Symmetric, non-reflexive relation I'm looking for an example of a mathematical relation that is symmetric but not reflexive.  A standard non-mathematical example is siblinghood.  

Here is an example that isn't really in the spirit of the question, but I'll provide it anyway.
Let $R$ be any symmetric relation (possibly even reflexive) on a set $X$. Define a new relation $R^\prime$ by
$$
R^\prime = R \setminus \{(x,x) \mid x \in X\}.
$$
What we've done is take a symmetric relation and simply remove all the reflexive bits. (In fact, we could have just removed a single pair $(x,x)$, since a single counterexample is enough to destroy reflexivity.)
The purpose of my example is to emphasize that, when constructing mathematical objects, you shouldn't feel restricted to the usual suspects. Feel free to really force the desired behavior explicitly.
