# Are there real-life relations which are symmetric and reflexive but not transitive?

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.

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