Editing Equivalence class

{{Short description|Mathematical concept}}
{{About|equivalency in mathematics|equivalency in music|equivalence class (music)}}
{{Redirect|Quotient map|Quotient map in topology|Quotient map (topology)}}

[[File:Congruent non-congruent triangles.svg|thumb|370px|[[Congruence (geometry)|Congruence]] is an example of an equivalence relation. The leftmost two triangles are congruent, while the third and fourth triangles are not congruent to any other triangle shown here. Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class.]]

In [[mathematics]], when the elements of some [[Set (mathematics)|set]] <math>S</math> have a notion of equivalence (formalized as an [[equivalence relation]]), then one may naturally split the set <math>S</math> into '''equivalence classes'''. These equivalence classes are constructed so that elements <math>a</math> and <math>b</math> belong to the same '''equivalence class''' [[if, and only if]], they are equivalent.

Formally, given a set <math>S</math> and an equivalence relation <math>\sim</math> on <math>S,</math> the {{em|equivalence class}} of an element <math>a</math> in <math>S</math> is denoted <math>[a]</math> or, equivalently, <math>[a]_{\sim}</math> to emphasize its equivalence relation <math>\sim</math>, and is defined as the set of all elements in <math>S</math> with which <math>a</math> is <math>\sim</math>-related. The definition of equivalence relations implies that the equivalence classes form a [[Partition of a set|partition]] of <math>S,
</math>  meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the '''quotient set''' or the '''quotient space''' of <math>S</math> by <math>\sim,</math> and is denoted by <math>S /{\sim}.</math>

When the set <math>S</math> has some structure (such as a [[group (mathematics)|group operation]] or a [[topological space|topology]]) and the equivalence relation <math>\sim,</math> is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include [[quotient space (linear algebra)|quotient spaces in linear algebra]], [[quotient space (topology)|quotient spaces in topology]], [[quotient group]]s, [[homogeneous space]]s, [[quotient ring]]s, [[quotient monoid]]s, and [[quotient category|quotient categories]].

==Definition and notation==

An [[equivalence relation]] on a set <math>X</math> is a [[binary relation]] <math>\sim</math> on <math>X</math> satisfying the three properties:{{sfn|Devlin|2004|p=122}}
* <math>a \sim a</math> for all <math>a \in X</math> ([[Reflexive relation|reflexivity]]),
* <math>a \sim b</math> implies <math>b \sim a</math> for all <math>a, b \in X</math> ([[Symmetric relation|symmetry]]),
* if <math>a \sim b</math> and <math>b \sim c</math> then <math>a \sim c</math> for all <math>a, b, c \in X</math> ([[Transitive relation|transitivity]]).

{{anchor|Notation and formal definition}}The equivalence class of an element <math>a</math> is defined as{{sfn|Devlin|2004|p=123}}
:<math>[a] = \{ x \in X : a \sim x \}.</math>
The word "class" in the term "equivalence class" may generally be considered as a synonym of "[[set (mathematics)|set]]", although some equivalence classes are not sets but [[proper class]]es. For example, "being [[group isomorphism|isomorphic]]" is an equivalence relation on [[group (mathematics)|groups]], and the equivalence classes, called [[isomorphism class]]es, are not sets.

The set of all equivalence classes in <math>X</math> with respect to an equivalence relation <math>R</math> is denoted as <math>X / R,</math> and is called <math>X</math> [[Modulo (mathematics)|modulo]] <math>R</math> (or the '''{{vanchor|quotient set}}''' of <math>X</math> by <math>R</math>).<ref>{{harvnb|Wolf|1998|loc=p. 178}}</ref> The [[surjective map]] <math>x \mapsto [x]</math> from <math>X</math> onto <math>X / R,</math> which maps each element to its equivalence class, is called the '''{{vanchor|canonical surjection}}''', or the '''canonical projection'''.

