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Equivalence relation

From Wikipedia, the free encyclopedia
(Redirected fromEquivalence relations)
Mathematical concept for comparing objects
This article is about the mathematical concept. For the patent doctrine, seeDoctrine of equivalents.
"Equivalency" redirects here. For other uses, seeEquivalence.
Transitive binary relations
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Total, Semiconnex Anti-
reflexive
Equivalence relationGreen tickY✗✗✗✗✗Green tickY✗✗
Preorder(Quasiorder)✗✗✗✗✗✗Green tickY✗✗
Partial order✗Green tickY✗✗✗✗Green tickY✗✗
Total preorder✗✗Green tickY✗✗✗Green tickY✗✗
Total order✗Green tickYGreen tickY✗✗✗Green tickY✗✗
Prewellordering✗✗Green tickYGreen tickY✗✗Green tickY✗✗
Well-quasi-ordering✗✗✗Green tickY✗✗Green tickY✗✗
Well-ordering✗Green tickYGreen tickYGreen tickY✗✗Green tickY✗✗
Lattice✗Green tickY✗✗Green tickYGreen tickYGreen tickY✗✗
Join-semilattice✗Green tickY✗✗Green tickY✗Green tickY✗✗
Meet-semilattice✗Green tickY✗✗✗Green tickYGreen tickY✗✗
Strict partial order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict weak order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict total order✗Green tickYGreen tickY✗✗✗✗Green tickYGreen tickY
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Definitions, for alla,b{\displaystyle a,b} andS≠∅:{\displaystyle S\neq \varnothing :}aRb⇒bRa{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}aRb and bRa⇒a=b{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}a≠b⇒aRb or bRa{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}minSexists{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}a∨bexists{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}a∧bexists{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}aRa{\displaystyle aRa}not aRa{\displaystyle {\text{not }}aRa}aRb⇒not bRa{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
Green tickY indicates that the column's property is always true for the row's term (at the very left), while✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated byGreen tickY in the "Symmetric" column and✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require thehomogeneous relationR{\displaystyle R} betransitive: for alla,b,c,{\displaystyle a,b,c,} ifaRb{\displaystyle aRb} andbRc{\displaystyle bRc} thenaRc.{\displaystyle aRc.}
A term's definition may require additional properties that are not listed in this table.

The52 equivalence relations on a 5-element set depicted as5×5{\displaystyle 5\times 5}logical matrices (colored fields, including those in light gray, stand for ones; white fields for zeros). The row and column indices of nonwhite cells are the related elements, while the different colors, other than light gray, indicate the equivalence classes (each light gray cell is its own equivalence class).

Inmathematics, anequivalence relation is abinary relation that isreflexive,symmetric, andtransitive. Theequipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any numbera{\displaystyle a} is equal to itself (reflexive). Ifa=b{\displaystyle a=b}, thenb=a{\displaystyle b=a} (symmetric). Ifa=b{\displaystyle a=b} andb=c{\displaystyle b=c}, thena=c{\displaystyle a=c} (transitive).

Each equivalence relation provides apartition of the underlying set into disjointequivalence classes. Two elements of the given set are equivalent to each otherif and only if they belong to the same equivalence class.

Notation

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Various notations are used in the literature to denote that two elementsa{\displaystyle a} andb{\displaystyle b} of a set are equivalent with respect to an equivalence relationR;{\displaystyle R;} the most common are "a∼b{\displaystyle a\sim b}" and "a ≡b", which are used whenR{\displaystyle R} is implicit, and variations of "a∼Rb{\displaystyle a\sim _{R}b}", "a ≡Rb", or "aR⁡b{\displaystyle {a\mathop {R} b}}" to specifyR{\displaystyle R} explicitly. Non-equivalence may be written "a ≁b" or "a≢b{\displaystyle a\not \equiv b}".

