This paper addresses the classical problem of determining the set of possible states of a linear discrete time SISO system subject to bounded disturbances, from measurements corrupted by bounded noise. These so-called uncertainty sets evolve with time as new measurements become available. We present two theorems which give a complete description of the relationship between uncertainty sets at two successive time instants, and this yields an efficient algorithm for recursively updating uncertainty sets. Numerical simulations demonstrate performance improvements over existing exact methods. (C) 2018 Elsevier Ltd. All rights reserved.