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[[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.]]
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* If {{mvar|X}} is the set of all cars, and {{math|~}} is the [[equivalence relation]] "has the same color as", then one particular equivalence class would consist of all green cars, and {{math|''X''/~}} could be naturally identified with the set of all car colors.
* Let {{mvar|X}} be the set of all rectangles in a plane, and {{math|~}} the equivalence relation "has the same area as", then for each positive real number ''A'', there will be an equivalence class of all the rectangles that have area ''A''.<ref>{{harvnb|Avelsgaard|1989|loc=p. 127}}</ref>
* Consider the [[modular arithmetic|modulo]] 2 equivalence relation on the set of [[integer]]s, {{math|ℤ}}, such that {{math|''x'' ~ ''y''}} if and only if their difference {{math|''x'' − ''y''}} 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]}}, {{math|[9]}}, and {{math|[1]}} all represent the same element of {{math|ℤ/~}}.<ref name="Devlin 2004 loc=p. 123">{{harvnb|Devlin|2004|loc=p. 123}}</ref>
* Let {{mvar|X}} be the set of [[
* If {{mvar|X}} consists of all the lines in, say, the [[Euclidean plane]], and ''L'' ~ ''M'' means that ''L'' and ''M'' 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]].
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==Graphical representation==
[[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''}}, where the vertices are the elements of {{math|''X''}}, and two vertices {{mvar|s}} and {{mvar|t}} are joined if and only if {{math|''s'' ~ ''t''}}. Among these graphs are the graphs of equivalence relations; they are characterized as the graphs such that the [[connected component (graph theory)|connected components]] are [[clique (graph theory)|cliques]].<ref
==Invariants==
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==Notes==
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==References==
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==External links==
*{{
{{Authority control}}
[[Category:Algebra]]
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