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)

Yıl 2019, Sayı: 29, 1 - 31, 30.12.2019

Florentin Smarandache

Öz

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.

Anahtar Kelimeler

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

Kaynakça

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