My question is not the same as this one : Are there real-life relations which are symmetric and reflexive but not transitive?. The answer " x has slept with y " was a good answer to this question ( more that 120 upvotes) . It wouldn't do for mine.

I'm not asking for arbitrary relations having such and such properties.

I'm looking for real life relations having a conceptual analogy with " membership". So here , the conceptual content of the examples am asking for really matters.

  • Let's admit, if you please, for the sake of the question, that the membership relation can hold between a concrete object and a set of concrete objects ( in spite of the fact that, properly, it only holds between sets, due to the fact that, in set theory, every object dealt with is a set).

I'm looking for real life analogous cases of the fact that the membership relation is not transitve.

  • Maybe these ones :

(1) I am a member a football club. My football club belongs to a football league. But I am not a member of the football league.

(2) I am a british citizen, meaning that I belong to the british poeple. We are, say, in 1943, so currently the british people belongs to the Allies. But I am not one of the Allies.

(3) Letter ' $a$ ' belongs to the word ' cat'. The word ' cat' belongs to the sentence ' the cat is on the map'. But letter ' a ' is not a member of the sentence ' the cat is on the map'. ( Arguably, a sentence is not a set of letters, nor of sounds, but a set of words).

  • I'm not totally satisfied with these examples. For they instantiate intransitivity, rather that non-transitivity strictly speaking.

Can you think of cases illustrating non-transitivity, I mean, the fact that in case $aRb$ and $bRc$ is true, $aRc$ can be true, but does not always nor automatically follow?

  1. My friend Jane and I are both members of the Mathematics faculty at NoName University. NoName is an institutional member of the Gondwanaland Mathematical Society (the GMS). Jane's a member of the GMS, but I'm not.

  2. Israel and Palestine are members of UNESCO. UNESCO is one of the constituent bodies of the United Nations. Israel is a member of the UN, but Palestine isn't.


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