Constructible Families of Subsets, and Their Properties

Abstract

The focus of this paper is to find properties inherent to constructible families of subsets. From a family of subsets U = {A1,A2,...}, U1 is defined to be U∪ {(Ai)c,Am ∪ An,Am ∩ An : Ai,Am,An ∈ U}. From U1, U2 can be defined as U1 ∪{(Ai)c,Am ∪ An,Am ∩ An : Ai,Am,An ∈U1}. Whether these constructible families can be expanded infinitely or if they terminate at a greatest constructible family for a starting U, certain properties hold in general, and will be proven in this paper. Additionally, this paper will prove that the minimum requirements for a family U to construct the powerset P(X)of a given finite universe X of order n arethatallelements inXbepairwiseseparable in U and that U is a family of order i = dlog2 ne. Further study on constructible families of subsets includes construction of the powerset for infinite sets X, as well as going in the opposite direction of construction. Initial work in these areas is presented. Lastly, potential areas for application of constructible families are discussed.