In this paper, implication (IMP) operations are demonstrated in a circuit with two SiOx-based memristors and a CMOS transistor. Specifically, a circuit with two one-diode and one-resistor (1D1R) memory elements and a transistor are designed to perform the IMP operations. A circuit consisting of a $4 \times 4$ crossbar 1D1R memristor array together with selection transistors is proposed and used to realize the functionality of a one-bit, full adder in 43 steps. Compared with CMOS logic circuits, the advantages and disadvantages of memristor-enabled logic circuits are discussed. This result suggests that the memristor-enabled logic circuit is most suitable for low-power and high-density applications, as well as a simple and robust approach to realize programmable memcomputing chips compatible with large-scale CMOS manufacturing technology.