Transitivity for set of 2 elements

[autodidude]

If we have a set of two elements, say S={0. 1} and we defined 0 to be less than 1, would this obey the transitivity axiom? (If a<b and b<c then a<c? )

To me, it seems looks like you need at least 3 elements but I'm not entirely sure.

 

[CompuChip]

I would say that it is vacuously true.

 

[autodidude]

Thanks...could you please elaborate a bit on what 'vacuous' means in this context?

EDIT: Nevermind, I wiki'd it and think I understand it. Of course, any additional thoughts would be appreciated.

 

[HallsofIvy]

For others who might be wondering, the implication "if P then Q" is true whenever P is false, whether Q is true or not. That is what is meant by "vacuously true". In this particular problem, because there are only two elements, "a< b and b< c" is never true, therefore the conclusion, that "<" for this set is transitive, is "vacuously true".