Definitions

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Abinary relation∼{\displaystyle \,\sim \,} on a setX{\displaystyle X} is said to be an equivalence relation, if it is reflexive, symmetric and transitive. That is, for alla,b,{\displaystyle a,b,} andc{\displaystyle c} inX:{\displaystyle X:}

X{\displaystyle X} together with the relation∼{\displaystyle \,\sim \,} is called asetoid. Theequivalence class ofa{\displaystyle a} under∼,{\displaystyle \,\sim ,} denoted[a],{\displaystyle [a],} is defined as[a]={x∈X:x∼a}.{\displaystyle [a]=\{x\in X:x\sim a\}.}[1][2]

Alternative definition using relational algebra

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Inrelational algebra, ifR⊆X×Y{\displaystyle R\subseteq X\times Y} andS⊆Y×Z{\displaystyle S\subseteq Y\times Z} are relations, then thecomposite relationSR⊆X×Z{\displaystyle SR\subseteq X\times Z} is defined so thatxSRz{\displaystyle x\,SR\,z} if and only if there is ay∈Y{\displaystyle y\in Y} such thatxRy{\displaystyle x\,R\,y} andySz{\displaystyle y\,S\,z}.[note 1] This definition is a generalisation of the definition offunctional composition. The defining properties of an equivalence relationR{\displaystyle R} on a setX{\displaystyle X} can then be reformulated as follows:

Examples

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Simple example

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On the setX={a,b,c}{\displaystyle X=\{a,b,c\}}, the relationR={(a,a),(b,b),(c,c),(b,c),(c,b)}{\displaystyle R=\{(a,a),(b,b),(c,c),(b,c),(c,b)\}} is an equivalence relation. The following sets are equivalence classes of this relation:[a]={a},    [b]=[c]={b,c}.{\displaystyle [a]=\{a\},~~~~[b]=[c]=\{b,c\}.}

The set of all equivalence classes forR{\displaystyle R} is{{a},{b,c}}.{\displaystyle \{\{a\},\{b,c\}\}.} This set is apartition of the setX{\displaystyle X}. It is also called thequotient set ofX{\displaystyle X} byR{\displaystyle R}.

Equivalence relations

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The following relations are all equivalence relations:

Relations that are not equivalences

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  • The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7.
  • The relation "has acommon factor greater than 1 with" betweennatural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1.
  • Theempty relationR (defined so thataRb is never true) on a setX isvacuously symmetric and transitive; however, it is not reflexive (unlessX 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 functionsf andg are approximately equal near some point if the limit off − g is 0 at that point, then this defines an equivalence relation.

Connections to other relations

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Well-definedness under an equivalence relation

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If∼{\displaystyle \,\sim \,} is an equivalence relation onX,{\displaystyle X,} andP(x){\displaystyle P(x)} is a property of elements ofX,{\displaystyle X,} such that wheneverx∼y,{\displaystyle x\sim y,}P(x){\displaystyle P(x)} is true ifP(y){\displaystyle P(y)} is true, then the propertyP{\displaystyle P} is said to bewell-defined or aclass invariant under the relation∼.{\displaystyle \,\sim .}

A frequent particular case occurs whenf{\displaystyle f} is a function fromX{\displaystyle X} to another setY;{\displaystyle Y;} ifx1∼x2{\displaystyle x_{1}\sim x_{2}} impliesf(x1)=f(x2){\displaystyle f\left(x_{1}\right)=f\left(x_{2}\right)} thenf{\displaystyle f} is said to be amorphism for∼,{\displaystyle \,\sim ,} aclass invariant under∼,{\displaystyle \,\sim ,} or simplyinvariant under∼.{\displaystyle \,\sim .} This occurs, e.g. in the character theory of finite groups. The latter case with the functionf{\displaystyle f} can be expressed by a commutative triangle. See alsoinvariant. Some authors use "compatible with∼{\displaystyle \,\sim }" or just "respects∼{\displaystyle \,\sim }" instead of "invariant under∼{\displaystyle \,\sim }".

More generally, a function may map equivalent arguments (under an equivalence relation∼A{\displaystyle \,\sim _{A}}) to equivalent values (under an equivalence relation∼B{\displaystyle \,\sim _{B}}). Such a function is known as a morphism from∼A{\displaystyle \,\sim _{A}} to∼B.{\displaystyle \,\sim _{B}.}

Related important definitions

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Leta,b∈X{\displaystyle a,b\in X}, and∼{\displaystyle \sim } be an equivalence relation. Some key definitions and terminology follow:

Equivalence class

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Main article:Equivalence class

A subsetY{\displaystyle Y} ofX{\displaystyle X} such thata∼b{\displaystyle a\sim b} holds for alla{\displaystyle a} andb{\displaystyle b} inY{\displaystyle Y}, and never fora{\displaystyle a} inY{\displaystyle Y} andb{\displaystyle b} outsideY{\displaystyle Y}, is called anequivalence class ofX{\displaystyle X} by∼{\displaystyle \sim }. Let[a]:={x∈X:a∼x}{\displaystyle [a]:=\{x\in X:a\sim x\}} denote the equivalence class to whicha{\displaystyle a} belongs. All elements ofX{\displaystyle X} equivalent to each other are also elements of the same equivalence class.

