Neutrosophic Set is a Generalization of Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set, Spherical Fuzzy Set, and q-Rung Orthopair Fuzzy Set, while Neutrosophication is a Generalization of Regret Theory, Grey System Theory, and Three-Ways Decision (revisited)

Abstract

In this paper, we prove that Neutrosophic Set (NS) is an extension of Intuitionistic Fuzzy Set (IFS) no matter if the sum of neutrosophic components is <1, or >1, or =1. For the case when the sum of components is 1 (as in IFS), after applying the neutrosophic aggregation operators, one gets a different result than applying the intuitionistic fuzzy operators, since the intuitionistic fuzzy operators ignore the indeterminacy, while the neutrosophic aggregation operators take into consideration the indeterminacy at the same level as truth-membership and falsehood-nonmembership are taken. NS is also more flexible and effective because it handles, besides independent components, also partially independent and partially dependent components, while IFS cannot deal with these. Since there are many types of indeterminacies in our world, we can construct different approaches to various neutrosophic concepts. Neutrosophic Set (NS) is a generalisation of Inconsistent Intuitionistic Fuzzy Set (IIFS) -which is equivalent to the Picture Fuzzy Set (PFS) and Ternary Fuzzy Set (TFS) -, Pythagorean Fuzzy Set (PyFS), Spherical Fuzzy Set (SFS), and q-Rung Orthopair Fuzzy Set (q-ROFS). Moreover, all these sets are more general than Intuitionistic Fuzzy Set. We prove that Atanassov’s Intuitionistic Fuzzy Set of the second type (IFS2), and Spherical Fuzzy Sets (SFS) do not have independent components. Furthermore, we show that Spherical Neutrosophic Set (SNS) and n-Hyper Spherical Neutrosophic Set (n-HSNS) are generalisations of IFS2 and SFS. The main distinction between Neutrosophic Set (NS) and all previous set theories are a) the independence of all three neutrosophic components - truth-membership (T), indeterminacy-membership (I), falsehood-nonmembership (F) - concerning each other in NS – while in the previous set theories their components are dependent on each other, and b) the importance of indeterminacy in NS - while in previous set theories indeterminacy is entirely or partially ignored. Also, Regret Theory, Grey System Theory, and Three-Ways Decision are particular cases of Neutrosophication and Neutrosophic Probability. We now extend the Three-Ways Decision to n-Ways Decision.

Keywords

• Neutrosophic set
• intuitionistic fuzzy set
• Pythagorean fuzzy set
• spherical fuzzy set
• q-rung orthopair fuzzy set

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