivity: if a ~ b and b ~ c then a ~ c. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b

The equivalence class a under "~", denoted [a], is the subset of X whose elements b are such that a~b. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X X together with "~" is called a setoid. In Mathematics, a setoid is a set (or type) equipped with an Equivalence relation.

Examples of equivalence relations

A ubiquitous equivalence relation is the equality ("=") relation between elements of any set. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric Other examples include:

• "Has the same birthday as" on the set of all people, given naive set theory. Naive set theory is one of several theories of sets used in the discussion of the Foundations of mathematics.
• "Is similar to" or "congruent to" on the set of all triangles. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line
• "Is congruent to modulo n" on the integers. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French
• "Has the same image under a function" on the elements of the domain of the function. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined
• Logical equivalence of logical sentences. In Logic, statements p and q are logically equivalent if they have the same logical content This article is a technical mathematical article in the area of predicate logic
• "Is isomorphic to" on models of a set of sentences. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models This article is a technical mathematical article in the area of predicate logic
• In some axiomatic set theories other than the canonical ZFC (e. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common g. , New Foundations and related theories):
  • Similarity on the universe of well-orderings gives rise to equivalence classes that are the ordinal numbers. In Mathematical logic, New Foundations ( NF) is an Axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set.
  • Equinumerosity on the universe of:
    • Finite sets gives rise to equivalence classes which are the natural numbers. In the field of Mathematics, two sets A and B are equinumerous if they have the same Cardinality, i The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an
    • Infinite sets gives rise to equivalence classes which are the transfinite cardinal numbers. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.
• Let a, b, c, d be natural numbers, and let (a, b) and (c, d) be ordered pairs of such numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry Then the equivalence classes under the relation (a, b) ~ (c, d) are the:
  • Integers if a + d = b + c;
  • Positive rational numbers if ad = bc. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions
• Let (r_n) and (s_n) be any two Cauchy sequences of rational numbers. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence The real numbers are the equivalence classes of the relation (r_n) ~ (s_n), if the sequence (r_n − s_n) has limit 0. In Mathematics, the real numbers may be described informally in several different ways
• Green's relations are five equivalence relations on the elements of a semigroup. In Mathematics, Green's relations are five Equivalence relations that characterise the elements of a Semigroup in terms of the Principal ideals In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation
• "Is parallel to" on the set of subspaces of an affine space. Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space.

Examples of relations that are not equivalences

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 does not imply that 5 ≥ 7. It is, however, a partial order. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement
• The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an (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 on a non-empty set X (i. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members e. aRb is never true) is vacuously symmetric and transitive, but not reflexive. A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If  A  then  (If X is also empty then R is reflexive. )
• The relation "is approximately equal to" between real numbers or other things, 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.
• The relation "is a sibling of" on the set of all human beings is not an equivalence relation. Although siblinghood is symmetric (if A is a sibling of B, then B is a sibling of A) it is neither reflexive (no one is a sibling of himself), nor transitive (since if A is a sibling of B, then B is a sibling of A, but A is not a sibling of A). Instead of being transitive, siblinghood is "almost transitive", meaning that if A ~ B, and B ~ C, and A ≠ C, then A ~ C.
• The concept of parallelism in ordered geometry is not symmetric and is, therefore, not an equivalence relation. Ordered geometry is a form of Geometry featuring the concept of intermediacy (or "betweenness" but like Projective geometry, omitting the basic notion
• An equivalence relation on a set is never an equivalence relation on a proper superset of that set. For example R = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)} is an equivalence relation on {1,2,3} but not on {1,2,3,4} or on the natural number. The problem is that reflexivity fails because (4,4) is not a member.

Connection to other relations

A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraic structure, and which respects the additional structure. See Congruence (geometry for the term as used in elementary geometry In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, 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 Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that 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. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.

