Equivalence relation in a simple directed graph $D$
I have written my proof, but I'm still not sure if it's rigth:
Determine if for all directed graphs $D$ the following relation $R$ defined in $V(D)$ is an equivalence relation: $xRy$ if and only if $x=y$ or there is an $xy$-path directed.
Reflexivity: It's reflexive because if $x = x'$ and then $x' = x$ then by definition it holds, but not sure about the path because since $D$ is simple, there can't be the $x$ $x$ edge
Simetry: It is not symetric because a directed path existing from $x$ to $y$ does not imply that there is some directed path from $y$ to $x$.
Transitivity : We uppose there is a directed path from $x$ to $y$ and a directed path from $y$ to $z$. Then we procede to cancontenate this paths and we get de $xz$ path. But here I'm not sure because this might not be a path because it could possibly have repeated edges if some edge or edges appeared in both paths involved in the concatenation
Any help?
$\endgroup$ 4 Reset to default
