Complete sets

TL;DR

This paper introduces the concept of completeness of sets within integers, explores its properties under various operations, and provides bounds on counting complete subsets in integer intervals.

Contribution

It defines the notion of complete sets, studies their properties, and establishes a lower bound for the number of such sets within integer intervals.

Findings

Established the concept of completeness for sets of integers.

Analyzed how completeness is preserved under set operations.

Derived a lower bound for the count of complete subsets, showing (N) \u226a N \u2217 log N.

Abstract

In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem of counting the number of complete subsets of any given set. That is, given any interval of integers $\mathcal{H}:=[1,N]$ and letting $\mathcal{C}(N)$ denotes the complete set counting function, we establish the lower bound $\mathcal{C}(N)\gg N\log N$.