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A relation $R$ on a set $X$ is called Euclidean if for all $a, b, c \in X$, if $aRb$ and $aRc$ then $bRc$.

This does not imply transitivity - the relation $\{(a, b), (b, c), (b, b), (c, c), (c, b)\}$ is non-transitive and Euclidean.

I was wondering if there are any "real life examples" of these kinds of relations, which are interesting too?

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I don't know if it counts as an interesting real-life example, but the relation on $\{0,1,-1\}$ defined by $xRy$ if $xy\leq0$ is at least an example. Further, it seems that this generalises to any set made up of an arbitrary amount of positive numbers, $0$, and precisely one negative number. It might generalise to other sets, but I'm pretty confident about that one.

Edit: I retract my statement, since $0R1$ and $0R1$ but $1\not R1$

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  • $\begingroup$ If your answer is essentialy wrong (that is, if it can't be easily fixed), you should delete it. You can edit and undelete a deleted answer, if you find a good answer. $\endgroup$ – ajotatxe Sep 5 '17 at 11:41
  • $\begingroup$ Good to know. I haven't been using this site long, so I'm not too familiar with the etiquette. That said, I don't actually see how to delete it. $\endgroup$ – themathandlanguagetutor Sep 5 '17 at 11:42
  • $\begingroup$ Simply click on "delete", the fourth word just beneath the body of your answer. $\endgroup$ – ajotatxe Sep 5 '17 at 11:43
  • $\begingroup$ I only have edit, share, and more, and under more is only flag $\endgroup$ – themathandlanguagetutor Sep 5 '17 at 11:45
  • $\begingroup$ Perhaps you can't delete it because of your reputation. Perhaps someone can delete it for you. I can't. $\endgroup$ – ajotatxe Sep 5 '17 at 11:46
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$x\text{R}y$ defined by $0≤x≤y+1≤2$ is not transitive, but right Euclidean on natural numbers.

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