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| {{Unreferenced|date=December 2009}}
| | External Auditor Giacobbe from Bloomfield, spends time with hobbies for instance butterflies, Growtopia Hack and operating on cars. Recently had a family voyage to The Banks of the Danube.<br><br>Feel free to visit my website ... [http://gamingcheats.foliodrop.com/pages/growtopia-hack-2014 Growtopia cheats] |
| :''This article is about equivalency in [[mathematics]]; for equivalency in [[music]] see [[equivalence class (music)]].''
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| In [[mathematics]], given a [[Set (mathematics)|set]] {{mvar|X}} and an [[equivalence relation]] {{math|~}} on {{mvar|X}}, the '''equivalence class''' of an element {{mvar|a}} in {{mvar|X}} is the [[subset]] of all elements in {{mvar|X}} which are equivalent to {{mvar|a}}. It follows from the definition of the equivalence relations that the equivalence classes form a [[Partition of a set|partition of {{mvar|X}}]]. The '''quotient set''' of {{mvar|X}} by {{math|~}} is the set of the equivalence classes. It is denoted as {{math|''X'' / ~}}.
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| When {{mvar|X}} is equipped with some structure, and the equivalence relation is defined in relation with this structure, the quotient set often inherits some related structure. Examples include [[quotient space (linear algebra)|quotient spaces in linear algebra]], [[quotient space (topology)|quotient spaces in topology]], [[quotient group]]s, [[homogenous space]]s, [[quotient ring]]s, [[quotient monoid]]s, and the [[quotient category]].
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| ==Notation and formal definition==
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| An equivalence relation is a [[binary relation]] {{math|~}} satisfying three properties:
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| *For every element {{mvar|a}} in {{mvar|X}}, {{math|''a'' ~ ''a''}} ([[Reflexive relation|reflexivity]]),
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| *For every two elements {{mvar|a}} and {{mvar|b}} in {{mvar|X}}, if {{math|''a'' ~ ''b''}}, then {{math|''b'' ~ ''a''}} ([[Symmetric relation|symmetry]])
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| *For every three elements {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} in {{mvar|X}}, if {{math|''a'' ~ ''b''}} and {{math|''b'' ~ ''c''}}, then {{math|''a'' ~ ''c''}} ([[Transitive relation|transitivity]]).
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| The equivalence class of an element {{mvar|a}} is denoted {{math|[''a'']}} and may be defined as the set
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| :<math>[a] = \{ x \in X \mid a \sim x \}</math>
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| of elements that are related to {{mvar|a}} by {{math|~}}. The alternative notation {{math|[''a'']<sub>''R''</sub>}} can be used to denote the equivalence class of the element {{mvar|a}} specifically with respect to the equivalence relation {{mvar|R}}. This is said to be the {{mvar|R}}-equivalence class of {{mvar|a}}.
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| The set of all equivalence classes in {{mvar|X}} given an equivalence relation {{math|~}} is denoted as {{math|''X''/~}} and called the '''quotient set''' of {{mvar|X}} by {{math|~}}. The [[surjective map]] <math> x\mapsto [x]</math> from {{mvar|X}} onto {{math|''X''/~}}, which maps each element to its equivalence class is called the '''canonical surjection''' or the '''canonical projection map'''.
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| When an element is chosen (often implicitly) in each equivalence class, this defines an [[injective map]] called ''[[section (category theory)|section]]''. If this section is denoted by {{math|''s''}}, one has {{math|1= [''s''(''c'')] = ''c''}} for every equivalence class {{math|''c''}}. The element {{math|''s''(''c'')}} is called a '''representative''' of {{math|''c''}}. Every element of a class may be chosen as a representative of the class, by choosing the section accordingly.
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| Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are called ''[[Canonical form|canonical]] representatives''. For example, in [[modular arithmetic]], one consider the equivalence relation on the integers defined by {{math|''a'' ~ ''b''}} if {{math|''a'' - ''b''}} if multiple of a given integer {{math| ''n''}}, called ''modulus''. Each class contains a unique non negative integer lower than {{math|''n''}}, and these integers are the canonical representatives. A witness that the class and its representative are more or less identified is the fact that the notation {{math| ''a'' mod ''n''}} may denote either the class or its canonical representative (which is the [[remainder]] of the [[Euclidean division]] of {{math| ''a''}} by {{math|''n''}}).
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| ==Analogy with division==
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| This operation can be thought of as the act of dividing the input set by the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division. One way in which the quotient set resembles division is that if {{mvar|X}} is finite and the equivalence classes are all [[equinumerous]], then the number of equivalence classes in {{math|''X''/~}} can be calculated by dividing the number of elements in {{mvar|X}} by the number of elements in each equivalence class. The quotient set {{math|''X''/~}} may be thought of as the set {{mvar|X}} with all the equivalent points identified.
