Why is this relation not transitive?

Question

Question: Let $A$ be $\{a,b,c\}$. Let the relation $R$ be $A \times A$ - $\{(a,a), (a,b), (b,b), (c,a)\}$ Which of the following statements about $R$ is true?
a. $R$ is reflexive, is symmetric, and is transitive.
b. $R$ is reflexive, is symmetric, and is not transitive.
c. $R$ is not reflexive, is not symmetric, and is not transitive.
d. $R$ is not reflexive, is symmetric, and is not transitive.
e. $R$ is reflexive, is not symmetric, and is transitive.
I understand why the relation is neither reflexive nor symmetric but as for whether it's not transitive I kind of understand why its not but cannot explain it.
For a relation to be transitive $(a R b) ∧ (b R z) → (a R z)$ but in the context of the question this becomes $(a R b) ∧ (b R b) → (a R b)$ however since I since I already used the ordered pair $(a R b)$ in the premise I understand I can't use it again so that being said how do I explain why it is not transitive?

Answer 1

There is no problem with the one you are discussing (note that the elements of $A\times A-\{(a,a),(a,b),(b,b),(c,a)\}$ are all the elements of $A\times A$ without those specified after the $"-"$, so you do not even have $(a,b)$ or $(b,b)$ in $\mathcal{R}$).

Rather note that you have $(c,b)$ and $(b,c)$ in $\mathcal{R}$ but not $(c,c)$.

Answer 2

$(c,a) \in R$, $(a.b) \in R$ but $(c,b)$ not in $R$