Showing a relation is transitive

Question

Problem:

Let $A = \{(a,b)\}$, determine whether the relation $R = \{(b,a)\}$ is transitive.

Claim:

No, $R$ is not transitive.

Proof:

Since $a,b\in A$ and $a\in R$ but $b,a\notin R$.

I am not sure if this is a sufficient proof. Any suggestions would help.

Answer 1

Well, it is transitive (with $A=\{a,b\}$). The definition is $$\forall a,b,c\in A: aRb \wedge bRc\Rightarrow aRc$$ where $R\subseteq A\times A$.

The point is that if the premise is not fulfilled, the relation is transitive (since then the implication becomes true). For instance, $R=\{(a,b)\}$ (where $A=\{a,b\}$) and $R=\{(a,b),(a,c)\}$ are all transitive, but $R=\{(a,b),(b,c)\}$ is not since $(a,c)$ is missing.