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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.

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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.

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  • $\begingroup$ If the empty relation is the empty set how can it be transitive if it has no elements? And how can it not be reflexive? How can the emptiness not be related to itself? $\endgroup$ – Gabu Nov 29 '15 at 2:50
  • $\begingroup$ 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. $\endgroup$ – Noah Schweber Nov 29 '15 at 3:08
  • $\begingroup$ @Gabu It's vacuously true. You should Google that phrase or search it on here. I'm on my phone otherwise I would post more. $\endgroup$ – user223391 Nov 29 '15 at 3:10

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