- Equivalence relation
In

mathematics , an**equivalence relation**is abinary relation between two elements of a set which groups them together as being "equivalent" in some way. 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 , andtransitive . In other words, the following must hold for "~" to be an equivalence relation on "X":*Reflexivity: "a" ~ "a"

*Symmetry: if "a" ~ "b" then "b" ~ "a"

*Transitivity: if "a" ~ "b" and "b" ~ "c" then "a" ~ "c".The

equivalence class "a" under "~", denoted ["a"] , is the subset of "X" whose elements "b" are such that "a"~"b". "X" together with "~" is called asetoid .**Examples of equivalence relations**A ubiquitous equivalence relation is the equality ("=") relation between elements of any set. Other examples include:

* "Has the same birthday as" on the set of all people, givennaive set theory .

* "Is similar to" or "congruent to" on the set of all triangles.

* "Is congruent to modulo "n" on theintegers .

* "Has the same image under a function" on the elements of the domain of the function.

*Logical equivalence of logical sentences.

* "Isisomorphic to" on models of a set of sentences.

* In some axiomatic set theories other than the canonicalZFC (e.g.,New Foundations and related theories):

**Similarity on theuniverse ofwell-ordering s gives rise toequivalence class es that are theordinal number s.

**Equinumerosity on theuniverse of:

***Finite sets gives rise toequivalence class es which are thenatural number s.

***Infinite sets gives rise to equivalence classes which are thetransfinite cardinal numbers .

* Let "a", "b", "c", "d" benatural number s, and let ("a, b") and ("c, d") beordered pair s of such numbers. Then theequivalence class es under the relation ("a, b") ~ ("c, d") are the:

**Integer s if "a" + "d" = "b" + "c";

**Positiverational number s if "ad" = "bc".

*Let ("r_{n"}) and ("s_{n"}) be any twoCauchy sequence s of rational numbers. Thereal number s are the equivalence classes of the relation ("r_{n"}) ~ ("s_{n"}), if the sequence ("r_{n"}− "s_{n"}) has limit 0.

*Green's relations are five equivalence relations on the elements of asemigroup .

* "Isparallel to" on the set ofsubspace s of anaffine 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.

* The relation "has a common factor greater than 1 with" betweennatural numbers greater than 1, is reflexive and symmetric, but not transitive. (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 anon-empty set "X" (i.e. "aRb" is never true) is vacuously symmetric and transitive, but not reflexive. (If "X" is also empty then "R" "is" reflexive.)

* 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.

* 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". However, the relation "A" is a sibling of "B" or "A" is "B" is an equivalence relation. (This applies only to "full" siblings. "A" and "B" could have the same mother, and "B" and "C" the same father, without "A" and "C" having a common parent.)

* The concept of parallelism inordered geometry is not symmetric and is, therefore, not an equivalence relation.

* 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 analgebraic structure , 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 thenormal subgroup s.

*Apartial order replaces symmetry with antisymmetry and is thus reflexive, antisymmetric, and transitive. Equality is the only relation that is both an equivalence relation and a partial order.

*Astrict partial order is irreflexive, transitive, andasymmetric .

*Apartial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexiveiff for all "a"∈"X" exists "b"∈"X" such that "a"~"b".

*Adependency relation is reflexive and symmetric.

*Apreorder is reflexive and transitive.**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 anof "X" by ~. 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" ] ).equivalence class

*The set of all possible equivalence classes of "X" by ~, denoted "X"/~ =: { ["x"] : "x" ∈ "X"}, is theof "X" by ~. If "X" is aquotient set topological space , there is a natural way of transforming "X"/~ into a topological space; seequotient space for the details.

* Theof ~ is the function π : "X" → "X"/~, defined by π("x") = ["x" ] , mapping elements of "X" into their respective equivalence classes by ~.projection :

**Theorem**onprojection s (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" = "g"π. If "f" is asurjection and "a" ~ "b" ↔ "f"("a") = "f"("b"), then "g" is abijection .*The

**equivalence kernel**of a function "f" is the equivalence relation, denoted "Ef", such that "xEfy" ↔ "f"("x") = "f"("y"). The equivalence kernel of aninjection is theidentity relation .

* 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". 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".

*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 anequivalence class of "X" by ~, each element of "X" belongs to a unique equivalence class of "X" by ~. Thus there is a naturalbijection from the set of all possible equivalence relations on "X" and the set of all partitions of "X"."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"}:: $B\_n\; =\; sum\_\{k=0\}^infty\; frac\{k^n\}\{ek!\}.$**Generating equivalence relations***Given any set "X", there is an equivalence relation over the set of all possible functions "X"→"X". Two such functions are deemed equivalent when their respective sets of

fixpoint s have the samecardinality , corresponding to cycles of length one in apermutation . Functions equivalent in this manner form an equivalence class on "X"^{2}, and these equivalence classes partition "X"^{2}.*An equivalence relation ~ on "X" is the equivalence kernel of its

surjective projection π : "X" → "X"/~. (Birkhoff and Mac Lane 1999: 33 Th. 18). Conversely, anysurjection between sets determines a partition on its domain, the set ofpreimage s of dn|singletons in thecodomain . 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" (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 "x"_{1}, "x"_{2}, ..., "x"_{"n"}in "X" such that "a" = "x"_{1}, "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 thesingleton sdn of "X".*Let "r" be any sort of relation on "X". Then "r" ∪ "r"

