Tricky transitive relations

$S$ is not transitive as @Conan noted. $T$ is transitive,because you can easily check that for $(1,1)$ the only pair which satisfy the definition is $(3,1)$: $$(1,1),(3,1)\in T\Rightarrow (3,1)\in T$$ For other pairs the antecedent of the definition is always wrong so the whole definition is satisfied for $T$.

deezy, in S, we have (3,1) and (1,2), but we do not have (3,2)