Based on the position and orientation characteristic (POC) set and the POC equations for serial mechanisms and parallel mechanisms proposed by authors, this paper presents a novel general degree of freedom (DOF) formula which is totally different from approaches based on the screw theory and the displacement group. It can be used to determine the full-cycle DOF of parallel mechanisms (PMs) and multiloop spatial mechanisms using symbolic “union” and “intersection” operations for POC sets. These operations involve only several rules and only simple mathematical tools (vector algebra, theory of sets, etc.) are used. Furthermore, criteria for determination of the inactive joints and selection of the actuating joints are proposed. The presented approach is illustrated with several examples.