A Lower Bound for Sets with Distinct Subset Sums

A finite set $A \\subset \\mathbb{N}$ has \\emph{distinct subset sums} if no two of its subsets share a sum. We give the elementary proof that such a set satisfies $\\sum_{a \\in A} a \\ge 2^{|A|} - 1$, hence $\\sum_{a \\in A} a \\ge \\tfrac12 \\cdot 2^{|A|}$: the subset-sum map is injective on the $2^{|A|}$ subsets of $A$, and every value it takes lies between $0$ and $\\sum_{a \\in A} a$. The bound is tight, achieved by the powers of two. It is also, by a wide margin, not the bound Erd\\H{o}s conjectured. This paper proves the counting bound and states precisely where it stops.