^{−1}is asymmetric relation . Thetransitive closure "s" of "r" ∪ "r"^{−1}assures that "s" is transitive and reflexive. Moreover, "s" is the "smallest" equivalence relation containing "r", and "r"/"s" partially orders "X"/"s".*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")). Then thequotient space "X"/~ can be naturally identified with atorus : 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 atorus .**Algebraic structure****Lattices**The possible equivalence relations on any set "X", when ordered by

set inclusion , form acomplete lattice , called**Con**"X" by convention. The canonical map**ker**: "X"∧"X" →**Con**"X", relates themonoid "X"^"X" of all functions on "X" and**Con**"X".**ker**issurjective but notinjective . 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 relation s. It is less known that transformation groups (some authors preferpermutation group s) and their orbits shed light on the mathematical structure of equivalence relations. Just asorder relation s are grounded inordered set s, sets closed under pairwisesupremum andinfimum , equivalence relations are grounded in partitioned sets, sets closed underbijection s preserving partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known aspermutation s.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": ∀"x" ∈ "A" ∀"g" ∈ "G" ("g"("x") ∈ ["x"] ). Then the following three connected theorems hold (Van Fraassen 1989: §10.3):

* ~ partitions "A" into equivalence classes. (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". (Wallace 1998: 202, Th. 6; Dummit and Foote 2004: 114, Prop. 2).In sum, given an equivalence relation ~ over "A", there exists atransformation 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 relation s. The arguments of the lattice theory operationsmeet andjoin are elements of some universe "A". Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a set ofbijections , "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" ~ "b" ↔ ("ab"^{−1}∈ "H"). The equivalence classes of ~—also called the orbits of the action of "H" on "G"—are the rightof "H" in "G". Interchanging "a" and "b" yields the left cosets.coset sFor 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. 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". The composition of any two elements of "G" exists, because the domain andcodomain of any element of "G" is "A". Moreover, the composition of bijections isbijective (Wallace 1998: 22, Th. 6);

*"Existence ofidentity element ". Theidentity function , "I"("x")="x", is an obvious element of "G";

*"Existence ofinverse function ". Everybijective function "g" has an inverse "g"^{−1}, such that "gg"^{−1}= "I";

*"Composition associates". "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 anautomorphism group ) because function composition preserves the partitioning of "A".**Relation with category theory and with groupoids**The composition of

morphism s central tocategory theory , denoted here by concatenation, generalizes the composition of functions central to transformation groups. The axioms ofcategory theory assert that the composition ofmorphism s associates, and that the left and rightidentity morphism s exist for any morphism.A morphism "f" can be said to have an inverse when "f" is an

isomorphism , i.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 ofgroupoid .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

congruence s 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 largercardinal number .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 anypreorder ;

* "Symmetric and transitive": The relation "R" on**N**, defined as "aRb" ↔ "ab" ≠ 0. Or anypartial 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 anydependency relation .Properties definable in

first-order logic that an equivalence relation may or may not possess include:

*The number ofequivalence classes is finite or infinite;

*The number of equivalence classes equals the (finite) natural number "n";

*All equivalence classes have infinitecardinality ;

*The number of elements in each equivalence class is the natural number "n".**Euclid anticipated equivalence**Euclid 's "The Elements" includes the following "Common Notion 1"::Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). The following theorem connects

Euclidean relation s and equivalence relations:**Theorem**. If a relation is Euclidean andreflexive , it is also symmetric and transitive."Proof":

*("aRc" ∧ "bRc") → "aRb" ["a/c"] = ("aRa" ∧ "bRa") → "aRb" ["reflexive"; erase**T**∧] = "bRa" → "aRb". Hence "R" issymmetric .

*("aRc" ∧ "bRc") → "aRb" ["symmetry"] = ("aRc" ∧ "cRb") → "aRb". Hence "R" istransitive . $square$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 indeductive system s.**ee 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 andSaunders Mac Lane , 1999 (1967). "Algebra", 3rd ed. Chelsea.

*Borceux, F. and Janelidze, G., 2001. "Galois theories", Cambridge University Press, ISBN 0521803098.

*Brown, R., 2006. [*http://www.bangor.ac.uk/r.brown/topgpds.html "Topology and Groupoids"*] , Booksurge LLC. ISBN 1419627228.

*Castellani, E., 2003, "Symmetry and equivalence" inKatherine Brading and E. Castellani (eds.), "Symmetries in Physics: Philosophical Reflections". Cambridge University 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.

*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 [*http://www.emis.de/journals/TAC/reprints/articles/7/tr7abs.html TAC Reprint, 2005*] .

*John Randolph Lucas, 1973. "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. "Laws and Symmetry". Oxford Univ. Press.

*Wallace, D. A. R., 1998. "Groups, Rings and Fields". Springer-Verlag.**External links***Bogomolny, A., " [

*http://www.cut-the-knot.org/blue/equi.shtml Equivalence Relationship*] "cut-the-knot . Accessed7 December 2007

*Wikimedia Foundation.
2010.*