Mobile edge computing (MEC) which enables communication, computation and storage capabilities at the network edge, is an emerging paradigm for 5G and beyond 5G networks. Resource-limited mobile devices can offload their tasks to the MEC server to reduce computation latency and save energy consumption. However, limited wireless spectrum resources make it hard for MEC to guarantee low latency when a large number of users offload the computation tasks or download the computed results. Full-duplex (FD) communication that allows simultaneous transmission and reception of signals over the same frequency band is a promising solution to the shortage of spectrum resources. In this paper, we study a multi-user FD-MEC system involving all the phases of uploading, computing, and downloading. With the help of an FD-capable base station (BS), two half-duplex (HD) users can form an FD pair and share the same time-frequency resource for task uploading and result downloading. We formulate a latency minimization problem considering power control, time scheduling, offloading decision, and user pairing strategy, which is a mixed-integer non-linear programming (MINLP) problem. The problem can be decomposed into two subproblems. The first subproblem is delay minimization for each possible FD pair, and the second one is to decide the offloading and user pairing strategy, and the user scheduling order. We analyze the problem in two cases: (1) the computing delay at the edge server is negligible and (2) the computing delay at the edge server is non-negligible. For both cases, the first subproblem is solved using a bisection algorithm. Then, we solve the second subproblem in the two cases by a relaxing-based algorithm and a concave-convex procedure (CCCP)-based algorithm, respectively. Finally, numerical results demonstrate that with our proposed strategies, the overall latency can be significantly reduced by integrating FD communication with the MEC system.