A remark on relatively prime sets

TL;DR

This paper simplifies the computation of certain subset-counting functions related to relatively prime sets, showing that only one needs explicit calculation as others follow, and introduces a more general, simpler method.

Contribution

It demonstrates that only one of four related functions needs explicit computation, providing a simpler and more general approach than previous methods.

Findings

Only one subset-counting function needs explicit calculation.

The new method is simpler than existing approaches.

The approach yields more general results.

Abstract

Four functions counting the number of subsets of $\{1, 2, ..., n\}$ having particular properties are defined by Nathanson and generalized by many authors. They derive explicit formulas for all four functions. In this paper, we point out that we need to compute only one of them as the others will follow as a consequence. Moreover, our method is simpler and leads to more general results than those in the literature.