In the properties of relations, the transitive relation is defined as follows:
If I read it informally, it says, "If $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$
What surprised me was when the author of the paper said that this definition meant, if a relation contains just $(a, b) \in R$ but not $(b, c)$, then R is transitive.
Also, in his examples, he shows that $R_1$, $R_2$ and $R_5$ are transitive relations, whereas to me, since one statement of the conjuction is false, the conjuction is false, and it doesn't proceed to the then part, so $R_1$, $R_2$ and $R_5$ are not transitive.
Q. Am I reading and applying this conjunction incorrectly?
Q. Isn't the author incorrectly translating the conjunction in the definition?
Let $p$ and $q$ be any propositions. Then the implication $p \rightarrow q$ is the proposition defined as follows: $$ \begin{matrix} p \ & q \ & p \rightarrow q \\ F \ & F \ & T \\ F \ & T \ & T \\ T \ & F \ & F \\ T \ & T \ & T \end{matrix} $$ The truth of $p \rightarrow q$ when $q$ is False is said to be vacuous.
Note that the implication is False when and only when $p$ is True but $q$ is False.
Now in your case, $p$ and $q$ are given as follows: $$ p \ \colon \ (a, b) \in R \land (b, c) \in R $$ and $$ q \ \colon \ (a, c) \in R. $$ In fact your $p$ and $q$ are the propositional functions defined by $$ p(a, b, c) \ \colon \ (a, b) \in R \land (b, c) \in R $$ and $$ q(a, c) \ \colon \ (a, c) \in R. $$
Thus for each $(a, b, c) \in R\times R \times R$, you are to show that $$ p(a, b, c) \rightarrow q(a, c) $$ is True.
Hope this helps.