On Sum-Distinct Elements of a Power Set

Abstract

We consider a subset-sum problem in $(2^{\mathcal{S}},\cup )$, $(2^{\mathcal{S}},\mathrm{\Delta })$, $(2^{\mathcal{S}},\uplus )$, and $({\mathcal{S}}_n,+)$, where $S$ is an $n$-element set, $\mathcal{S}\triangleq \{0,1,2,\dots ,2^n-1\}$, and $\cup$, $\mathrm{\Delta }$, $\uplus$, 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 $\{n^{-1}|\mathcal{Z}|=2,3,\dots \}$, 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.