Algebraic Properties in Hybrid Cosine-Jaccard Measure for Neutrosophic Set

Abstract

The recent hybrid cosine-Jaccard measure for neutrosophic sets proposed to handle uncertainty, indeterminacy, and inconsistency in the complex data more effectively. The three main properties: boundedness, symmetry, and reflexivity have been established; therefore, algebraic characteristics should be explored in depth for the establishment of advanced computational frameworks. This paper is the extension of the theoretical foundation by introducing monotonicity and a distributive-like property over weighted combinations. Mathematical proof is provided for each property, followed by numerical examples to demonstrate their validity and computational consistency. This research establishes a theoretical framework for a hybrid cosine-Jaccard measure for neutrosophic sets, facilitating its facilitating reliable implementation in intricate algebraic operations related to neutrosophic multi-criteria decision-making, clustering algorithms, and information retrieval systems.