Inspired by Halmos (Naive Set Theory) . . .
For each of these three possible properties [reflexivity, symmetry, and transitivity], find a relation that does not have that property but does have the other two.
One can construct each of these relations and, in particular, a relation that is
symmetric and reflexive but not transitive:
$$R=\{(a,a),(a,b),(b,a),(b,b),(c,c),(b,c),(c,b)\}.$$
It is clearly not transitive since $(a,b)\in R$ and $(b,c)\in R$ whilst $(a,c)\notin R$. On the other hand, it is reflexive since $(x,x)\in R$ for all cases of $x$: $x=a$, $x=b$, and $x=c$. Likewise, it is symmetric since $(a,b)\in R$ and $(b,a)\in R$ and $(b,c)\in R$ and $(c,b)\in R$. However, this doesn't satisfy me.
Are there real-life examples of $R$?
In this question, I am asking if there are tangible and not directly mathematical examples of $R$: a relation that is reflexive and symmetric, but not transitive. For example, when dealing with relations which are symmetric, we could say that $R$ is equivalent to being married. Another common example is ancestry. If $xRy$ means $x$ is an ancestor of $y$, $R$ is transitive but neither symmetric nor reflexive.
I would like to see an example along these lines within the answer. Thank you.
Best Answer
$\quad\quad x\;$ has slept with $\;y$ ${}{}{}{}{}$