The question is, "Is $(S,R)$ a poset if $S$ is the set of all people in the world and $(a, b)∈R$, where a and b are people, if
a) a is taller than b?
b)a is not taller than b?
c) $a=b$ or a is an ancestor of b?
d) a and b have a common friend?"
The only ones that I am having trouble with are c) and d).
For c, do both conditions have to be meet, or only one?
For d, I can see how it is reflexive; and I can see that isn't antisymmetric. But is it transitive? To me, it would seem like it wasn't, because both a and b could have a different group of friends, the intersection of the groups being the null set.
I would appreciate the help. Thank you!
In $(c)$ You have a disjunctive statement (OR), which holds if either, or both, parts hold.
For $(d)$ Let's say $a \sim b$. That is, Ann ($a$) is friends with Bob ($b$). Can there be another person $c =$ Curt who is friends with Bob, but not with Ann? That is can it be the case that $a\sim b$ AND $b\sim c$ but $a \not\sim c$?
What does that tell you about whether relation $(d)$ is transitive?