Complete sets

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 $\mathcal{C}(N)\gg N\log N$.