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I would like to understand the transitive property in relations...I just cant get it in my brain. I mean the definition is crystal clear. However I still struggle. For example:

Given the set $A=\{0,1,2\}$ the $R=\{(0,0),(0,1),(1,1),(2,2)\}$

According to the definition if $(a,b) \in R$ and $(b,c) \in R \to (a,c)\in R$

So $0\sim0$ and $0\sim 1$ then I need $(0,1)$ again? I makes no sense for me, I mean the numbers are the same..I mean is $a=b$ also possible?

Vladimir Vargas
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Mamba
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  • Your notation is weird. Instead of writing $a,b \in R$, which is usually shorthand for "the two elements $a$ and $b$ are in the set $R$", you should write $(a,b) \in R$, which says "the element/ordered pair $(a,b)$ is in the set $R$". What you currently have written, which is $a, b \in R$, would make people think $a$ is the name of one ordered pair, and $b$ is the name of another ordered pair, and both ordered pairs $a$ and $b$ are in $R$ (since the elements of $R$ are ordered pairs). – layman Jan 19 '15 at 17:47

1 Answers1

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If $\sim$ is a transitive relation on $A$ then if $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$. You have $0\sim 0$ and $0\sim 1$ which implies $0\sim 1$. Therefore $(0,1)\in R$, which is something that's true. Recall that$\{a,a,a,a,a\}=\{a\}$ so you don't need $(0,1)$ to be repeated in A, you already have it.

Vladimir Vargas
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  • Thank you. I dont get the second part. i mean the relation is reflexive. is this a hint somehow? but the basic message is that I dont have to repead the elements since they alreade a part of my relation?! – Mamba Jan 19 '15 at 17:47
  • @Mamba It's not a hint, I read wrong, please reload the page. I read "equivalence relations" instead of transitive. – Vladimir Vargas Jan 19 '15 at 17:48
  • aah ok. thx again !!! – Mamba Jan 19 '15 at 17:49
  • @Mamba with respect to the second question, you want $(0,1)$ to be in $R$, but it is already there, so there's no problem and the relation is transitive. If you have something like $0\sim 0$ and $0\sim 0$ therefore $0\sim 0$ therefore $(0,0)$ must be in $R$, and you repeat this process infinitely you would end up needing $R={(0,0),(0,0),(0,0),...}$ but this makes no sense because you only need to write that $(0,0)\in R$, repeating would be redundant. – Vladimir Vargas Jan 19 '15 at 17:51
  • Do I have only to find one transitive tuple? I mean ${2,2}$ doesnt fulfil this property. Is this not a violation of transitivity? – Mamba Jan 19 '15 at 18:00
  • @Mamba Notation is important: $(a,b)\in R$ means that $a\sim b$. ${a,b}$ is a set with $a$ and $b$ as elements. What property isn't $(2,2)$ fulfilling? – Vladimir Vargas Jan 19 '15 at 18:05