# Existence of reflexive, symmetric and intransitive relations on sets containing more than one element

Are there any examples of reflexive, symmetric and intransitive relations on sets containing more than one element (not explicitly defined by their elements? (edited due to imprecision, I apologize.)

For a big class of examples, consider an undirected graph $$G$$ on a set of vertices $$V$$, and say two vertices $$v_1, v_2 \in V$$ are related if $$v_1=v_2$$ or there is an edge from $$v_1$$ to $$v_2$$ in $$G$$.

This defines a reflexive, symmetric relation on $$V$$, but will not typically be transitive. (It will be transitive if $$G$$ is complete, or a union of complete subgraphs, but not in general.)

• Thank you @Clive Newstead! I even forgot to write I was looking for relations that are not necessarily defined by their elements counted, but you gave me an interesting answer anyway! – Cheesecake Nov 21 '19 at 17:23

Sure. How about $$R=\{(1,1),(2,2),(3,3), (1,2), (2,1), (2,3), (3,2)\}$$ on $$\{1,2,3\}$$?

It's reflexive because of $$(1,1), (2,2),$$ and $$(3,3)$$, and it's symmetric,

but it's not transitive, because $$(1,2),(2,3)\in R$$ but $$(1,3)\not\in R$$.

Addendum in response to edit of the question:

If you want such a relation defined with some properties,

rather than explicitly in terms of the elements,

define $$(x,y)\in R\;$$ by $$\;|x-y|\le1$$.

• W.Tanner, I forgot to ask (precisely), my big mistake, if there exist such relations defined with some operation or number properties. – Cheesecake Nov 21 '19 at 17:19
• @VerkhotsevaKatya: you mean like $(x,y)\in R$ defined by $|x-y|\le1$ ? – J. W. Tanner Nov 21 '19 at 17:57
• Yes, thank you! I have totally forgotten it. – Cheesecake Nov 21 '19 at 18:06