I have the relation $xRy$ is equivalent to "There is a path between $x$ and $y$"
If I now want to check symmetry, $xRy$ is equivalent to $yRx$, I read that this is true. I thought this is only the case when you switch "path" to "road" since the definition of path is "A path is a road without repetition" and a road is a sequence $w = \{x_0,x_1,x_2,\cdots,x_k\}$.
So I assume that if two nodes are bound with a road you can go either way but with a path it's only into one direction (like in a directional graph). Can you explain the symmetry?