Quotient set

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Main article:Quotient set

The set of all equivalence classes ofX{\displaystyle X} by∼,{\displaystyle \sim ,} denotedX/∼:={[x]:x∈X},{\displaystyle X/{\mathord {\sim }}:=\{[x]:x\in X\},} is thequotient set ofX{\displaystyle X} by∼.{\displaystyle \sim .} IfX{\displaystyle X} is atopological space, there is a natural way of transformingX/∼{\displaystyle X/\sim } into a topological space; seeQuotient space for the details.[undue weight? –discuss]

Projection

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Main article:Projection (relational algebra)

Theprojection of∼{\displaystyle \,\sim \,} is the functionπ:X→X/∼{\displaystyle \pi :X\to X/{\mathord {\sim }}} defined byπ(x)=[x]{\displaystyle \pi (x)=[x]} which maps elements ofX{\displaystyle X} into their respective equivalence classes by∼.{\displaystyle \,\sim .}

Theorem onprojections:[5] Let the functionf:X→B{\displaystyle f:X\to B} be such that ifa∼b{\displaystyle a\sim b} thenf(a)=f(b).{\displaystyle f(a)=f(b).} Then there is a unique functiong:X/∼→B{\displaystyle g:X/\sim \to B} such thatf=gπ.{\displaystyle f=g\pi .} Iff{\displaystyle f} is asurjection anda∼b if and only if f(a)=f(b),{\displaystyle a\sim b{\text{ if and only if }}f(a)=f(b),} theng{\displaystyle g} is abijection.

Equivalence kernel

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Theequivalence kernel of a functionf{\displaystyle f} is the equivalence relation ~ defined byx∼y if and only if f(x)=f(y).{\displaystyle x\sim y{\text{ if and only if }}f(x)=f(y).} The equivalence kernel of aninjection is theidentity relation.

Partition

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Main article:Partition of a set

Apartition ofX is a setP of nonempty subsets ofX, such that every element ofX is an element of a single element ofP. Each element ofP is acell of the partition. Moreover, the elements ofP arepairwise disjoint and theirunion isX.

Counting partitions

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LetX be a finite set withn elements. Since every equivalence relation overX corresponds to a partition ofX, and vice versa, the number of equivalence relations onX equals the number of distinct partitions ofX, which is thenthBell numberBn:

Bn=1e∑k=0∞knk!{\displaystyle B_{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}\quad } (Dobinski's formula).

Fundamental theorem of equivalence relations

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A key result links equivalence relations and partitions:[6][7][8]

  • An equivalence relation ~ on a setX partitionsX.
  • Conversely, corresponding to any partition ofX, there exists an equivalence relation ~ onX.

In both cases, the cells of the partition ofX are the equivalence classes ofX by ~. Since each element ofX belongs to a unique cell of any partition ofX, and since each cell of the partition is identical to an equivalence class ofX by ~, each element ofX belongs to a unique equivalence class ofX by ~. Thus there is a naturalbijection between the set of all equivalence relations onX and the set of all partitions ofX.

