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I'm sorry if this seems like a question asked before doing any research, but the discrete math textbook I'm using doesn't mention it, and I tried googling it, but I don't even know the name of it, so I couldn't find anything either.

The original question is: Let A be a set and R $\subseteq A \times A$ be a binary relation on A. Prove that R is reflexive $\Leftarrow\Rightarrow R^0\subseteq R$

I would just like to know what $R^0$ means here.

Tony Tarng
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  • Maybe ${(x,x), x\in A}$... – Augustin Jul 02 '16 at 16:51
  • @Augustin that was my first guess too. But it'd be strange if it were denote $R^0$ as it has not relation to $R$. – quid Jul 02 '16 at 16:52
  • No but it's still a relation. Actually it's the smallest reflexive relation. – Augustin Jul 02 '16 at 16:54
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    If the exponent is composition, then $R^0$ should be the identity, which I would expect to be the diagonal. – pjs36 Jul 02 '16 at 16:55
  • Well, it is true that a relation $R$ is reflexive if and only if $\Delta:={(x,x)\ :\ x\in A}\subseteq R$, so it makes sense for it to mean that. Also, it is true that $\Delta\circ R=R\circ\Delta=R=R^{1+0}$. –  Jul 02 '16 at 16:55
  • Well I guess that makes sense. – Tony Tarng Jul 02 '16 at 16:56
  • I did not think of that meaning of the exponent. That is indeed a plausible explanation. – quid Jul 02 '16 at 17:01
  • @GEdgar The book containing the question contains only practice problems. The textbook I'm referring to is Discrete Mathematics And Its Application by K. H. Rosen. So I assume it's just a weird (uncommon) notation. – Tony Tarng Jul 02 '16 at 17:31
  • So, I guess there is a book by Rosen, which contains this definition, but you have only the practice problems for the book? – GEdgar Jul 02 '16 at 17:37

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