Why is this relation $R=\{(a,b), (b,c), (c,a)\}$ transitive?

Question

I have a set of relations, shown below:

$R=\{(a,b), (b,c), (c,a)\}$ for $A= \{a,b,c\}$

According to my professor, this relation is transitive but I don't understand why. I was under the impression that any given three relations of a set of relations have to follow the $aRb, bRc, aRc$ pattern but since $aRc$ isn't in the set then it'd fail the test since $aRc$ and $cRa$ aren't equivalent. I'm sure I'm missing something, but I don't understand what.

Answer 1

As expressed, the relation $R$ fails to be transitive. Indeed, we need $(a, c) \in R$ and instead of $(c, a) \in R$, as $(c, a)$ poses a number of problems in terms of the relation being transitive:

As written:

• $(a, b), (b, c) \in R,$ but $(a, c) \notin R$, as you observe.
• Furthermore: $(c, a), (a, b) \in R$ but $(c, b) \notin R$.
• And, $(b, c), (c, a) \in R$, but $(b, a)\notin R$.

For each reason above, transitivity fails.

So $(c, a) \in R$ seems to be the ill-placed, perhaps misprinted pair in $R$, which if replaced by $(a, c)$ would alleviate these failures, and the relation would then be transitive.