Order and equivalence relations are both transitive, but only equivalence relations are symmetric as well. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying If symmetry is weakened to antisymmetry, the result is a partial order. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

A partial equivalence relation is transitive and symmetric, but not reflexive. In Mathematics, a partial equivalence relation (often abbreviated as PER) R on a set X is a relation which is symmetric

• Transitive and symmetric imply reflexive iff for all a∈X exists b∈X such that a~b. ↔
  

A dependency relation is reflexive and symmetric, but not transitive. In Mathematics and Computer science, a dependency relation is a Binary relation that is finite symmetric, and reflexive.
A preorder is reflexive and transitive, but neither symmetric nor antisymmetric. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions
A strict partial order is transitive alone. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

Equivalence relations can thus be seen as the culmination of a hierarchy of order relations.

Equivalence class, quotient set, partition

Let X be a nonempty set with typical elements a and b. Some definitions:

• The set of all a and b for which a ~ b holds make up an equivalence class of X by ~. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X Let [a] =: {x ∈ X : x ~ a} denote the equivalence class to which a belongs. Then all elements of X equivalent to each other are also elements of the same equivalence class: ∀a, b ∈ X (a ~ b ↔ [a ] = [b ]).
• The set of all possible equivalence classes of X by ~, denoted X/~ =: {[x] : x ∈ X}, is the quotient set of X by ~. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient space for the details. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying
• The projection of ~ is the function π : X → X/~, defined by π(x) = [x ], mapping elements of X into their respective equivalence classes by ~.
  

Theorem on projections (Birkhoff and Mac Lane 1999: 35, Th. 19): Let the function f: X → B be such that a ~ b → f(a) = f(b). Then there is a unique function g : X/~ → B, such that f = gπ. If f is a surjection and a ~ b ↔ f(a) = f(b), then g is a bijection. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

• The equivalence kernel of a function f is the equivalence relation, denoted Ef, such that xEfy ↔ f(x) = f(y). The equivalence kernel of an injection is the identity relation. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric
• A partition of X is a set P of subsets of X, such that every element of X is an element of a single element of P. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their union is X.

Theorem ("Fundamental Theorem of Equivalence Relations": Wallace 1998: 31, Th. 8; Dummit and Foote 2004: 3, Prop. 2):

• An equivalence relation ~ partitions X. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks "
• Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks "

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 ~. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X Thus there is a natural bijection from the set of all possible equivalence relations on X and the set of all partitions of X. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

Counting possible 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 possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B_n:

Generating equivalence relations

• Given any set X, there is an equivalence relation over the set of all possible functions X→X. In combinatorial Mathematics, the n th Bell number, named in honor of Eric Temple Bell, is the number of partitions of a set Two such 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 fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" In several fields of Mathematics the term permutation is used with different but closely related meanings Functions equivalent in this manner form an equivalence class on X², and these equivalence classes partition X².

• An equivalence relation ~ on X is the equivalence kernel of its surjective projection π : X → X/~. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every (Birkhoff and Mac Lane 1999: 33 Th. 18). Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, the codomain, or target, of a function f: X → Y is the set 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 two equivalence relations over X (viewed as a subset of X × 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 generated by R is the smallest equivalence relation containing R. Concretely, R generates the equivalence relation a ~ b iff there exist elements x₁, x₂, . ↔ . . , x_n in X such that a = x₁, b = x_n, and (x_i,x_{i+ 1})∈R or (x_i+1,x_i)∈R, i = 1, . . . , n-1.

Note that the equivalence relation generated in this manner can be trivial. For instance, the equivalence relation ~ generated by:

• The binary relation ≤ has exactly one equivalence class, X itself, because x ~ y for all x and y;
• An antisymmetric relation has equivalence classes that are the singletons of X.

