Question

This is not homework but has a direct relation to my homework. In other words, I need to know this information to be able to do my homework.

Is R transitive: R = {(a,b),(b,a),(c,c)}? I would think that it would also need to include (a,a),(b,b) but I am unsure.

Is the empty set {} irreflexive?

These are cases which have not been explained clearly and I would appreciate clarification.

Answer 1

If you look for example at Wikipedia: Transitive relation you have this nice quantified expression that becomes true if your relation is transitive.

Because it’s universally quantified it’s correct for the empty set (because universally quantified expressions about the empty set are true by definition).
And you are absolutely right. If there is (a,b) and (b,a) in R, then there also has to be (a,a) for R to be transitive.

The irreflexivity is also universally quantified (“It is a binary relation on a set where no element is related to itself.” => ∀x:~(xRx) or ~∃x:xRx), so it holds for the empty set.