Reconfigurable intelligent surfaces (RIS) have drawn considerable attention recently due to their controllable scattering elements that are able to direct electromagnetic waves into desirable directions. Although RISs share some similarities with relays, the two have fundamental differences impacting their performance. To harness the benefits of both relaying and RISs, a multi-user communication system is proposed in this paper wherein a relay and an RIS cooperate to improve performance in terms of energy efficiency. Using singular value decomposition (SVD), semidefinite programming (SDP), and function approximations, we propose different solutions for optimizing the beamforming matrices at the base-station (BS), the relay, and the phase shifts at the RIS to minimize the total transmit power subject to quality-of-service (QoS) constraints. The problem is solved in different cases when the relay operates in half-duplex and full-duplex modes and when the reflecting elements have continuous and discrete phase shifts. Simulation results are provided to compare the performance of the system with and without the RIS or the relay in both full-duplex and half-duplex modes, under different optimization solutions. Generally, the results show that the system with full-duplex relay and RIS cooperation outperforms all the other scenarios, and the contribution of full-duplex relay is higher than that of the RIS. However, an RIS performs better than a half-duplex relay when the required QoS is high. The results also show that increasing the number of RIS reflecting elements improves performance better in the presence of a relay than in its absence.