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

Question

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.)

Answer 1

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.)

Answer 2

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 $(1,3)\not\in R$.

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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 $\;|x-y|\le1$.