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The | In [[mathematics]], '''Euclidean relations''' are a class of [[binary relation]]s that satisfy a weakened form of [[transitive relation|transitivity]] that formalizes [[Euclid]]'s "Common Notion 1" in ''[[Euclid's Elements|The Elements]]'': ''things which equal the same thing also equal one another.'' | ||
==Definition== | |||
A [[binary relation]] ''R'' on a [[set (mathematics)|set]] ''X'' is '''Euclidean''' (sometimes called '''right Euclidean''') if it satisfies the following: for every ''a'', ''b'', ''c'' in ''X'', if ''a'' is related to ''b'' and ''c'', then ''b'' is related to ''c''.<ref name="fagin">{{citation|title=Reasoning About Knowledge|first=Ronald|last=Fagin|authorlink=Ronald Fagin|publisher=MIT Press|year=2003|isbn=978-0-262-56200-3|page=60|url=http://books.google.com/books?id=xHmlRamoszMC&pg=PA60}}.</ref> | |||
To write this in [[predicate logic]]: | |||
:<math>\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c).</math> | |||
Dually, a relation ''R'' on ''X'' is '''left Euclidean''' if for every ''a'', ''b'', ''c'' in ''X'', if ''b'' is related to ''a'' and ''c'' is related to ''a'', then ''b'' is related to ''c'': | |||
:<math>\forall a, b, c\in X\,(b\,R\, a \land c \,R\, a \to b \,R\, c).</math> | |||
== Relation to transitivity == | |||
The property of being Euclidean is different from [[transitive relation|transitivity]]: both the Euclidean property and transitivity infer a relation between ''b'' and ''c'' from relations between ''a'' and ''b'' and between ''a'' and ''c'', but with different argument orderings in the relations. However, if a relation is [[symmetric relation|symmetric]], then the argument orders do not matter; thus a symmetric relation with any one of these three properties (transitive, right Euclidean, left Euclidean) must have all three.<ref name="fagin"/> | |||
If a relation is Euclidean and [[reflexive relation|reflexive]], then it must also be symmetric and hence transitive (following the previous paragraph), and so it must be an [[equivalence relation]]. Consequently, equivalence relations are exactly the reflexive Euclidean relations.<ref name="fagin"/> | |||
==References== | |||
{{reflist}} | |||
[[Category:Mathematical relations]] | |||
[[Category:Euclid|Relation]] | |||
Revision as of 15:56, 19 December 2013
In mathematics, Euclidean relations are a class of binary relations that satisfy a weakened form of transitivity that formalizes Euclid's "Common Notion 1" in The Elements: things which equal the same thing also equal one another.
Definition
A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every a, b, c in X, if a is related to b and c, then b is related to c.[1]
To write this in predicate logic:
Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b is related to c:
Relation to transitivity
The property of being Euclidean is different from transitivity: both the Euclidean property and transitivity infer a relation between b and c from relations between a and b and between a and c, but with different argument orderings in the relations. However, if a relation is symmetric, then the argument orders do not matter; thus a symmetric relation with any one of these three properties (transitive, right Euclidean, left Euclidean) must have all three.[1]
If a relation is Euclidean and reflexive, then it must also be symmetric and hence transitive (following the previous paragraph), and so it must be an equivalence relation. Consequently, equivalence relations are exactly the reflexive Euclidean relations.[1]
References
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