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Lets say I have {a,b,c,d}

I though of (a,b)(b,c) - but this is antisymmetric, right?

Then I though of (a,b)(b,a)(b,c) but this time is transitive

Finally I tried with (a,b)(b,c)(c,d) but again, is this antisymmetric?

Rodrigo Schillaci
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    Welcome to MSE. Here's how to ask a good question. Follow these guidelines to get help in this forum. For example, there you'll find the following: "Your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title." – jjagmath Feb 19 '24 at 18:56
  • @Jean-ArmandMoroni You mean $(a,b)$ and $(b,a)$ would imply $(a,a)$ and $(b,b)$; $(a,b)$ and $(b,c)$ would imply $(a,c)$. – Robert Israel Feb 19 '24 at 19:01
  • @RobertIsrael Yes, thanks, but nevermind, I deleted the comment as I could not correct it any more. – Jean-Armand Moroni Feb 19 '24 at 19:05
  • I mean, while I understand what you are trying to say, the way I got teached this, if the relation is not in brackets, then it doesnt exist and we dont imply anything. So if I have (a,b) and (b,a) even tho it implies (a,a) we dont write it. So to come up with a relation with {a,b,c,d} that is nothing, reflexive, symmetric, antisymmetric nor transitive I can write as many relations as I want, but I cant seem to get it right – Rodrigo Schillaci Feb 19 '24 at 19:08
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    @RodrigoSchillaci "If I have $(a,b)$ and $(b,a)$ even tho it implies $(a,a)$ we dont write it". Of course if your teaching assumes that all relations are transitive even when not writing explicitely all pairs, that's gonna be difficult to come up with a non-transitive relation. :-) – Jean-Armand Moroni Feb 19 '24 at 19:12

1 Answers1

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$R=\{(a,b), (b,c), (b,a)\}$ will do.

It is obviously non reflexive, also non symmetric because $(c,b)\notin R$; not antisymmetric because $(a,b),(b,a)\in R$ but $a\neq b$, and not transitive beacause $(a,b),(b,c)\in R$ but $(a,c)\notin R$.

Julio Puerta
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