This article proposes a novel robust state estimation method for discrete linear time-invariant systems by using the positive system theory and the set theory to overcome the disadvantages of interval observers and set-valued observers to finally achieve an expected balance between computational complexity and robust state estimation conservatism. The proposed method first transforms the system into an equivalent system with its system matrix in the canonical companion form. Then, based on this canonical companion form, we partition the equivalent system into two subsystems, where the first subsystem owns a nonnegative subsystem matrix and the second subsystem has a subsystem matrix including possible negative elements of the system matrix of the equivalent system. In this way, the design of interval observers is divided into two steps. The first step is to design an interval observer for the first subsystem based on the positive system theory. The second step is to design a zonotopic set-valued observer for the second subsystem based on the set theory. Consequently, a so-called set-theoretic interval observer (SIO) of the whole system can be synthesized by integrating the interval estimations and set-valued estimations of the two subsystems together. At the end, a numerical system is used to illustrate the effectiveness of the proposed SIO.