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

Question

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.

Answer 1

My favorite example is synonymy: certainly any word is synonymous with itself, and if you squint you can imagine that if a word appears in the thesaurus entry for another, then the latter will symmetrically appear in the thesaurus entry for the former. But synonymy is not transitive.

However this and many other examples are special cases of vertices joined by edges in graphs which is a canonical example of Tolerance:

Tolerance relations are binary reflexive, symmetric but generally not transitive relations historically introduced by Poincare', who distinguished the mathematical continuum from the physical continuum, then studied by Halpern, and most notably the topologist Zeeman.

Recent surveys include:

Peters & Wasilewski's "Tolerance spaces: origins, theoretical aspects and applications" Info Sci 2012, and Sossinsky's "Tolerance Space Theory" Acta App Math 1986, which mentions these examples:

• Metric space with distance between points less than $\epsilon$

• Topological space with a fixed covering and 2 points both contained in one element of the cover

• Vertices in the same simples of a simplicial complex

• Vertices joined by an edge in an undirected graph

• Sequences that differ by 1 (or 2, or 3) binary digits

• Cosets in a group with nonempty intersection

An intersting textbook that discusses tolerances is Pirlot & Vincke's Semiorders, 1997.

Sossinsky's paper goes on to mention:

(i) tolerance spaces appear quite naturally in the most varied branches of mathematics;

(ii) the tolerance setting is very convenient for the use of many existing powerful mathematical tools;

(iii) only results 'within tolerance' are usually required in practical applications.

and that "tolerance, in a way, is a trick for avoiding the specific hazards of infinite-dimensional-function spaces, eg their local noncompactness; moreover, in a certain sense, in tolerance spaces, you can't have large finite dimensions"

Answer 2

$x$ lives within one mile of $y$.

This is reflexive and symmetric, but not transitive.

Answer 3

$x$ has lived with $y$ at some point (whether in the same building or same location on the streets).

Alternately, $x$ and $y$ have at least one biological parent in common.

Answer 4

$x$ has the same number of legs and/or the same number of teeth as $y$.

Answer 5

$\quad\quad x\;$ has slept with $\;y$ ${}{}{}{}{}$

Answer 6

• $x$ has had body contact with $y$.
• $x$ and $y$ were once nationals of the same country.

Answer 7

There exists a question on math.SE that both $x$ and $y$ have answered.

Answer 8

What about

• $\,xRy\Longleftrightarrow\,\,x\,,\,y\,$ are blood related?

Answer 9

$x$ is indistinguishable from $y$.

The non-transitivity of this relation is my favorite way to account for the non-intuitiveness of the theory of evolution.

Answer 10

On the set of countries: $x$ and $y$ share a border.

Answer 11

Several of the examples given have in common some similarity between things (if I resemble John and John resembles Mike, I do not necessarily resemble Mike: I and J. might have some common features different from those J. has in common with M.).

And, sure enough, a reflexive, symmetric, non-transitive relation has been called a “similarity relation”; see for instance this search, and several other hits in (especially fuzzy) set theory.

Answer 12

$x$ and $y$ are foods that go well together (with respect to a fixed person's palate, I suppose).