Let R be a relation on a set A that is reflexive and transitive but not symmetric. Let R(x) = {y: xRy}. Does the set a = {R(x): x ∈ A} always form a partition of A?
I really don't know where to start with this one. I know that R(x) is the same as x/R except R is not an equivalence relation.
HINT: What happens if $R$ is the relation $\le$ on $\Bbb Z$, say? Start by finding $R(0)=\{n\in\Bbb Z:0\le n\}$ and $R(1)$.