Reliability Function and Boolean Ring

Abstract

Reliability functions are fundamental mathematical models in engineering reliability theory, giving the probability that a system or component functions over a given time period. Boolean rings are algebraic structures with addition, multiplication, and complementation operations that follow certain axioms. This paper shows that collections of reliability functions form Boolean rings under natural definitions of the Boolean operators. Although reliability theory and Boolean rings are mature subjects, the formal connection between them has not been fully recognized. Establishing this link allows the extensive body of results from Boolean ring theory to be applied in analyzing reliability functions. The Boolean perspective leads to simplified calculations and new reliability theorems. We provide mathematical background on reliability functions and Boolean rings before presenting the formal proof that reliability functions satisfy Boolean ring properties. Several examples demonstrate how the Boolean viewpoint yields insight into reliability calculations, system structure functions, and other areas. The relationships revealed here unify reliability theory and Boolean algebra, while offering opportunities for further theoretical developments in both fields. This work elucidates the underlying Boolean structure inherent in reliability modeling.