Uncertainty vs Indeterminacy: A Journey from Fuzziness to Neutrosophy

Michael Gr. Voskoglou^1,

¹Department of Mathematical Sciences, Graduate Technological Educational Institute of Western Greece, Patras, Greece

Cite this paper

Michael Gr. Voskoglou. Uncertainty vs Indeterminacy: A Journey from Fuzziness to Neutrosophy. American Journal of Applied Mathematics and Statistics. 2022; 10(2):65-68. doi: 10.12691/AJAMS-10-2-4

Abstract

The present paper reviews the process that led from Zadeh’s fuzziness to Smarandache’s neutrosophy and discusses the future perspectives of the corresponding theories. Starting from the definitions of fuzzy set and of Atanassov’s intuitionistic fuzzy set, it proceeds to a detailed study of the concept of neutrosophic set, which takes in account the existing in real life indeterminacy. The basic operations of complement, union and intersection between neutrosophic sets are defined and the classical notion of topological space is extended to the notion of neutrosophic topological space. It is further shown that the fundamental concepts of convergence, continuity, compact topological space and Hausdorff topological space can be naturally extended to neutrosophic topological spaces.

Keywords

Fuzzy Set (FS), Fuzzy Logic (FL), Intuitionistic Fuzzy Set (IFS), indeterminacy, Neutrosophic Set (NS), Neutrosophic Topological Space (NTS)

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Kosko, B., Fuzziness Vs Probability, Int. J. of General Systems, 17(2-3), 211-240, 1990.
[2]	Zadeh, L.A., Fuzzy Sets, Information and Control, 8, 338-353, 1965
[3]	Zadeh, L.A., Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern., 3, 28-44, 1973.
[4]	Voskoglou, M.Gr., Generalizations of Fuzzy Sets and Related Theories, in M. Voskoglou (Ed.), An Essential Guide to Fuzzy Systems, 345-352, Nova Publishers, N.Y., 2019.
[5]	Atanassov, K.T., Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 20(1), 87-96, 1986.
[6]	Smarandache, F., Neutrosophy/Neutrosophic probability, set, and logic, Amer. Res. Press, Rehoboth, USA, 1998.
[7]	Klir, G. J. & Folger, T. A., Fuzzy Sets, Uncertainty and Information, Prentice-Hall, London, 1988.
[8]	Kosko, B., Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion: New York, 1993.
[9]	Zadeh, L.A., The Concept of a Linguistic Variable and itsApplication to Approximate Reasoning, Information Science, 8, 199-249, 1975.
[10]	Atanassov, K.T., Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, , N.Y, 1999.
[11]	Smarandache, F., Indeterminacy in Neutrosophic Theories and their Applications, International Journal of Neutrosophic Science, 15(2), 89-97, 2021.
[12]	Cuong, B., Picture Fuzzy Sets, J. of Computer Science and Cybernetics, 20(4), 409-420, 2014.
[13]	Deng, J., Control Problems of Grey Systems, Systems and Control Letters, 288-294, 1982.
[14]	Pawlak, Z., Rough Sets: Aspects of Reasoning about Data, Kluer Academic Publishers, Dordrecht, 1991.
[15]	Molodtsov, D., Soft Set Theory-First Results, Computers and Mathematics with Applications, 37(4-5), 19-31, 1999.
[16]	Voskoglou, M.Gr., A Combined Use of Soft Sets and Grey Numbers in Decision Making, Journal of Computational and Cognitive Engineering, 2022.
[17]	Wang, H., Smarandanche, F., Zhang, Y., Sunderraman, R., Single Valued Neutrosophic Sets, Review of the Air Force Academy (Brasov), 1(16), 10-14, 2010.
[18]	Willard, S., General Topology, Dover Publ. Inc., N.Y., 2004.
[19]	Chang, S.L., Fuzzy Topological Spaces, Journal of Mathematical Analysis and Applications, 24(1), 182-190, 1968.
[20]	Luplanlez, F.G, On intuitionistic fuzzy topological spaces, Kybernetes, 35(5), 743-747, 2006.
[21]	Shabir, M. & Naz M., On Soft Topological Spaces, Computers and Mathematics with Applications, 61, 1786-1799, 2011.
[22]	Salama, A.A., Alblowi, S.A., Neutrosophic Sets and Neutrosophic Topological Spaces, IOSR Journal of Mathematics, 3(4), 31-35, 2013.