# Do there exist interesting binary relations satisfying reflexivity and symmetry, but not transitivity?

Given the usual set-theoretic definition of a binary relation, along with the usual notions of

• reflexivity
• symmetry
• transitivity

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.

• Here is one you met before first year. Let $A$ be the set of points on the edges of a certain triangle. Define a relation $R$ on $A$ by $(x,y)\in R$ if $x$ and $y$ are on the same edge of the triangle. – André Nicolas Jul 16 '11 at 23:29
• is friends with – isomorphismes Jan 28 '14 at 21:12
• The three axioms put together form an equivalence relation; here are some examples of relaxing at least one of the three conditions. – isomorphismes Jan 30 '14 at 3:15
• One interesting kind of intransitive thing plato.stanford.edu/entries/nonwellfounded-set-theory satisfies a>b>c>...>a. – isomorphismes Jan 30 '14 at 3:18

For $x, y \in \mathbb{R}$, put $(x,y)$ in the relation if $|x-y|<1$. Or do the same thing in $\mathbb{R}^2$. Or else replace $1$ by some $\epsilon>0$.
• @François G. Dorais: Graph theory was already taken. I was thinking geometry/analysis. The colouring problem for $|x-y|=1$ in $\mathbb{R}^2$, and its relatives, are not directly connected to inequalities. – André Nicolas Jul 17 '11 at 5:57