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In this paper, a novel neuro-fuzzy computing system is proposed where its learning is based on the creation of fuzzy relations by using a new implication method without utilizing any exact mathematical techniques. Then, a simple memristor crossbar-based analog circuit is designed to implement this neuro-fuzzy system which offers very interesting properties. In addition to high connectivity between neurons and being fault tolerant, all synaptic weights in our proposed method are always non-negative, and there is no need to adjust them precisely. Finally, this structure is hierarchically expandable, and it can do fuzzy operations in real time since it is implemented through analog circuits. Simulation results confirm the efficiency and applicability of our neuro-fuzzy computing system. They also indicate that this system can be a good candidate to be used for creating artificial brain.