Relation that is only symmetric, reflexive, antisymmetric or transitive?

Question

What could be a possible example of a relation that's symm, reflex, antisymm, transitive?

I am working on practice problems on the unit about Sets and Relations. The question asks me to give a example of a relation that is symm, reflexi, antisym, and transitive. I don't seem to find any possible example that suits this conditions.

Thanks in advance guys!

Answer 1

I would start by making sure that it’s not transitive. Let $R$ be the relation, and suppose that $a\,R\,b$, where $a\ne b$. Symmetry will require that $b\,R\,a$, so you’ll have to have $a\,R\,b\,R\,a$; transitivity would tell you that $a\,R\,a$, so if you make sure that $a\,\not R\,a$, you’ll kill two birds with one stone by ensuring both that $R$ is not transitive and that $R$ is not reflexive. That leaves just antisymmetry to deal with.

At this point you have a set $A=\{a,b\}$ and a relation $R=\{\langle a,b\rangle,\langle b,a\rangle\}$ on $A$. Is this $R$ antisymmetric? If it’s not, you’re done. If it is, can you add something to it (and possible to $A$) to make kill off antisymmetry?

Answer 2

Integers that add up to an odd number.