Is it possible to have a symmetric and transitive relation on $\mathbb Z$ that isn't reflexive?

Question

I was wondering this the other day and I have been trying to come up with an example for a bit but can't produce one. If there is a proof that such a thing couldn't exist I would be interested in seeing it. Seeing an example of such a set would also be cool as well. Thanks in advance !

Answer 1

Well it is simple that is must be reflexive.Let R be relation in Z such that it is both symmetric and transitive.

Let xRy and yRx,but then by transitivity it must hold that xRx.

Thus such relation can not exist.

Answer 2

The empty relation - nothing relates to anything else - is symmetric and transitive, but not reflexive.