Discrete Mathematics - Transitivity

Question

an extremely basic question that I would like to clarify.

Question :

Let R be a relation.

if aRb and bRa but $a \not R a$ and $b \not R b$, is it true that the R is not transitive? (Since aRb and bRa requires aRa to be considered transitive and vice versa)

Thanks for the help.

Answer 1

If by "a is not reflexive", you mean $a\not Ra$, then yes. The three statements $aRb$ and $bRa$ and $a\not Ra$ together disprove transitivity of $R$. Same with $aRb, bRa$ and $b\not Rb$.

Transitivity means "For any three elements $a, b, c$, if we have $aRb$ and $bRc$, then $aRc$." It does not require $a, b, c$ to be distinct, and particularily $c = a$ is allowed.