I'm trying to figure out which one of these relations are transitive relation(s)
{(a,a), (a,b), (b,b), (b,c)}
{(a,a), (a,b), (b,b)}
{(a,a), (a,b), (b,a), (b,b), (c,c)}
{(a,a), (b,c), (c,b)}
As I've understood the definition of transitive relation is; if (a,b) and (b,c), then (a,c)
I can't seem to find any (a,c) in any of those relations, would that make none of those transitive relation?
Best Answer
The first and last relations are not transitive. Can you see why? You're right that in the first case, since $(a, b), (b, c) \in R$, we'd need $(a,c)\in R$ for the relation to be transitive. $(a, c)\notin R$, so the relation is not transitive.
What's missing in the last relation? We have $(b, c)$ and $(c, b)$ in the relation, so $b\mapsto c \mapsto b$, but $(b, b)\notin R$. Hence, the relation cannot be transitive.
For transitivity, if $(a, b)$ and $(b, c)$ are in the relation, then we must have $(a, c)$ in the relation. But $a, b, c$ need not be distinct. So the problem in the last case isn't about no relation between $a, c$. $a$ is related only to itself, and no other element is related to $a$, so since $(a, a)\in R$, $a$ is "off the hook" here. The problem in the last case is that if it were transitive, then it would be true that $\Big((b, c)\in R \land (c, b)\in R\Big) \implies (b, b) \in R$. It's not, so the relation is not transitive.