Homological Invariants of Boolean Set Operations

This paper explores the intricate relationship between Boolean set operations and algebraic topology, focusing on how homological invariants can quantitatively characterize the topological structure of sets derived from these operations. We investigate the impact of union, intersection, and complementation on Betti numbers and other homology groups. By constructing simplicial complexes from Boolean combinations of sets, we demonstrate how changes in connectivity, cycles, and voids are reflected in these invariants. The study delves into both theoretical underpinnings and practical implications, offering a novel framework for analyzing the complexity and structural properties of composite topological spaces in fields ranging from data analysis to theoretical computer science.