# If a relation is euclidean, is it necessarily asymmetric?

$$R$$ is relation on set $$A$$, that is $$R\subseteq A \times A$$. $$R$$ is euclidean if $$(\forall x,y,z\in A)(xRy\land xRz \Rightarrow yRz)$$. $$R$$ is asymmetric if $$(\forall x,y\in A)(xRy\Rightarrow \lnot(yRx)).$$

For example, if R euclidean relation on A, and $$(1,2)\in R$$, then because $$R$$ euclidean and $$(1,2), (1,2)\in R \Rightarrow (2,2)\in R$$, which means it isn't asymmetric (because every asymmetric relation is necessarily not reflexive).

But if $$R$$ is an empty relation, then it's both asymmetric and euclidean, which means an euclidean relation is not necessarily asymmetric. Or am I thinking too much into it?

• What is the euclidean relation? Commented Oct 22, 2020 at 9:03
• @Wuestenfux: en.wikipedia.org/wiki/Euclidean_relation Commented Oct 22, 2020 at 9:05
• @Wuestenfux I edited the post so that it contains the definitions of euclidean and asymmetric relations now. Thanks for pointing it out :) Commented Oct 22, 2020 at 9:12
• No. From $xRy ∧ xRz$, by commutativity of $\land$ both $yRz$ and $zRy$ follow. Commented Oct 22, 2020 at 9:16
• What I mean is: An Euclidean relation is not necessarily asymmetric. This means that the def of Euclidean does not imply asymm. We can have "degenerate" case... Commented Oct 22, 2020 at 9:34

Let R be an Euclidean relation on A. and let $$(x,y) \in R$$
$$xRy \land xRy \Rightarrow yRy$$ which means Euclidean relation cant be asymmetric if there exists an $$(x,y) \in R$$ in case of Empty Relation we know that it doesnt have any elements so this proposition doesnt contain it