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.

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