Comparing equivalence relations

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See also:Partition of a set § Refinement of partitions

If∼{\displaystyle \sim } and≈{\displaystyle \approx } are two equivalence relations on the same setS{\displaystyle S}, anda∼b{\displaystyle a\sim b} impliesa≈b{\displaystyle a\approx b} for alla,b∈S,{\displaystyle a,b\in S,} then≈{\displaystyle \approx } is said to be acoarser relation than∼{\displaystyle \sim }, and∼{\displaystyle \sim } is afiner relation than≈{\displaystyle \approx }. Equivalently,

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 "∼{\displaystyle \sim } is finer than≈{\displaystyle \approx }" on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection ageometric lattice.[9]

Generating equivalence relations

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a∼b{\displaystyle a\sim b} if there exists anatural numbern{\displaystyle n} and elementsx0,…,xn∈X{\displaystyle x_{0},\ldots ,x_{n}\in X} such thata=x0{\displaystyle a=x_{0}},b=xn{\displaystyle b=x_{n}}, andxi−1Rxi{\displaystyle x_{i-1}\mathrel {R} x_{i}} orxiRxi−1{\displaystyle x_{i}\mathrel {R} x_{i-1}}, fori=1,…,n.{\displaystyle i=1,\ldots ,n.}
The equivalence relation generated in this manner can be trivial. For instance, the equivalence relation generated by anytotal order onX has exactly one equivalence class,X itself.

Algebraic structure

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Much of mathematics is grounded in the study of equivalences, andorder relations.Lattice theory captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily ongroup theory and, to a lesser extent, on the theory of lattices,categories, andgroupoids.

Group theory

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Just asorder relations are grounded inordered sets, sets closed under pairwisesupremum andinfimum, equivalence relations are grounded inpartitioned sets, which are sets closed underbijections that preserve partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known aspermutations. Hencepermutation groups (also known astransformation groups) and the related notion oforbit shed light on the mathematical structure of equivalence relations.

Let '~' denote an equivalence relation over some nonempty setA, called theuniverse or underlying set. LetG denote the set of bijective functions overA that preserve the partition structure ofA, meaning that for allx∈A{\displaystyle x\in A} andg∈G,g(x)∈[x].{\displaystyle g\in G,g(x)\in [x].} Then the following three connected theorems hold:[11]

  • ~ partitionsA into equivalence classes. (This is theFundamental Theorem of Equivalence Relations, mentioned above);
  • Given a partition ofA,G is a transformation group under composition, whose orbits are thecells of the partition;[15]
  • Given a transformation groupG overA, there exists an equivalence relation ~ overA, whose equivalence classes are the orbits ofG.[16][17]

In sum, given an equivalence relation ~ overA, there exists atransformation groupG overA whose orbits are the equivalence classes ofA under ~.

This transformation group characterisation of equivalence relations differs fundamentally from the waylattices characterize order relations. The arguments of the lattice theory operationsmeet andjoin are elements of some universeA. Meanwhile, the arguments of the transformation group operationscomposition andinverse are elements of a set ofbijections,A →A.

Moving to groups in general, letH be asubgroup of somegroupG. Let ~ be an equivalence relation onG, such thata∼b if and only if ab−1∈H.{\displaystyle a\sim b{\text{ if and only if }}ab^{-1}\in H.} The equivalence classes of ~—also called the orbits of theaction ofH onG—are the rightcosets ofH inG. Interchanginga andb yields the left cosets.

Related thinking can be found in Rosen (2008: chpt. 10).

Categories and groupoids

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LetG be a set and let "~" denote an equivalence relation overG. Then we can form agroupoid representing this equivalence relation as follows. The objects are the elements ofG, and for any two elementsx andy ofG, there exists a unique morphism fromx toyif and only ifx∼y.{\displaystyle 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 afree groupoid on adirected 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 calledcongruences, are important. This leads to the notion of an internal groupoid in acategory.[18]

Lattices

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The equivalence relations on any setX, when ordered byset inclusion, form acomplete lattice, calledConX by convention. The canonicalmapker :X^X →ConX, relates themonoidX^X of allfunctions onX andConX.ker issurjective but notinjective. Less formally, the equivalence relationker onX, takes each functionf :X →X to itskernelkerf. Likewise,ker(ker) is an equivalence relation onX^X.

Equivalence relations and mathematical logic

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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 ofmodel 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 ≤ onN. Or anypreorder;
  • Symmetric and transitive: The relationR onN, defined asaRb ↔ab ≠ 0. Or anypartial equivalence relation;
  • Reflexive and symmetric: The relationR onZ, defined asaRb ↔ "a −b is divisible by at least one of 2 or 3." Or anydependency relation.

Properties definable infirst-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 numbern;
  • All equivalence classes have infinitecardinality;
  • The number of elements in each equivalence class is the natural numbern.

