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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 modern mathematics ...
, an equivalence relation is a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
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 Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
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 a ...
es. 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 In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
\,\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 definit ...
). * 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 a ...
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 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 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 definit ...
). * 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 Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
of the set X with respect to R.


Equivalence relations

The following relations are all equivalence relations: * "Is equal to" on the set of numbers. For example, \tfrac is equal to \tfrac. * "Has the same birthday as" on the set of all people. * "Is similar to" on the set of all triangles. * "Is congruent to" on the set of all triangles. * Given a natural number n, "is congruent to,
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...
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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
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 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 ...
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 In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
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 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 A substitution reaction (also known as single displacement reaction or single substitution reaction) is a chemical reaction during which one functional group in a chemical compound is replaced by another functional group. Substitution reactions ar ...
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 In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, the ...
is transitive and symmetric. Such a relation is reflexive if and only if it is
total Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are comp ...
, 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 In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relation of collinearity amon ...
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 and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
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, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
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 ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
, 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 G i ...
s). *Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in
classical mathematics In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory. It stands in contrast to other types of mathematics such as constructive m ...
(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 Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
. 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
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
s: 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, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
.


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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, r ...
.


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
fixpoint A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the ...
s have the same
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, corresponding to cycles of length one in a
permutation In mathematics, a permutation of a 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 to the act or proc ...
. * 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 that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
\pi : X \to X / \sim. Conversely, any surjection between sets determines a partition on its domain, the set of
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s 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 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 on ''X'' has exactly one equivalence class, ''X'' itself. * Equivalence relations can construct new spaces by "gluing things together." Let ''X'' be the unit
Cartesian square In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
\times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
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. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/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. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
Then the quotient space X / \sim can be naturally identified ( homeomorphism) 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.


Algebraic structure

Much of mathematics is grounded in the study of equivalences, and
order relation Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
s. 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 and, to a lesser extent, on the theory of lattices, categories, and groupoids.


Group theory

Just as
order relation Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
s are grounded in ordered sets, sets closed under pairwise
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 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 to the act or proc ...
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 it ...
s (also known as
transformation groups In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the ...
) and the related notion of orbit shed light on the mathematical structure of equivalence relations. Let '~' denote an equivalence relation over some nonempty set ''A'', called the universe 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 Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
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 Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
and join are elements of some universe ''A''. Meanwhile, the arguments of the transformation group operations composition and
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
are elements of a set of bijections, ''A'' → ''A''. Moving to groups in general, let ''H'' be a subgroup 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.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 In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally 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 In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
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; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. 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. An implication of
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
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 and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
; * ''Symmetric and transitive'': The relation ''R'' on N, defined as ''aRb'' ↔ ''ab'' ≠ 0. Or any
partial equivalence relation In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, the ...
; * ''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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
; *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 Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say ...
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 Raymond Louis Wilder (3 November 1896 in Palmer, Massachusetts – 7 July 1982 in Santa Barbara, California) was an American mathematician, who specialized in topology and gradually acquired philosophical and anthropological interests. Life Wilde ...
(1965) ''Introduction to the Foundations of Mathematics'' 2nd edition, Chapter 2-8: Axioms defining equivalence, pp 48–50, John Wiley & Sons.


External links

* * Bogomolny, A.,
Equivalence Relationship
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
. Accessed 1 September 2009
Equivalence relation
at PlanetMath * {{DEFAULTSORT:Equivalence Relation Binary relations Equivalence (mathematics) Reflexive relations Symmetric relations Transitive relations