Is the empty relation antireflexive?

Question

I have read on this post that the empty relation is not reflexive. Does it mean that it is neither reflexive nor antireflexive. Because I saw in an article that a relation is called nonreflexive if it is neither of them.

If it is indeed antireflexive, how can we prove it?

Answer 1

Recall what it means for a relation $R$ on a set $S$ to be antireflexive. You can define it in two equivalent ways:

• For all $x \in S$, $xRx$ never holds
• For all $x \in S$, $(x,x) \notin R \subseteq S \times S$

The empty relation is the case where $R = \emptyset$. $(x,x)$ is never in $R$ in such a case (nor any other pairing, because it is the empty set), and thus it is antireflexive.