Journal of New Theory (Dec 2019)
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, weprove 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 theneutrosophic aggregation operators, one gets a different result than applyingthe intuitionistic fuzzy operators, since the intuitionistic fuzzy operatorsignore the indeterminacy, while the neutrosophic aggregation operators takeinto consideration the indeterminacy at the same level as truth-membership andfalsehood-nonmembership are taken. NS is also more flexible and effectivebecause it handles, besides independent components, also partially independentand partially dependent components, while IFS cannot deal with these. Sincethere are many types of indeterminacies in our world, we can constructdifferent approaches to various neutrosophic concepts. Neutrosophic Set (NS) is a generalisation of Inconsistent Intuitionistic Fuzzy Set (IIFS) -which isequivalent to the Picture Fuzzy Set (PFS) and Ternary Fuzzy Set (TFS) -,Pythagorean Fuzzy Set (PyFS), Spherical Fuzzy Set (SFS), and q-Rung OrthopairFuzzy Set (q-ROFS). Moreover, all these sets are more general thanIntuitionistic Fuzzy Set. We prove that Atanassov’s Intuitionistic Fuzzy Set ofthe second type (IFS2), and Spherical Fuzzy Sets (SFS) do not have independentcomponents. Furthermore, we show that Spherical Neutrosophic Set (SNS) andn-Hyper Spherical Neutrosophic Set (n-HSNS) are generalisations of IFS2 andSFS. The main distinction between Neutrosophic Set (NS) and all previous settheories are a) the independence of allthree neutrosophic components - truth-membership (T),indeterminacy-membership (I), falsehood-nonmembership (F) - concerning each other in NS – while in the previous set theories theircomponents are dependent on each other, and b) the importance of indeterminacyin NS - while in previous set theories indeterminacy is entirely or partiallyignored. Also, Regret Theory, Grey System Theory, and Three-Ways Decision areparticular cases of Neutrosophication and Neutrosophic Probability. We nowextend the Three-Ways Decision to n-Ways Decision.
