$R$ 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?
Best Answer
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