Closures of a relation

The reflexive-transitive closure of a given relation is the same as the reachability relation through that relation. And what of other types of closures?

Let us continue with our study of relational logic, beginning first by establishing the independence of the properties of reflexivity, symmetry, and transitivity. After this, we shall explore the various closure conceptions of a relation, the reflexive-transitive closure, the symmetric closure, and so forth.

Independence of reflexity, symmetry, transitivity

We have introduced three fundamental properties of a binary relation—reflexive, symmetric and transitive—which together define the concept of an equivalence relation. Did we really need all three notions, or are these properties perhaps redundant in some way? Perhaps two of these notions imply the third? Let us investigate.