## Relations That Are Symmetric and Antisymmetric

When I teach relations and their properties, the question of whether a relation can be both symmetric and antisymmetric always seems to arise.  This post addresses that question.

First, a reminder of the definitions here: A relation $\rho$ on a set S is symmetric if, for all $x,y \in S$, $(x,y) \in \rho$ implies that $(y,x) \in \rho$.  A relation $\rho$ on a set S is antisymmetric if, for all $x,y \in S$, $(x,y) \in \rho$ and $(y,x) \in \rho$ implies that $x = y$.

At first glance the definitions look a bit strange, in the sense that we would expect antisymmetric to mean “not symmetric.”  But that’s not quite what antisymmetric means. In fact, it is possible for a relation to be both symmetric and antisymmetric.

The boring (trivial) example is the empty relation $\rho$ on a set.  In this case, the antecedents in the definitions of symmetric ( $(x,y) \in \rho$) and antisymmetric ( $(x,y) \in \rho$ and $(y,x) \in \rho$) are never satisfied.  Thus the conditional statements in the definitions of the two properties are vacuously true, and so the empty relation is both symmetric and antisymmetric.

More interestingly, though, there are nontrivial examples.  Let’s think through what those might be. Suppose $(x,y) \in \rho$.  If $\rho$ is symmetric, then we require $(y,x) \in \rho$ as well.  If $\rho$ is antisymmetric, then, since $(x,y) \in \rho$ and $(y,x) \in \rho$, we must have $x = y$.  And this gives us a characterization of relations that are both symmetric and antisymmetric:

If a relation $\rho$ on a set $S$ is both symmetric and antisymmetric, then the only ordered pairs in $\rho$ are of the form $(x,x)$, where $x \in S$.

Thus, for example, the equality relation on the set of real numbers is both symmetric and antisymmetric.

Perhaps it’s clearer now the sense in which antisymmetric is opposing symmetric.  Symmetric means that there cannot be one-way relationships between two different elements.  Antisymmetric means that there cannot be two-way relationships between two different elements.  Both definitions allow for a relationship between an element and itself.