Re: Testing for the equivalence relation

Dan wrote:
>
> I'm seeking some help on a rather basic and trivial question.
>
> Is there a case where a relation's extension proves to be symmetric and
> transitive, but the relation is not an equivalence relation? In other
> words, can a relation be symmetric and transitive, but not be
> reflexive?

Yes. Consider the empty relation over a certain domain D. It's clearly symmetric and transitive, but if the domain D is non-empty then it is of course not reflexive. That is exactly the part that was missing in your argumentation; it is indeed the case that if every element in D is involved in at least one pair of the relation then transitivity and symmetry will imply reflexivity and therefore be sufficient conditions for an equivalence relation.

• Jan Hidders