In this research study, the importance of set theory and logic as the building blocks of artificial intelligence (AI) that can be explained and the representation of knowledge are examined. We render AI more understandable through formalization of knowledge with the help of set-theoretical and logical schemes. The research examines the roles played by the use of the space of vectors, orthogonal sets and sequences in a manner that streamlines knowledge bases in a manner that is mathematically sound. It further demonstrates the importance of logical reasoning methods in making decision-making processes of AI systems transparent, which is significant standards of explainability. By integrating AI models in such a manner, they will be simpler to comprehend, more credible, and will address complex issues more efficiently.