Why is the following relation not transitive?

Question

Let $A$ be $\{a,b,c\}$. Let the relation $R$ be $\{(a,a),(c,c),(a,b),(b,a)\}$.

Since $(a,b) \land (b,a) \to (a,a)$ this shows transitivity. Furthermore $(c,c)$ doesn't have anything else it relates to so, therefore, you can assume the premises are false as well as the conclusion being false so, therefore, showing transitivity.

Answer 1

It seems the question has been answered in comments, but just to make sure it has an answer:

A relation $R$ on a set $A$ is called transitive provided for all $x$, $y$, and $z$ in $A$, $x\mathrel Ry \wedge y\mathrel Rz \implies x\mathrel Rz$.

Let $x=b$, $y=a$, and $z=b$. Then $x \mathrel{R} y$ and $y \mathrel{R} z$. But $x \mathrel{R} z$ is equivalent to $b \mathrel{R} b$, which is not true. So $R$ is not transitive.

Answer 2

for transitivity, for all (a,b) and (b,c) belongs to the relation (a,c) should belong to the relation. This interpretation forces us to check (b,a) , (a,b) implies (b,b).