For many properties of binary relations, like symmetry, asymmetry, and transitivity, it does not depend on what domain of the relation we are considering. More precisely, given a binary relation $R$ on $A$ and a superset $B$ of $A$, $R$ is symmetric/asymmetric/transitive on $A$ iff it is so on $B$. However, reflexivity does depend on the domain. Why? Also, bonus question, is there a logical characterization of those properties which does not depend on the domain? That is, given a formula $F$ of first-order logic, what syntactic property must it have so that $R$ satisfies $F$ in a domain-independent way?