Abstract: Certain problems concerning state constraints and control constraints can often be reduced to the study of positive invariance of some subsets of the state space of dynamical systems. In this paper, the robust positive invariance of given linear state constraint sets of uncertain discrete-time systems is studied by means of the mixed monotone decomposition method. Necessary and sufficient conditions of the robust positively invariant sets are established for the linear discrete-time system family specified by matrix convex polyhedron and additive interval disturbances. Also, sufficient conditions are derived for nonlinear system family. All the conditions are given in terms of the parameters of vertex systems of the system family and are therefore easy to check. Finally, an illustrating example is given.