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relation (mathematics)

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relation (mathematics)
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{{Short description|Relationship between two sets, defined by a set of ordered pairs}}{{about|basic notions of (homogeneous binary) relations in mathematics|a more in-depth treatment|Binary relation|relations on any number of elements|Finitary relation}}(File:Relación binaria 01.svg|thumb|300px|Illustration of an example relation on a set {{math|1=A = {{mset| a, b, c, d }}}}. An arrow from {{mvar|x}} to {{mvar|y}} indicates that the relation holds between {{mvar|x}} and {{mvar|y}}. The relation is represented by the set {{math|{ (a,a),}} {{math|(a,b),}} {{math|(a,d),}} {{math|(b,a),}} {{math|(b,d),}} {{math|(c,b),}} {{math|(d,c),}} {{math|(d,d) } }} of ordered pairs.)In mathematics, a relation on a set may, or may not, hold between two given members of the set.As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values {{math|1}} and {{math|3}} (denoted as {{math|1 < 3}}), and likewise between {{math|3}} and {{math|4}} (denoted as {{math|3 < 4}}), but not between the values {{math|3}} and {{math|1}} nor between {{math|4}} and {{math|4}}, that is, {{math|3 < 1}} and {{math|4 < 4}} both evaluate to false.As another example, "is sister of{{-"}} is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa.Set members may not be in relation "to a certain degree" – either they are in relation or they are not.Formally, a relation {{mvar|R}} over a set {{mvar|X}} can be seen as a set of ordered pairs {{math|(x,y)}} of members of {{mvar|X}}.{{sfn|Codd|1970|ps=}}The relation {{mvar|R}} holds between {{mvar|x}} and {{mvar|y}} if {{math|(x,y)}} is a member of {{mvar|R}}.For example, the relation "is less than" on the natural numbers is an infinite set {{math|1=Rless}} of pairs of natural numbers that contains both {{math|(1,3)}} and {{math|(3,4)}}, but neither {{math|(3,1)}} nor {{math|(4,4)}}.The relation "is a nontrivial divisor of{{-"}} on the set of one-digit natural numbers is sufficiently small to be shown here:{{math|1=Rdv = {{mset| (2,4), (2,6), (2,8), (3,6), (3,9), (4,8) }}}}; for example {{math|2}} is a nontrivial divisor of {{math|8}}, but not vice versa, hence {{math|1=(2,8) ∈ Rdv}}, but {{math|1=(8,2) ∉ Rdv}}.If {{mvar|R}} is a relation that holds for {{mvar|x}} and {{mvar|y}} one often writes {{math|xRy}}. For most common relations in mathematics, special symbols are introduced, like "{{math|

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