Question 60

Relation Definition:
For distinct subsets $U, V \subseteq S$:

Note that if $m = \min(U \Delta V) \in U$, then $m \notin V$. This defines a strict total linear order (anti-lexicographic order) on the power set $\mathcal{P}(S)$.

Candidate for Smallest Subset (S2):

• Consider the set $S$ itself (the entire set).
• For any other subset $V \subsetneq S$, $S \Delta V = S \setminus V$.
• Since $S \setminus V \subseteq S$, the minimum element of $S \Delta V$ is an element of $S$, which means $\min(S \Delta V) \in S$.
• Therefore, by definition, $S < V$ holds for every subset $V \ne S$.
• Thus, $S$ is smaller than every other subset of $S$. Statement $S2$ is TRUE.

Candidate for Largest Subset (S1):

• Consider the empty set $\emptyset$.
• For any non-empty subset $U \subseteq S$, $U \Delta \emptyset = U$.
• The minimum element of $U$ is obviously an element of $U$, so $\min(U \Delta \emptyset) \in U$.
• By definition, $U < \emptyset$ holds for every non-empty subset $U$.
• This means that every other subset is smaller than $\emptyset$, which equivalently means that $\emptyset$ is larger than every other subset! Statement $S1$ is TRUE.

Conclusion:
Both S1 and S2 are true, corresponding to option (A).