Solution

1. Let the preorder be given by this diagram (with implicit reflexive arrows):

   

2. Let \(Q\) be given by the following diagram

   • 

   • and let \(Q'=S\times S\)

3. ?\(Cl(Q) = \{11,12,22,33\}\) \(\sqsubset\) ?\(\leq = \{11,22,33,12,23,13\}\)

4. ?\(Cl(Q') = Q' = S \times S\) \(\not \sqsubset\) ?\(\leq\) (reason: ?\((3,1) \in S \times S\) but \((3,1) \not \in \leq\))