Set theory relation: irreflexive and transitive

Question

Which of the following relations on $T = \{1, 2, 3\}$ is irreflexive and transitive.

1. $\{(2, 1), (2, 3)\}$
2. $\{(1, 1), (2, 1), (3, 2)\}$
3. $\{(2, 1), (1, 2), (3, 2), (2, 3)\}$
4. $\{(1, 1), (2, 2), (3, 3), (2, 1), (1, 2)\}$

From my understanding 2 and 4 are excluded because for irreflexitivity $ x \in T $ then $(x, x) \notin R$

but I don't see how either 1 or 3 can be transitive... which I understand as if $(x, y) \in R$ and $(y, z) \in R$ then $(x, z) \in R$

am I missing something here?

Answer 1

$(1)$ is transitive, because the condition of transitivity is vacuously satisfied. There are no elements related in such a way for transitivity to fails, hence, by default, the relation is transitive.

$(3)$ is not transitive, because $(3, 2), (2, 3) \in R$ but $(3, 3)\notin R$.

Answer 2

Only one of them is transitive, and vacuously so, since there are no $x,y,z$ such that both $(x,y)$ and $(y,z)$ are in the relation. The other has a pair $(x,y)$ and a pair $(y,z)$ such that $(x,z)$ is not an element of the relation.