Volume 27 , Issue 2 , PP: 511-541, 2026 | Cite this article as | XML | Html | PDF | Full Length Article
Adel Mohammed Al-Odhari 1 *
Doi: https://doi.org/10.54216/IJNS.270242
This article presents a focused investigation of type-1 neutrosophic sets, derived from classical sets by introducing an indeterminacy component, I. type-1 neutrosophic sets generalize classical set theory by incorporating four-valued logic, which was generated by Boolean logic in our work. This work will appear in the future. As we know, a neutrosophic set is based on a many-valued logic defined by three independent membership functions: truth, indeterminacy, and falsehood. This work systematically re-examines and consolidates foundational research conducted between 2024 and 2025, isolating type-1 structures from the broader frameworks of type-2 and type-3 neutrosophic sets for clearer axiomatic and theoretical development. We establish core concepts, terminology, operations, and properties specific to type-1 neutrosophic sets, constructing and analyzing the type-1 neutrosophic Cartesian product. In addition, we introduce and investigate the properties of type-1 neutrosophic ordered pairs and their corresponding products. This foundation formally defines type-1 neutrosophic relations and neutrosophic partially ordered relations, establishing their core properties. Furthermore, the article explores type-1 neutrosophic functions, detailing their various types, including injective,surjective, and bijective functions and their respective properties. A significant focus is placed on invertible neutrosophic functions, where we examine the conditions for invertibility and prove key related theorems.By focusing exclusively on type-1, we aim to create a more dynamic and effective foundation for application across diverse neutrosophic fields, including neutrosophic algebra, number theory, and logic. This focused approach is intended to open new research pathways within the neutrosophic science.
Type-1 Neutrosophic Set and their properties , Operations on Type-1 Neutrosophic Set and Their Properties , Cartesian Product of Type-1 , Neutrosophic Relations of Type-1 , Neutrosophic Functions of Type-1 , Invertible Netrosophic of Type-1
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