• Let r be any sort of relation on X. Then r ∪ r⁻¹ is a symmetric relation. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that The transitive closure s of r ∪ r⁻¹ assures that s is transitive and reflexive. In Mathematics, the transitive closure of a Binary relation R on a set X is the smallest Transitive relation on X Moreover, s is the "smallest" equivalence relation containing r, and r/s partially orders X/s. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

• Equivalence relations can construct new spaces by "gluing things together. " Let X be the unit Cartesian square [0,1] × [0,1], and let ~ be the equivalence relation on X defined by ∀a, b ∈ [0,1] ((a, 0) ~ (a, 1) ∧ (0, b) ~ (1, b)). Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. Then the quotient space X/~ can be naturally identified with a torus: 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. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

Algebraic structure

Modular lattices

The possible equivalence relations on any set X, when ordered by set inclusion, form a modular lattice, called Con X by convention. In the branch of mathematics called Order theory, a modular lattice is a lattice that satisfies the following self-dual condition Modular law: x The canonical map ker: X∧X → Con X, relates the monoid X^X of all functions on X and Con X. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function ker is surjective but not injective. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every 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.

Group theory

It is very well known that lattice theory captures the mathematical structure of order relations. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying It is less known that transformation groups (some authors prefer permutation groups) and their orbits shed light on the mathematical structure of equivalence relations. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, sets closed under bijections preserving partition structure. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying Ordered set is used with distinct meanings in Order theory. A set with a Binary relation R on its elements that is reflexive (for In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. In several fields of Mathematics the term permutation is used with different but closely related meanings

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. In Mathematical logic, the universe of a structure (or model) is its domain. Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). Then the following three connected theorems hold (Van Fraassen 1989: §10. 3):

• ~ partitions A into equivalence classes. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " (This is the Fundamental Theorem of Equivalence Relations, 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 Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks " In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. (Wallace 1998: 202, Th. 6; Dummit and Foote 2004: 114, Prop. 2).

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 ~. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying 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. In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

Moving to groups in general, let H be a subgroup of some group G. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Let ~ be an equivalence relation on G, such that a ~ b ↔ (ab⁻¹ ∈ H). The equivalence classes of ~—also called the orbits of the action of H on G—are the right cosets of H in G. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH Interchanging a and b yields the left cosets.

For more on group theory and equivalence relations, see Lucas (1973: §31).

‡Proof (adapted from Van Fraassen 1989: 246). Let function composition interpret group multiplication, and function inverse interpret group inverse. In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B Then G is a group under composition, meaning that ∀x ∈ A ∀g ∈ G ([g(x)] = [x]), because G satisfies the following four conditions:

• G is closed under composition. In Mathematics, a composite function represents the application of one function to the results of another The composition of any two elements of G exists, because the domain and codomain of any element of G is A. In Mathematics, the codomain, or target, of a function f: X → Y is the set Moreover, the composition of bijections is bijective (Wallace 1998: 22, Th. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 6);
• Existence of identity element. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that The identity function, I(x)=x, is an obvious element of G;
• Existence of inverse function. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B Every bijective function g has an inverse g⁻¹, such that gg⁻¹ = I;
• Composition associates. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, associativity is a property that a Binary operation can have f(gh) = (fg)h. This holds for all functions over all domains (Wallace 1998: 24, Th. 7).

Let f and g be any two elements of G. By virtue of the definition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that [g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function composition preserves the partitioning of A. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

Relation with category theory and with groupoids

The composition of morphisms central to category theory, denoted here by concatenation, generalizes the composition of functions central to transformation groups. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a composite function represents the application of one function to the results of another The axioms of category theory assert that the composition of morphisms associates, and that the left and right identity morphisms exist for any morphism. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

A morphism f can be said to have an inverse when f is an isomorphism, i. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective e. , there exists a morphism g such that fg and gf are the approrpiate identity morphisms. Hence the category-theoretic concept nearest to an equivalence relation is a (small) category whose morphisms are all isomorphisms. This is just the concept of groupoid. In Mathematics, especially in Category theory and Homotopy theory

In a groupoid G, two objects x,y are 'equivalent' if there is an element g of the groupoid from x to y. There may be many such g, and they can be regarded as different `proofs' that x is equivalent to y.

Regarding an equivalence relation as a special case of a groupoid has many implications: one is that whereas we do not have a notion of `free equivalence relation' we do have a notion of free groupoid on a directed graph. Thus we can talk of a `presentation of an equivalence relation', meaning a presentation of the corresponding groupoid. The other advantage is that it views bundles of groups, group actions, sets, and equivalence relations, as special cases of the same notion, that of groupoid, and so allows analogies between these theories and concepts.

