NEW OPERATIONS ON PYTHAGOREAN NEUTROSOPHIC FUZZY SETS

Pythagorean Neutrosophic Fuzzy Sets (PNFS) as a significant breakthrough in handling uncertainty and indeterminacy, offering a comprehensive framework that synthesizes the strengths of neutrosophic sets and Pythagorean fuzzy sets. This study meticulously investigates fundamental set operations within PNFS, encompassing Additive, Product, Scalar Product, Scalar Power and Operation @, intricately tailored to accommodate the unique characteristics of PNFS, capturing degrees of truth, indeterminacy, and falsity associated with Pythagorean Fuzzy environment. The paper introduces novel operations explicitly designed for PNFS, including Scalar Power and Operation @, thereby expanding the toolkit for managing uncertainty within mathematical frameworks. A robust foundation is laid through meticulous presentations of mathematical formulations and properties of PNFS operations, covering aspects like commutativity, idempotency, absorption law, associativity, De Morgan's rules, and distributivity over complement. This contributes significantly to the theoretical underpinning of PNFS. The efficacy of the proposed operations is demonstrated through illustrative examples, showcasing their practical utility in navigating complex and ambiguous information. This positions PNFS as a valuable tool in decision-making, pattern recognition, and other domains where uncertainty is a critical factor.The study makes a substantial contribution to the dynamic field of neutrosophic and fuzzy set theories by providing a versatile framework for managing uncertainty. PNFS's adaptability renders it applicable to a diverse range of real-world scenarios, facilitating the seamless integration of advanced mathematical concepts into practical applications. In conclusion, this exploration of Pythagorean Neutrosophic Fuzzy Sets not only advances theoretical understanding but also offers practical solutions for addressing complexity in real-world applications. The proposed operations represent a valuable contribution to the broader scientific and engineering community, fostering innovative approaches to comprehensively manage uncertainty across various contexts.