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


*

*$\{(2, 1), (2, 3)\}$

*$\{(1, 1), (2, 1), (3, 2)\}$

*$\{(2, 1), (1, 2), (3, 2), (2, 3)\}$

*$\{(1, 1), (2, 2), (3, 3), (2, 1), (1, 2)\}$


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?

$(1)$ 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 $(3, 3)\notin R$.
