I am going through "Seven Sketches in Compositionality: An Invitation to Applied Category Theory", by Brendan Fong and David I. Spivak and have trouble understanding the concept of upper set preorders. A definition is provided on page 17 as:
an upper set in $P$ is a subset $U$ of $P$ satisfying the condition that if $p \in U$ and $p \leq q$, then $q \in U$.
As I understand it, an upper set can be built from a preorder by taking one element and including all other elements greater or equal to this one.
The application in exercise 1.57 requires building a Hasse diagram from the product preorder of $ \{ a \leq b, a \leq c \}$ and $ \{ 1 \leq 2 \} $. I successfully completed this—answers are provided page 263, so I could check.
However, the next step is the construction of the upper set preorder, and I do not understand the provided result. Notably, why is $ \{ (b, 1), (b, 2), (c, 1), (c, 2) \} $ not a valid upper set, along with $ \{ (a, 2), (b, 1), (b, 2), (c, 2) \} $ and $ \{ (a, 2), (b, 2), (c, 1), (c, 2) \} $?
{ (b, 1), (b, 2), (c, 1), (c, 2) },
{ (b, 2), (c, 1), (c, 2) },
{ (b, 1), (b, 2), (c, 2) },
{ (b, 2), (c, 2) }
and the empty set are upper sets.