Given the usual set-theoretic definition of a binary relation, along with the usual notions of
Do there exist any interesting (i.e. surprising, yielding novel results, worth studying etc.) binary relations (across the various fields of study) satisfying reflexivity and symmetry, but not transitivity? If so, could you provide a non-trivial example?
In my (limited) experience (< 1 year of undergraduate study) I've not come across an example satisfying this constraint, but I'm also relatively new to studying Mathematics.
 A binary relation on sets $A$ and $B$ is defined as a subset of the cartesian product $A \times B$, that is, a collection of ordered pairs
 A really simple example would be the relation over sets of people encoding
had a conversation with. That is, we've all debated with ourselves granting reflexivity, and the symmetry is similarly obvious, while transitivity is not guaranteed.