{{anchor|representative}}Every element of an equivalence class characterizes the class, and may be used to ''represent'' it. When such an element is chosen, it is called a '''representative''' of the class. The choice of a representative in each class defines an [[injective function|injection]] from <math>X / R</math> to {{mvar|X}}. Since its [[function composition|composition]] with the canonical surjection is the identity of <math>X / R,</math> such an injection is called a [[Section (category theory)|section]], when using the terminology of [[category theory]]. 

Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are called {{em|[[Canonical form|canonical]] representatives}}. For example, in [[modular arithmetic]], for every [[integer]] {{mvar|m}} greater than {{math|1}}, the [[congruence modulo m|congruence modulo {{mvar|m}}]] is an equivalence relation on the integers, for which two integers {{mvar|a}} and {{mvar|b}} are equivalent—in this case, one says ''congruent''—if {{mvar|m}} divides <math>a-b;</math> this is denoted <math DISPLAY=inline>a\equiv b \pmod m.</math> Each class contains a unique non-negative integer smaller than <math>m,</math> and these integers are the canonical representatives. 

The use of representatives for representing classes allows avoiding considering explicitly classes as sets. In this case, the canonical surjection that maps an element to its class is replaced by the function that maps an element to the representative of its class. In the preceding example, this function is denoted <math>a \bmod m,</math> and produces the remainder of the [[Euclidean division]] of {{mvar|a}} by {{mvar|m}}.

==Properties==

For a set <math>X</math> with an [[equivalence relation]] <math>\sim</math>, every element <math>x</math> of <math>X</math> is a member of the equivalence class <math>[x]</math> by [[Reflexive relation|reflexivity]] (<math>a \sim a</math> for all <math>a \in X</math>). Every two equivalence classes <math>[x]</math> and <math>[y]</math> are either equal if <math>x \sim y</math>, or [[disjoint sets|disjoint]] otherwise.  Therefore, the set of all equivalence classes of <math>X</math> forms a [[partition of a set|partition]] of <math>X</math>: every element <math>x</math> of <math>X</math> belongs to one and only one equivalence class.<ref>{{harvnb|Maddox|2002|loc=p. 74, Thm. 2.5.15}}</ref>

Conversely, for a set <math>X</math>, every partition comes from an equivalence relation in this way, and different relations give different partitions. Thus <math>x \sim y</math> if and only if <math>x</math> and <math>y</math> belong to the same set of the partition.<ref>{{harvnb|Avelsgaard|1989|loc=p. 132, Thm. 3.16}}</ref>

It follows from the properties in the previous section that if <math>\,\sim\,</math> is an equivalence relation on a set <math>X,</math> and <math>x</math> and <math>y</math> are two elements of <math>X,</math> the following statements are equivalent:
* <math>x \sim y</math>,
* <math>[x] = [y]</math>, and 
* <math>[x] \cap [y] \ne \emptyset.</math>

==Examples==
* Let <math>X</math> be the set of all rectangles in a plane, and <math>\,\sim\,</math> the equivalence relation "has the same area as", then for each positive real number <math>A,</math> there will be an equivalence class of all the rectangles that have area <math>A.</math><ref>{{harvnb|Avelsgaard|1989|loc=p. 127}}</ref>
* Consider the [[Modular arithmetic|modulo]] 2 equivalence relation on the set of [[integer]]s, <math>\Z,</math> such that <math>x \sim y</math> if and only if their difference <math>x - y</math> is an [[even number]]. This relation gives rise to exactly two equivalence classes: one class consists of all even numbers, and the other class consists of all odd numbers. Using square brackets around one member of the class to denote an equivalence class under this relation, <math>[7], [9],</math> and <math>[1]</math> all represent the same element of <math>\Z /{\sim}.</math>{{sfn|Devlin|2004|p=123}}
* Let <math>X</math> be the set of [[ordered pair]]s of integers <math>(a, b)</math> with non-zero <math>b,</math> and define an equivalence relation <math>\,\sim\,</math> on <math>X</math> such that <math>(a, b) \sim (c, d)</math> if and only if <math>a d = b c,</math> then the equivalence class of the pair <math>(a, b)</math> can be identified with the [[rational number]] <math>a / b,</math> and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.<ref>{{harvnb|Maddox|2002|loc=pp. 77–78}}</ref> The same construction can be generalized to the [[field of fractions]] of any [[integral domain]].
* If <math>X</math> consists of all the lines in, say, the [[Euclidean plane]], and <math>L \sim M</math> means that <math>L</math> and <math>M</math> are [[parallel lines]], then the set of lines that are parallel to each other form an equivalence class, as long as a [[parallel (geometry)#Reflexive variant|line is considered parallel to itself]]. In this situation, each equivalence class determines a [[point at infinity]].

