Prove that the upper sets on a discrete preorder for some set \(X\) is simply the power set \(P(X)\)
The upper set criterion is satisfied by any subset, thus all possible subsets are upper sets.
The binary relation is equality, thus the upper subset criterion becomes \(p \in U \land p = q \implies q \in U\) or alternatively \(p \in U \implies p \in U\) which is always satisfied.