See also

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Notes

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  1. ^Sometimes the compositionSR⊆X×Z{\displaystyle SR\subseteq X\times Z} is instead written asR;S{\displaystyle R;S}, or asRS{\displaystyle RS}; in both cases,R{\displaystyle R} is the first relation that is applied. See the article onComposition of relations for more information.
  1. ^If: Givena,{\displaystyle a,} leta∼b{\displaystyle a\sim b} hold using totality, thenb∼a{\displaystyle b\sim a} by symmetry, hencea∼a{\displaystyle a\sim a} by transitivity. —Only if: Givena,{\displaystyle a,} chooseb=a,{\displaystyle b=a,} thena∼b{\displaystyle a\sim b} by reflexivity.
  1. ^Weisstein, Eric W."Equivalence Class".mathworld.wolfram.com. Retrieved2020-08-30.
  2. ^abc"7.3: Equivalence Classes".Mathematics LibreTexts. 2017-09-20. Retrieved2020-08-30.
  3. ^Halmos, Paul Richard (1914).Naive Set Theory. New York: Springer. p. 41.ISBN 978-0-387-90104-6.{{cite book}}:ISBN / Date incompatibility (help)
  4. ^Lena L. Severance (1930)The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis, link from HathiTrust
  5. ^Garrett Birkhoff andSaunders Mac Lane, 1999 (1967).Algebra, 3rd ed. p. 35, Th. 19. Chelsea.
  6. ^Wallace, D. A. R., 1998.Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.
  7. ^Dummit, D. S., and Foote, R. M., 2004.Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.
  8. ^Karel Hrbacek &Thomas Jech (1999)Introduction to Set Theory, 3rd edition, pages 29–32,Marcel Dekker
  9. ^Birkhoff, Garrett (1995),Lattice Theory, Colloquium Publications, vol. 25 (3rd ed.), American Mathematical Society,ISBN 9780821810255. Sect. IV.9, Theorem 12, page 95
  10. ^Garrett Birkhoff andSaunders Mac Lane, 1999 (1967).Algebra, 3rd ed. p. 33, Th. 18. Chelsea.
  11. ^Rosen (2008), pp. 243–45. Less clear is §10.3 ofBas van Fraassen, 1989.Laws and Symmetry. Oxford Univ. Press.
  12. ^Bas van Fraassen, 1989.Laws and Symmetry. Oxford Univ. Press: 246.
  13. ^Wallace, D. A. R., 1998.Groups, Rings and Fields. Springer-Verlag: 22, Th. 6.
  14. ^Wallace, D. A. R., 1998.Groups, Rings and Fields. Springer-Verlag: 24, Th. 7.
  15. ^Proof.[12] Letfunction composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning thatx∈A{\displaystyle x\in A} andg∈G,[g(x)]=[x],{\displaystyle g\in G,[g(x)]=[x],} becauseG satisfies the following four conditions:Letf andg be any two elements ofG. By virtue of the definition ofG, [g(f(x))] = [f(x)] and [f(x)] = [x], so that [g(f(x))] = [x]. HenceG is also a transformation group (and anautomorphism group) because function composition preserves the partitioning ofA.◼{\displaystyle A.\blacksquare }
  16. ^Wallace, D. A. R., 1998.Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.
  17. ^Dummit, D. S., and Foote, R. M., 2004.Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.
  18. ^Borceux, F. and Janelidze, G., 2001.Galois theories, Cambridge University Press,ISBN 0-521-80309-8

References

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  • Brown, Ronald, 2006.Topology and Groupoids. Booksurge LLC.ISBN 1-4196-2722-8.
  • Castellani, E., 2003, "Symmetry and equivalence" inBrading, Katherine, and E. Castellani, eds.,Symmetries in Physics: Philosophical Reflections. Cambridge Univ. Press: 422–433.
  • Robert Dilworth and Crawley, Peter, 1973.Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusses how equivalence relations arise inlattice theory.
  • Higgins, P.J., 1971.Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint.
  • John Randolph Lucas, 1973.A Treatise on Time and Space. London: Methuen. Section 31.
  • Rosen, Joseph (2008)Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag. Mostly chapters. 9,10.
  • Raymond Wilder (1965)Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axioms defining equivalence, pp 48–50,John Wiley & Sons.

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