Is (S, R) a poset if S is the set of all people in the world and , where a and b are people if a = b or a is an ancestor of b?
I know that I have to determine if this is reflexive, antisymmetric and transitive.
So I start with reflexivity and imagine that I have two people, I can either pick two people who are identical (a = b) or I can pick a person a who is an ancestor of a person b.
If I pick two who are identical then it is trivially reflexive, but if I choose a person a who is an ancestor of b, then it is not reflexive, as obviously a cannot be ancestor of him or herself.
Thus it is not reflexive and is not a poset either.
Yet, I see that in the solution for this problem that it is indeed reflexive, how is that possible?
I did but I think I fail to understand something really fundamental; is it sufficient that only some of the people maintain the a = b property for R to be reflexive? If not, how does the definition ensure that they are all equal when the very same definition states that they might not be (i.e. because of the 'or')?
Let 'Rab' stand for '<a b> in R'.
Let 'Cab' stand for 'a is an ancestor of b'.
The problem, as you stated it, is a bit deceptive.
Ordinarily, we would think we are given:
(1) Rab if and only if (a=b or Cab)
But the problem, as you stated it, actually gives only:
(2) If a=b or Cab, then Rab.
Now we have to check:
Reflexive: Is it the case that, for any a, we have Raa?
For (1) or (2) it's easy.
Antisymmetric: Is it the case that, for any a and b, we have 'if Rab and Rba, then a=b'?
With (1) it's easy. But with (2) we get a different answer.
Transitive: Is it the case that, for any a, b, and c, we have 'Rab and Rbc, then Rac'?
With (1) it's easy. But with (2) we get a different answer.
Ah, I see ... however, I thought I finally understood this but even when it means that a and b should only be related, I still don't see how this can be true.
There are two cases, if a = a, then they are obviously related, no problem.
But the case a = a does not necessarily hold for every element in the set? For at least some of them there is the possibility that a is the ancestor of b, and for this particular a, how does the reflexive property hold? Doesn't it have to hold for every element in the set?
You're mixed up about the tests for the relation.
Go back to the precise tests (here, a, b, and c are assumed to be people):
Reflexive: Is it true that, for any a, we have Raa?
That is, is every person either him(her)self or an ancestor of him(her)self?
Antisymmetric: Is it true that, for any a and b, we have 'if Rab and Rba, then a=b'?
That is, is it true that if either a and b are ancestors of each other or a and b are the same person, then a and b are the same person?
Transitive: Is it the true that, for any a, b, and c, we have 'Rab and Rbc, then Rac'?
That is, it true that if a, b, and c are the same person or a is an ancestor of b is an ancestor of c then either a and c are the same person or a is an ancestor of c?
That's all there is to it.