Editing Equivalence class (section)

==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}}.