Knowledge Bank

The purpose of this article is to present a solution to an interesting problem suggested by Richard Little of Baldwin-Wallace College involving "subset-sum-distinctness." Paul Erdős introduced the notion of "subset-sum-distinctness;" however, the focus of this article is instead on the problem and solution rather than Erdős. A set of counting numbers is subset-sum-distinct if and only if there is a one-to-one correspondence between the power set and the set of subset sums. In this article, we establish that no 10-element subset of the first 100 counting numbers is subset-sum-distinct by demonstrating the impossibility of a bijection from the power set of a 10-element subset to the set of sums. In order to prove a bijection between the sets is not possible, it is clearly sufficient to prove that any mapping from the power set to the set of sums is not injective. Lastly, we list several results about subset-sum-distinct sets in instances in which one is trying to find k-element subsets of the first n counting numbers.