==Graphical representation==
{{main|Cluster graph}}
[[File:Equivalentie.svg|thumb|160px|Graph of an example equivalence with 7 classes]]
An [[undirected graph]] may be associated to any [[symmetric relation]] on a set <math>X,</math> where the vertices are the elements of <math>X,</math> and two vertices <math>s</math> and <math>t</math> are joined if and only if <math>s \sim t.</math> Among these graphs are the graphs of equivalence relations. These graphs, called [[cluster graph]]s, are characterized as the graphs such that the [[Connected component (graph theory)|connected components]] are [[Clique (graph theory)|cliques]].{{sfn|Devlin|2004|p=123}}

==Invariants==

If <math>\,\sim\,</math> is an equivalence relation on <math>X,</math> and <math>P(x)</math> is a property of elements of <math>X</math> such that whenever <math>x \sim y,</math> <math>P(x)</math> is true if <math>P(y)</math> is true, then the property <math>P</math> is said to be an [[Invariant (mathematics)|invariant]] of <math>\,\sim\,,</math> or [[well-defined]] under the relation <math>\,\sim.</math>

A frequent particular case occurs when <math>f</math> is a function from <math>X</math> to another set <math>Y</math>; if  <math>f\left(x_1\right) = f\left(x_2\right)</math> whenever <math>x_1 \sim x_2,</math> then <math>f</math> is said to be {{em|class invariant under}} <math>\,\sim\,,</math> or simply {{em|invariant under}} <math>\,\sim.</math> This occurs, for example, in the [[character theory]] of finite groups. Some authors use "compatible with <math>\,\sim\,</math>" or just "respects <math>\,\sim\,</math>" instead of "invariant under <math>\,\sim\,</math>".

Any [[Function (mathematics)|function]] <math>f : X \to Y</math> is ''class invariant under'' <math>\,\sim\,,</math> according to which <math>x_1 \sim x_2</math> if and only if <math>f\left(x_1\right) = f\left(x_2\right).</math> The equivalence class of <math>x</math> is the set of all elements in <math>X</math> which get mapped to <math>f(x),</math> that is, the class <math>[x]</math> is the [[inverse image]] of <math>f(x).</math> This equivalence relation is known as the [[Kernel of a function|kernel]] of <math>f.</math>

More generally, a function may map equivalent arguments (under an equivalence relation <math>\sim_X</math> on <math>X</math>) to equivalent values (under an equivalence relation <math>\sim_Y</math> on <math>Y</math>). Such a function is a [[morphism]] of sets equipped with an equivalence relation.

==Quotient space in topology==

In [[topology]], a [[Quotient space (topology)|quotient space]] is a [[topological space]] formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes.

In [[abstract algebra]], [[congruence relation]]s on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a [[Quotient (universal algebra)|quotient algebra]]. In [[linear algebra]], a [[Quotient space (linear algebra)|quotient space]] is a vector space formed by taking a [[quotient group]], where the quotient homomorphism is a [[linear map]]. By extension, in abstract algebra, the term quotient space may be used for [[quotient module]]s, [[quotient ring]]s, [[quotient group]]s, or any quotient algebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a group action.

The orbits of a [[Group action (mathematics)|group action]] on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right [[coset]]s of a subgroup of a group, which arise from the action of the subgroup on the group by left translations, or respectively the left cosets as orbits under right translation.

A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously.

