Euclidean relation

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In mathematics, Euclidean relations are a class of binary relations that formalizes Euclid's "Common Notion 1" in The Elements: things which equal the same thing also equal one another.

Definition[edit]

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[edit]

The property of being Euclidean is different from transitivity. A transitive relation is Euclidean only if it is also symmetric. Only a symmetric Euclidean relation is transitive.

A relation which is both Euclidean and reflexive is also symmetric and therefore an equivalence relation.[1]

References[edit]

  1. ^ a b Fagin, Ronald (2003), Reasoning About Knowledge, MIT Press, p. 60, ISBN 978-0-262-56200-3 .