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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.

Brian Tung
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Phoebus Apollo
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  • The empty relation has all properties you wish, since you can't find a counter-example. – Bernard Sep 15 '17 at 17:40
  • @Bernard: True, but the relation is not empty in this case. OP still must show that the relation is transitive; transitivity is not monotonic. – Brian Tung Sep 15 '17 at 18:25

2 Answers2

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

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

Kenny Lau
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  • Thanks for your reply. I'm still not quite clear about it, because if, for example, we take France as x and Austria as y, there is no xRy? (sorry I am very new to logic) – Phoebus Apollo Sep 15 '17 at 18:00
  • @PhoebusApollo so the LHS of the $\implies$ will be false, which makes the whole thing true (false implies anything) – Kenny Lau Sep 15 '17 at 18:01
  • Okay, thanks! I think I understand. Just to check my understanding - if the relation was {(France, France), (Italy, Italy), (Austria, Austria)} it would be reflexive on the set {France, Italy, Austria}, right? – Phoebus Apollo Sep 15 '17 at 18:29
  • Yes.$ $$ $$ $$ $ – Kenny Lau Sep 15 '17 at 18:29