To prove that it's impossible to have a set B of 5 distinct positive single-digit integers where every nonempty subset has a different sum, the discussion highlights the application of the pigeonhole principle. With 5 distinct integers, there are 31 possible nonempty subset sums. However, the maximum sum achievable with the largest 4 distinct single-digit integers (9, 8, 7, 6) is 28. This indicates that there are more sums than possible distinct values, leading to the conclusion that at least two subsets must share a sum. The approach using the pigeonhole principle effectively demonstrates the impossibility of such a set existing.
