The implementation of two-terminal memristors could decrease the dependency on the transistors in logic circuits, enabling high-density circuits and reducing power consumption due to the memristor's smaller feature size and non-volatility, respectively. We propose universal logic function designs using ZnO-based transparent memristors. A highly accurate fitting of set and reset curves is achieved by employing Verilog model and used for circuit-level simulations. OR, NOR, NAND, and AND gates can be designed by simply tuning the memristor and pullup resistor while using a similar circuit configuration. Based on this facile configuration, more complex circuits, such as XOR and XNOR gates as well as half-adder and half-subtractor combinational logic circuits can be achieved. This work highlights a promising alternative for reconfigurable logical integrated circuits.