On Sum-Distinct Elements of a Power Set

We consider a subset-sum problem in \((2^\mathcal{S}, \cup)\), \((2^\mathcal{S}, \Delta)\), \((2^\mathcal{S}, \uplus)\), and \((\mathcal{S}_n, +)\), where \(S\) is an \(n\)-element set, \(\mathcal{S} \triangleq \{0,1,2,\ldots,2^n-1\}\), and \(\cup\), \(\Delta\), \(\uplus\), and \(+\) stand for set-union, symmetric set-difference, multiset-union, and real-number addition, respectively. Simple relationships between compatible pairs of sum-distinct sets in these structures are established. The behavior of a sequence \(\{n^{-1} |\mathcal{Z}| = 2, 3, \ldots\}\), where \(\mathcal{Z}\) is the maximum cardinality sum-distinct subset of \(\mathcal{S}\) (or \(\mathcal{S}_n\)), is described in each of the four structures.