# What is the empty relation?

I was reading the Wikipedia article on equivalence relations and one section says that "the empty relation R on a non-empty set X is vacuosly symmetric and transitive but not reflexive."

What is the empty relation? And what is vacuosly symmetric?

Thank you very much.

A relation on a set $A$ is by definition a subset $R\subseteq A\times A$. Then "$a$ is related to $b$" means "$(a,b)\in R$. The empty relation is then just the empty set, so that "$a$ is related to $b$ is always false.
• It's not reflexive - assuming $X$ is nonempty - since if $a\in X$, then $(a, a)$ is not an element of the empty relation. It is transitive, since - for every $a, b, c$ such that $(a, b)$ and $(b, c)$ are in the empty relation - we have $(a, c)$ is in the empty relation; this is because there are no such $a, b, c$ at all, so transitivity holds vacuously. Nov 29, 2015 at 3:08