This research delves into the transformative realm of neuromorphic computing, inspired by the intricate neural architecture of the human brain. A focal point of this exploration lies in the utilization of memristor crossbars, where recent breakthroughs underscore the exceptional performance potential within the domain of very large-scale integration (VLSI) design. This paper introduces pioneering circuits meticulously designed for on-chip training and the implementation of backpropagation in multi-layered neural networks. Through compelling demonstrations, the study showcases the training of two- and three-layered memristor-based neural networks, tackling diverse tasks ranging from full adder and parity checking to more complex operations like full subtractor, OR operations, and an ALU adept at executing these functions across both two- and three-layered neural network configurations. Employing SPICE simulations with the Technion Israel University developed VTEAM memristor/resistive random access memory (RRAM) model, the proposed approach not only accommodates multiple brain layers and output neurons but also emphasizes its practicality and adaptability. As a part of future endeavors, the research aims to further enrich its findings by comparing the proposed RRAM crossbar array with conventional CMOS circuits, providing a comprehensive understanding of specific performance metrics and efficiency benchmarks. It has been observed that power consumed by the full adder using neuromorphic implementation is less as compared to CMOS implementation while the delay of the full adder using neuromorphic implementation is bit more.