Editing Equivalence class (section)

==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]].