Covering and elimination inequalities are central to combinatorial optimization, yet their role has largely been studied in problem-specific settings or via no-good cuts. This paper introduces a unified perspective that treats these inequalities as primitives for set system approximation in binary integer programs (BIPs). We show that arbitrary set systems admit tight inner and outer monotone approximations, exactly corresponding to covering and elimination inequalities. Building on this, we develop a toolkit that both recovers classical structural correspondences (e.g., paths vs. cuts, spanning trees vs. cycles) and extends polyhedral tools from set covering to general BIPs, including facet conditions and lifting methods. We also propose new reformulation techniques for nonlinear and latent monotone systems, such as auxiliary-variable-free bilinear linearization, bimonotone cuts, and interval decompositions. A case study on distributionally robust network site selection illustrates the frameworkâs flexibility and computational benefits. Overall, this unified view clarifies inner/outer approximation criteria, extends classical polyhedral analysis, and provides broadly applicable reformulation strategies for nonlinear BIPs.