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)

Year 2019, Issue: 29, 1 - 31, 30.12.2019

Florentin Smarandache

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

References

• F. Smarandache, Definition of Neutrosophic Logic – A Generalization of the Intuitionistic Fuzzy Logic, Proceedings of the Third Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2003, September 10-12, 2003, Zittau, Germany, University of Applied Sciences at Zittau/Goerlitz, (2003) 141-146.
• F. Smarandache, Neutrosophic Set, A Generalization of the Intuitionistic Fuzzy Set, International Journal of Pure and Applied Mathematics 24(3) (2005) 287-297.
• F. Smarandache, Neutrosophic Set, A Generalization of the Intuitionistic Fuzzy Set, in Proceedings of 2006 IEEE International Conference on Granular Computing, edited by Yan-Qing Zhang and Tsau Young Lin, Georgia State University, Atlanta, (2006) 38-42.
• F. Smarandache, Neutrosophic Set, A Generalization of the Intuitionistic Fuzzy Set, Journal of Defense Resources Management, Brasov, Romania, (1) (2010) 107-116.
• F. Smarandache, A Geometric Interpretation of the Neutrosophic Set – A Generalization of the Intuitionistic Fuzzy Set, 2011 IEEE International Conference on Granular Computing, edited by Tzung-Pei Hong, Yasuo Kudo, Mineichi Kudo, Tsau-Young Lin, Been-Chian Chien, Shyue-Liang Wang, Masahiro Inuiguchi, GuiLong Liu, IEEE Computer Society, National University of Kaohsiung, Taiwan, (2011) 602-606. http://fs.unm.edu/IFS-generalized.pdf
• K. Atanassov, P. Vassilev, Intuitionistic Fuzzy Sets and other Fuzzy Sets Extensions Representable by Them, Journal of Intelligent and Fuzzy Systems DOI: 10.3233/JIFS-179426 (In press).
• F. Smarandache, Neutrosophy, A New Branch of Philosophy, Multiple-Valued Logic 8(3) (2002) 297-384.
• F. Smarandache, Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic Over-/Under-/Off- Logic, Probability, and Statistics, Pons Editions, Bruxelles, Belgique, (2016) 168 p. https://arxiv.org/ftp/arxiv/papers/1607/1607.00234.pdf
• F. Smarandache, n-Valued Refined Neutrosophic Logic and Its Applications in Physics, Progress in Physics 4 (2013) 143-146. https://arxiv.org/ftp/arxiv/papers/1407/1407.1041.pdf
• F. Smarandache, Law of Included Multiple-Middle & Principle of Dynamic Neutrosophic Opposition, by Florentin Smarandache, EuropaNova & Educational, Brussels-Columbus (Belgium-USA), (2014) 136 p. http://fs.unm.edu/LawIncludedMultiple-Middle.pdf
• F. Smarandache, Introduction to Neutrosophic Measure, Neutrosophic Integral and Neutrosophic Probability, Sitech, 2013. http://fs.gallup.unm.edu/NeutrosophicMeasureIntegralProbability.pdf
• F. Smarandache, Introduction to Neutrosophic Statistics, Sitech, 2014. http://fs.gallup.unm.edu/ NeutrosophicStatistics.pdf
• F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, ProQuest Information and Learning, Ann Arbor, Michigan, USA, (1998, 2000, 2002, 2005, 2006) 105 p.
• F. Smarandache, A Unifying Field in Logics: Neutrosophic Logic, Multiple-Valued Logic 8(3) (2002) 385-438. http://fs.unm.edu/eBook-Neutrosophics6.pdf
• F. Smarandache, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics, Proceedings of the First International Conference on Neutrosophy, University of New Mexico, Gallup Campus, Xiquan, Phoenix, 2002 147 p. http://fs.unm.edu/NeutrosophicProceedings.pdf
• K. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems 20 (1986) 87-96.
• K. Atanassov, Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, N.Y., 1999.
• K. Atanassov, Intuitionistic Fuzzy Sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Sci. - Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian); Reprinted: Int J Bioautomation 20(S1) (2016) 1-6.
• C. Hinde, R. Patching, Inconsistent Intuitionistic Fuzzy Sets. Developments in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics 1 (2008) 133-153.
• B. C. Cuong, V. Kreinovich, Picture Fuzzy Sets - a New Concept for Computational Intelligence Problems, Proceedings of the Third World Congress on Information and Communication Technologies WICT’2013, December 15-18, Hanoi, Vietnam, (2013) 1-6.
• C. Wang, M. Ha, X. Liu, A Mathematical Model of Ternary Fuzzy Set for Voting, Journal of Intelligent and Fuzzy Systems 29 (2015) 2381-2386.
• F. Smarandache, Degree of Dependence and Independence of the (Sub) Components of Fuzzy Set and Neutrosophic Set, Neutrosophic Sets and Systems 11 (2016) 95-97. doi.org/10.5281/zenodo.571359, http://fs.unm.edu/NSS/DegreeOfDependenceAndIndependence.pdf