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| ==Examples==
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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 consists of all green cars. {{math|''X''/~}} could be naturally identified with the set of all car colors ([[cardinality]] of {{math|''X''/~}} would be the number of all car colors)
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| * Consider the [[modular arithmetic|modulo]] 2 equivalence relation on the set {{math|'''Z'''}} of [[integer]]s: {{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 consisting of all even numbers, and the other consisting of all odd numbers. Under this relation {{math|[7]}}, {{math|[9]}}, and {{math|[1]}} all represent the same element of {{math|'''Z'''/~}}.
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| * Let {{mvar|X}} be the set of ordered pairs of integers {{math|(''a'',''b'')}} with {{mvar|b}} not zero, and define an equivalence relation {{math|~}} on {{mvar|X}} according to which {{math|(''a'',''b'') ~ (''c'',''d'')}} if and only if {{math|1=''ad'' = ''bc''}}. Then the equivalence class of the pair {{math|(''a'',''b'')}} can be identified with the [[rational number]] {{math|''a''/''b''}}, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers. The same construction can be generalized to the [[field of fractions]] of any [[integral domain]].
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| ==Properties==
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| Every element {{mvar|x}} of {{mvar|X}} is a member of the equivalence class {{math|[''x'']}}. Every two equivalence classes {{math|[''x'']}} and {{math|[''y'']}} are either equal or [[disjoint sets|disjoint]]. Therefore, the set of all equivalence classes of {{mvar|X}} forms a [[partition of a set|partition]] of {{mvar|X}}: every element of {{mvar|X}} belongs to one and only one equivalence class. Conversely every partition of {{mvar|X}} comes from an equivalence relation in this way, according to which {{math|''x'' ~ ''y''}} if and only if {{mvar|x}} and {{mvar|y}} belong to the same set of the partition.
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| It follows from the properties of an equivalence relation that
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| :: {{math|''x'' ~ ''y''}} if and only if {{math|1=[''x''] = [''y'']}}.
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| In other words, if {{math|~}} is an equivalence relation on a set {{math|X}}, and {{mvar|x}} and {{mvar|y}} are two elements of {{mvar|X}}, then these statements are equivalent:
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| * <math>x \sim y</math>
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| * <math>[x] = [y]</math>
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| * <math>[x] \cap [y] \ne \emptyset</math>.
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| ==Invariants==
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| If {{math|~}} is an equivalence relation on {{mvar|X}}, and {{math|''P''(''x'')}} is a property of elements of {{mvar|X}} such that whenever {{math|''x'' ~ ''y''}}, {{math|''P''(''x'')}} is true if {{math|''P''(''y'')}} is true, then the property {{mvar|P}} is said to be an [[Invariant (mathematics)|invariant]] of {{math|~}}, or [[well-defined]] under the relation {{math|~}}.
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| A frequent particular case occurs when {{mvar|f}} is a function from {{mvar|X}} to another set {{mvar|Y}}; if {{math|1=''f''(''x''<sub>1</sub>) = ''f''(''x''<sub>2</sub>)}} whenever {{math|''x''<sub>1</sub> ~ ''x''<sub>2</sub>}}, then {{mvar|f}} is said to be a ''[[morphism]]'' for {{math|~}}, a ''class invariant under'' {{math|~}}, or simply ''invariant under'' {{math|~}}. This occurs, e.g. in the character theory of finite groups. Some authors use "compatible with {{math|~}}" or just "respects {{math|~}}" instead of "invariant under {{math|~}}".
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| Any [[function (mathematics)|function]] {{math|''f'' : ''X'' → ''Y''}} itself defines an equivalence relation on {{mvar|X}} according to which {{math|''x''<sub>1</sub> ~ ''x''<sub>2</sub>}} if and only if {{math|1=''f''(''x''<sub>1</sub>) = ''f''(''x''<sub>2</sub>)}}. The equivalence class of {{mvar|x}} is the set of all elements in {{mvar|X}} which get mapped to {{math|''f''(''x'')}}, i.e. the class {{math|[''x'']}} is the [[inverse image]] of {{math|''f''(''x'')}}. This equivalence relation is known as the [[kernel of a function|kernel]] of {{mvar|f}}.
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| More generally, a function may map equivalent arguments (under an equivalence relation {{math|~<sub>''X''</sub>}} on {{mvar|X}}) to equivalent values (under an equivalence relation {{math|~<sub>''Y''</sub>}} on {{mvar|Y}}). Such a function is known as a morphism from {{math|~<sub>''X''</sub>}} to {{math|~<sub>''Y''</sub>}}.
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| ==See also==
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| *[[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
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| *[[Quotient group]], a construction of mathematical [[group theory|groups]] from equivalence classes of larger groups
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| *[[Homogenous space]], the quotient space of [[Lie group]]s.
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| {{DEFAULTSORT:Equivalence Class}}
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| [[Category:Mathematical relations]]
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| [[de:Äquivalenzrelation#Äquivalenzklassen]]
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| [[nl:Equivalentierelatie#Equivalentieklasse]]
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External Auditor Giacobbe from Bloomfield, spends time with hobbies for instance butterflies, Growtopia Hack and operating on cars. Recently had a family voyage to The Banks of the Danube.
Feel free to visit my website ... Growtopia cheats