A product’s architecture affects the ability of a company to customize, assemble, service, and recycle the product. Much of the flexibility to address these issues is locked into the product’s design during the configuration design stage when the architecture is determined. The concepts of modules and modularity are central to the description of an architecture, where a module is a set of components that share some characteristic. Modularity is a measure of the correspondence between the modules of a product from different viewpoints, such as functionality and physical structure. The purpose of this paper is to investigate formal foundations for configuration design. Since product architectures are discrete structures, discrete mathematics, including set theory and combinatorics, is used for the investigation. A Product Module Reasoning System (PMRS) is developed to reason about sets of product architectures, to translate design requirements into constraints on these sets, to compare architecture modules from different viewpoints, and to directly enumerate all feasible modules without generate-and-test or heuristic search approaches. The PMRS is described mathematically and applied to the design of architectures for a hand-held tape recorder. Life cycle requirements are used as design criteria.