Why is this a transitive relation?

Question

The relation is:

{(France, Italy), (Italy, Austria), (France, France), (Italy, Italy), (Austria, Austria)}

I don't understand how it is transitive on the set {France, Austria} - surely you would need the ordered pair (France, Austria) (or (Austria, France)) for it to be transitive on this set?

I would be grateful if someone could explain this.

Answer 1

It's important to note that the transitive property holds so long as there are no counterexamples. Among other things, that means that the empty relation is transitive, and symmetric for that matter; it only fails to be reflexive on any non-empty set.

In this case, the relation is not empty; it consists of the relation as given, restricted to the set $\{(\text{France}, \text{Austria})\}$: namely,

$$ \{(\text{France}, \text{France}), (\text{Austria}, \text{Austria})\} $$

As you can verify for yourself, there are no counterexamples: The absence of $(\text{France}, \text{Austria})$ is not a problem because there is no $x$ for which $(\text{France}, x)$ and $(x, \text{Austria})$ are both in the relation. Similarly for $(\text{Austria}, \text{France})$.

Answer 2

Because $\forall x,y,z \in \{\text{France},\text{Austria}\}: x R y \land y R z \implies x R z$.