On the Structure and Properties of Order (K) and Degree (J) Classes of Sets

Abstract

In this paper, we introduce a new class of sets called (k, j)-picture sets, which arise from studying collections of information subsets within a universe set. Let X be a universe set and consider k subsets S1, S2, …Sk of X. These subsets represent different portions of information contained in the universe. Our focus is on the relationships among these subsets, particularly how they share or do not share information. We assume that S1 is a reference subset used for comparison with the remaining subsets. Among the subsets S2, …Sk there exists at least one subset Sj such that it does not share information with S1. This condition creates a distinct structural configuration among the subsets. Based on this idea, we define the concept of a (k, j)-picture set. The definition captures situations where at least one subset remains independent of S1. Such configurations help in understanding patterns of separation among subsets. We further explore the structural and theoretical properties of these (k, j)-picture sets.