# Euclidean relation

In mathematics, **Euclidean relations** are a class of binary relations that satisfy a modified 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. 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

- 1 2 Fagin, Ronald (2003),
*Reasoning About Knowledge*, MIT Press, p. 60, ISBN 978-0-262-56200-3.