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?