In transitive relations, $aRb$ and $bRc$ implies $aRc$. But what if there are no $bRc$, can we say that the relation is transitive?

For example, are relations $R\subseteq V\times V$, corresponding to graphs like this one, transitive? $\require{AMScd}$ $$ \begin{CD} \bullet @>>> \bullet@<<<\bullet\\ @VVV @AAA @VVV\\ \bullet @<<< \bullet@>>> \bullet\\ @AAA @VVV @AAA\\ \bullet @>>> \bullet@<<<\bullet \end{CD} $$


A relation can be trivially transitive, so yes.

The condition for transitivity is:

Whenever $aRb$ and $bRc$ $-$ then it must be true that $aRc$.

That is, the only time a relation is not transitive is when $\exists \; a,b,c$ with $aRb$ and $bRc$, but $aRc$ does not hold.

So the relation corresponding to the graph is trivially transitive.

You can learn a lot about transitivity here


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