Can a relation be reflexive on the empty set

Question

I have the following binary relation that is asymmetric: $$ \{\langle A,B\rangle , \langle B, C\rangle\} $$ I have the definition

A binary relation $R$ is reflexive on a set $S$ iff for all elements $d\in S$ the pair $\langle d, d\rangle$ is an element of $R$

Does this mean that the binary relation is reflexive on $S$?

Answer 1

Reflexivity holds for the empty set: Reflexivity means $\forall x, xRx$. If there is no $x$, then it is vacuously true. However, your relation $\{(A,B),(B,C)\}$ is not over the empty set, so I don't see how this applies.

Answer 2

Usually when we say that $R$ is a relation on a set $S$, we implicitly require that $R\subseteq S\times S$. So when we say that $R$ is a reflexive relation on $\varnothing$, we also require that $R\subseteq\varnothing\times\varnothing=\varnothing$, which implies that $R=\varnothing$.

Of course, $\varnothing$ is reflexive as a relation on $\varnothing$, since if $x\in\varnothing$, then $\langle x,x\rangle\in\varnothing$, is true vacuously.

But it is not true that $\{\langle A,B\rangle,\langle B,C\rangle\}$ is reflexive on $\varnothing$ because it is not a relation on $\varnothing$.