Transitivity of a relation

Question

If a relation contains (a,b) and (a,c), in order for it to be transitive, is (b,c) required, or is (c,b) also required?

In my mind, it should require both (b,c) and (c,b), but I'm not certain.

Answer 1

If you mean a relation, then transitivity requires nothing out of $(a, b), (a, c)$. If you instead had $(a, b), (b, c)$, then $(a, c)$ would be needed. So the relation $$ R = \{(a, b), (a, c)\} $$ is vacuously transitive.

Take $<$, for instance. If $a<b$ and $a<c$, you cannot compare $b$ and $c$ from this information, even though $<$ is well-known to be transitive.

Answer 2

If you mean "If a relation contains (a,b) and (c,a)", then for transitivity only (b,c) is needed. (c,b) is needed when you need to prove symmetry.

If not, I think symmetry, i.e. (a,c) $\Leftrightarrow$ (c,a) is needed to imply (b,c) with transitivity.