Although the term can be used for any equivalence relation's set of equivalence classes, possibly with further structure, the intent of using the term is generally to compare that type of equivalence relation on a set <math>X,</math> either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on <math>X,</math> or to the orbits of a group action. Both the sense of a structure preserved by an equivalence relation, and the study of [[Invariant (mathematics)|invariants]] under group actions, lead to the definition of [[#Invariants|invariants]] of equivalence relations given above.

==See also==

* [[Equivalence partitioning]], a method for devising test sets in [[software testing]] based on dividing the possible program inputs into equivalence classes according to the behavior of the program on those inputs
* [[Homogeneous space]], the quotient space of [[Lie group]]s
* {{annotated link|Partial equivalence relation}}
* {{annotated link|Quotient by an equivalence relation}}
* {{annotated link|Setoid}}
* {{annotated link|Transversal (combinatorics)}}

==Notes==

{{reflist|30em}}

==References==

* {{citation|first=Carol|last=Avelsgaard|title=Foundations for Advanced Mathematics|publisher=Scott Foresman|year=1989|isbn=0-673-38152-8}}
* {{citation|last=Devlin|first=Keith|title=Sets, Functions, and Logic: An Introduction to Abstract Mathematics|year=2004|publisher=Chapman & Hall/ CRC Press|edition=3rd|isbn=978-1-58488-449-1}}
* {{citation|last=Maddox|first=Randall B.|title=Mathematical Thinking and Writing|year=2002|publisher=Harcourt/ Academic Press|isbn=0-12-464976-9}}
* {{cite book | last=Stein | first=Elias M. | last2=Shakarchi | first2=Rami | title=Functional Analysis: Introduction to Further Topics in Analysis | publisher=Princeton University Press | date=2012 | isbn=978-1-4008-4055-7 | doi=10.1515/9781400840557}}
* {{citation|last=Wolf|first=Robert S.|title=Proof, Logic and Conjecture: A Mathematician's Toolbox|year=1998|publisher=Freeman|isbn=978-0-7167-3050-7}}

==Further reading==
* {{citation|last=Sundstrom|title=Mathematical Reasoning: Writing and Proof|year=2003|publisher=Prentice-Hall}}
* {{citation|last1=Smith|last2=Eggen|last3=St.Andre|title=A Transition to Advanced Mathematics |edition=6th |year=2006|publisher=Thomson (Brooks/Cole)}}
* {{citation|last=Schumacher|first=Carol|author-link= Carol Schumacher |title=Chapter Zero: Fundamental Notions of Abstract Mathematics|year=1996|publisher=Addison-Wesley|isbn=0-201-82653-4}}
* {{citation|last=O'Leary|title=The Structure of Proof: With Logic and Set Theory|year=2003|publisher=Prentice-Hall}}
* {{citation|last=Lay|title=Analysis with an introduction to proof|year=2001|publisher=Prentice Hall}}
* {{citation|last=Morash|first=Ronald P.|title=Bridge to Abstract Mathematics|publisher=Random House|year=1987|isbn=0-394-35429-X}}
* {{citation|last1=Gilbert|last2=Vanstone|title=An Introduction to Mathematical Thinking|year=2005|publisher=Pearson Prentice-Hall}}
* {{citation|last1=Fletcher|last2=Patty|title=Foundations of Higher Mathematics|publisher=PWS-Kent}}
* {{citation|last1=Iglewicz|last2=Stoyle|title=An Introduction to Mathematical Reasoning|publisher=MacMillan}}
* {{citation|last1=D'Angelo|last2=West|title=Mathematical Thinking: Problem Solving and Proofs|year=2000|publisher=Prentice Hall}}
* {{citation|last=Cupillari|author-link= Antonella Cupillari |title=The Nuts and Bolts of Proofs|publisher=Wadsworth}}
* {{citation|last=Bond|title=Introduction to Abstract Mathematics|publisher=Brooks/Cole}}
* {{citation|last1=Barnier|last2=Feldman|title=Introduction to Advanced Mathematics|year=2000|publisher=Prentice Hall}}
* {{citation|last=Ash|title=A Primer of Abstract Mathematics|publisher=MAA}}

==External links==

*{{Commons category-inline|Equivalence classes}}

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