• R. Princy, K. Mohana, Spherical Bipolar Fuzzy Sets and Its Application in Multi Criteria Decision Making Problem, Journal of New Theory, 2019 (In press).
• R. R. Yager, Pythagorean Fuzzy Subsets, Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), Edmonton, AB, Canada, pp. 57-61.
• F. K. Gündoğdu, C. Kahraman, A Novel Spherical Fuzzy QFD Method and Its Application to the Linear Delta Robot Technology Development, Engineering Applications of Artificial Intelligence 87 (2020) 103348.
• A. Guleria, R.K. Bajaj, T-Spherical Fuzzy Graphs: Operations and Applications in Various Selection Processes, Arabian Journal for Science and Engineering. https://doi.org/10.1007/s13369-019-04107-y
• F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras, and Applications, Second extended and improved edition, Pons Publishing House Brussels, 2017, http://fs.unm.edu/NeutrosophicPerspectives-ed2.pdf
• R. R. Yager, Generalized Orthopair Fuzzy Sets, IEEE Trans on Fuzzy Systems 25(5) (2017) 1222-1230.
• H. Bleichrodt, A. Cillo, E. Diecidue, A Quantitative Measurement of Regret Theory, Management Science 56(1) (2010) 161-175.
• J. L. Deng, Introduction to Grey System Theory, The Journal of Grey System 1(1) (1989) 1-24.
• Y. Yao, Three-Way Decision: An Interpretation of Rules in Rough Set Theory, in Proceeding of 4th International Conference on Rough Sets and Knowledge Technology, LNAI, Vol. 5589, Springer Berlin Heidelberg, (2009) 642–649.
• F. Smarandache, Three-Ways Decision is a Particular Case of Neutrosophication, in volume Nidus Idearum. Scilogs, VII: superluminal physics, vol. VII, Pons Ed., Brussels, (2019) 97-102.
• P. K. Singh, Three-Way n-valued Neutrosophic Concept Lattice at Different Granulation, International Journal of Machine Learning and Cybernetics 9(11) (2018) 1839-1855.
• P. K. Singh, Complex Neutrosophic Concept Lattice and Its Applications to Air Quality Analysis, Chaos, Solitons and Fractals, Elsevier, 109 (2018) 206-213.
• P. K. Singh, Interval-Valued Neutrosophic Graph Representation of Concept Lattice and Its (α,β,γ)-Decomposition, Arabian Journal for Science and Engineering 43(2) (2018) 723-740.
• P. K. Singh, Three-Way Fuzzy Concept Lattice Representation using Neutrosophic Set, International Journal of Machine Learning and Cybernetics 8(1) (2017) 69-79.
• F. Smarandache, Neutrosophic Set as Generalization of Intuitionistic Fuzzy Set, Picture Fuzzy Set and Spherical Fuzzy Set, and its Physical Applications, 2019 Joint Fall Meeting of the Texas Sections of American Physical Society (APS), AAPT and Zone 13 of the SPS, Friday–Saturday, October 25–26, 2019; Lubbock, Texas, USA, (2019). http://meetings.aps.org/Meeting/TSF19/scheduling?ukey=1480464-TSF19-otUQvu
• J. Dezert, Open Questions to Neutrosophic Inferences, Multiple-Valued Logic 8(3) (2002) 439-472.
• D. Dubois, S. Gottwald, P. Hajek, H. Prade, Terminological Difficulties in Fuzzy Set Theory – The Case of Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems 156 (2005) 585-491.
• W. B. V. Kandasamy, F. Smarandache, Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, ProQuest Information and Learning, Ann Arbor, Michigan, USA, 2003.
• F. Smarandache, Symbolic Neutrosophic Theory, Europa Nova, Bruxelles, (2015) 194p. https://arxiv.org/ftp/arxiv/papers/1512/1512.00047.pdf
• H. Wang, F. Smarandache, R. Sunderraman, A. Rogatko, Neutrosophic Relational Data Models, Society for Mathematics of Uncertainty, Creighton University, 2 (2008) 19-35.
• F. Smarandache, M. Ali, The Neutrosophic Triplet Group and its Application to Physics, presented by F. S. to Universidad Nacional de Quilmes, Department of Science and Technology, Bernal, Buenos Aires, Argentina, 02 June 2014.
• F. Smarandache, M. Ali, Neutrosophic Triplet Group, Neural Computing and Applications, Springer, 2016 1-7. http://fs.unm.edu/NeutrosophicTriplets.htm
• F. Smarandache, M. Ali, Neutrosophic Triplet as Extension of Matter Plasma, Unmatter Plasma, and Antimatter Plasma, 69th annual gaseous electronics conference, Bochum, Germany, Veranstaltungszentrum & Audimax, Ruhr-Universitat, 10–14 Oct. 2016, http://meetings.aps.org/Meeting/GEC16/Session/HT6.111
• F. Smarandache, Plithogeny, Plithogenic Set, Logic, Probability, and Statistics, Infinite Study Publ. Hse., GoogleLLC, Mountain View, California, USA, (2017) https://arxiv.org/ftp/arxiv/papers/1808/1808.03948.pdf Harvard SAO/NASA ADS: http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:1808. 03948