This article presents new methods for computing tight interval enclosures of the reachable sets of discrete-time nonlinear systems subject to bounded uncertainties. These methods are motivated by effective methods for continuous-time systems based on differential inequalities (DI). Modern DI methods, particularly those using refinement techniques based on redundant model equations, have demonstrated very sharp enclosures at low cost for several challenging test cases. However, they rely on key properties of continuous-time systems that do not hold generally in discrete time. Nevertheless, we show that discrete-time analogues of these methods do provide valid enclosures for systems satisfying certain monotonicity conditions. We show that these conditions are always satisfied for systems obtained by forward Euler discretization of continuous-time models with step sizes below a computable upper limit. New refinement algorithms are presented for exploiting redundant model equations in the discrete-time setting. The resulting discrete-time DI methods are compared to existing algorithms using several case studies.