This also applies in many other contexts where `quotienting', and so the appropriate equivalence relations, often called congruences are important. This leads to the notion of internal groupoid in a category. For this, see the book `Galois theories' cited below.

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. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

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. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models 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. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions 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. In Mathematics, a partial equivalence relation (often abbreviated as PER) R on a set X is a relation which is symmetric " Or any dependency relation. In Mathematics and Computer science, a dependency relation is a Binary relation that is finite symmetric, and reflexive.

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. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, the cardinality of a set is a measure of the "number of elements of the set"

Euclid anticipated equivalence

Euclid's The Elements includes the following "Common Notion 1":

Things which equal the same thing also equal one another. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). In Mathematics, a Binary relation R over a set X is euclidean if it holds for all a, b, and c in The following theorem connects Euclidean relations and equivalence relations:

Theorem. In Mathematics, a Binary relation R over a set X is euclidean if it holds for all a, b, and c in If a relation is Euclidean and reflexive, it is also symmetric and transitive.

Proof:

• (aRc ∧ bRc) → aRb [a/c] = (aRa ∧ bRa) → aRb [reflexive; erase T∧] = bRa → aRb. Hence R is symmetric. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or
• (aRc ∧ bRc) → aRb [symmetry] = (aRc ∧ cRb) → aRb. Hence R is transitive.

Hence an equivalence relation is a relation that is Euclidean and reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention. If this (and taking "equality" as an all-purpose abstract relation) is granted, a charitable reading of Common Notion 1 would credit Euclid with being the first to conceive of equivalence relations and their importance in deductive systems. A deductive system (also called a deductive apparatus of a Formal system) consists of the Axioms (or Axiom schemata and Rules of inference

See also

• Automorphism
• Automorphism group
• Congruence relation
• Directed set
• Equivalence
• Equivalence class
• Euclidean relation
• Group action
• Partial equivalence relation
• Symmetry group
• Total order
• Transformation group
• Up to

References

• Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself See Congruence (geometry for the term as used in elementary geometry In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, a Binary relation R over a set X is euclidean if it holds for all a, b, and c in In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a partial equivalence relation (often abbreviated as PER) R on a set X is a relation which is symmetric The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose Garrett Birkhoff ( January 19, 1911, Princeton, New Jersey, USA – November Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Algebra, 3rd ed. Chelsea.
• Borceux, F. and Janelidze, G. , 2001. Galois theories, Cambridge University Press, ISBN 0521803098.
• Brown, R. , 2006. Topology and Groupoids, Booksurge LLC. ISBN 1419627228.
• Castellani, E. , 2003, "Symmetry and equivalence" in Katherine Brading and E. Castellani (eds. ), Symmetries in Physics: Philosophical Reflections. Cambridge University Press: 422-433.
• Robert Dilworth and Crawley, Peter, 1973. Robert Palmer Dilworth (December 2 1914 – October 29 1993 was an American Mathematician. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusses how equivalence relations arise in lattice theory. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements'
• Dummit, D. S. , and Foote, R. M. , 2004. Abstract Algebra, 3rd ed. John Wiley & Sons.
• Higgins, P. J. , 1971. Categories and groupoids, van Nostrand, downloadable as TAC Reprint, 2005.
• John Randolph Lucas, 1973. John Randolph Lucas FBA (born 18 June, 1929) is a British philosopher A Treatise on Time and Space. London: Methuen. Section 31.
• Rosen, Joseph, 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag.
• Bas van Fraassen, 1989. Bastiaan Cornelis van Fraassen (born Goes, the Netherlands, 5 April 1941) is a member of the Princeton University Philosophy Laws and Symmetry. Oxford Univ. Press.
• Wallace, D. A. R. , 1998. Groups, Rings and Fields. Springer-Verlag.

External links

• Bogomolny, A. , "Equivalence Relationship" cut-the-knot. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. Accessed 7 December 2007