Uncertain Over-Under-Off-Set/Logic/Measure/Probability/Statistics

        The uncertain Set was extended by F. Smarandache in 2007 to:

- uncertain OverSet (when some  component is > 1), since he observed that, for example, an employee working overtime deserves a degree of membership > 1, with respect to an employee that only works regular full-time and whose degree of membership = 1;
- uncertain UnderSet (when some neutrosophic component is < 0), since, for example, an employee making more damage than benefit to his company deserves a degree of membership < 0, with respect to an employee that produces benefit to the company and has the degree of membership > 0;
- and to uncertain OffSet (when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and some neutrosophic component < 0).

Then, similarly, the uncertain Logic/Measure/Probability/Statistics etc. he extended to respectively uncertain Over-/Under-/Off- Logic / Measure / Probability / Statistics etc.

By "uncertain" we mean all types of fuzzy and fuzzy-extensions (intuitionistic fuzzy, spherical fuzzy, neutrosophic, refined neutrosophic, MultiNeutrosophic, plithogenic, etc.) Set/Logic/Measure/Probability/Statistics.    
   

     Applications in:
     Artificial Intelligence, Information Systems, Computer Science, Cybernetics, Theory Methods, Mathematical Algebraic Structures, Applied Mathematics, Automation, Control Systems, Big Data, Engineering, Electrical, Electronic, Philosophy, Social Science, Psychology, Biology, Biomedical, Genetics, Engineering, Medical Informatics, Operational Research, Management Science, Imaging Science, Photographic Technology, Instruments, Instrumentation, Physics, Optics, Economics, Mechanics, Neurosciences, Radiology Nuclear, Medicine, Medical Imaging, Interdisciplinary Applications, Multidisciplinary Sciences etc. [ Xindong Peng and Jingguo Dai, A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017, Artificial Intelligence Review, Springer, 18 August 2018; https://fs.unm.edu/BibliometricNeutrosophy.pdf ]

    References

[1] Florentin Smarandache, A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics, ProQuest Info & Learning, Ann Arbor, MI, USA, pp. 92-93, 2007, http://fs.gallup.unm.edu/eBook-Neutrosophics6.pdf ; first edition reviewed in Zentralblatt für Mathematik (Berlin, Germany): https://zbmath.org/?q=an:01273000 .

[2] Florentin Smarandache Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset Similarly for Neutrosophic Over-/Under-/OffLogic, Probability, and Statistics, book, 170 pages, Pons Editions, Brussels, 2016, https://fs.unm.edu/NeutrosophicOversetUndersetOffset.pdf   

[3 ] Florentin Smarandache, Operators on Single-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets, Journal of Mathematics and Informatics, Vol. 5, 2016, 63-67, Published online 29 June 2016, https://fs.unm.edu/SVNeutrosophicOverset-JMI.pdf  

[4] Florentin Smarandache, Interval-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets, International Journal of Science and Engineering Investigations vol. 5, issue 54, July 2016, https://fs.unm.edu/IV-Neutrosophic-Overset-Underset-Offset.pdf

[5] Florentin Smarandache, Degrees of Membership > 1 and < 0 of the Elements with Respect to a Neutrosophic OffSet, Neutrosophic Sets and Systems, Vol. 12, 3-8, 2016, https://fs.unm.edu/NSS/DegreesOf-Over-Under-Off-Membership.pdf

[6] Florentin Smarandache, Operators for Uncertain Over/Under/Off-Sets/-Logics/-Probabilities/-Statistics, Neutrosophic Sets and Systems, Vol. 79, 2025, pp. 493-500, DOI: 10.5281/zenodo.14560156, https://fs.unm.edu/NSS/29OperatorsUncertain.pdf

[7] Qiaoling Xu, A Neutrosophic Paradox-Based Assessment Framework with Over/Under/Off-Indeterminacy for Green Transformation in the Industrial Economy of Resource-Based Regions, Neutrosophic Sets and Systems, Vol. 86, 2025, pp. 590-598, DOI: 10.5281/zenodo.15514444, https://fs.unm.edu/NSS/36Paradox.pdf   

[8] Ting Yu, A Refined Neutrosophic Over/Under Probability Model with Off-Norm Operators for Evaluating the Efficiency of AI-Based Digital Tools in Korean Language Teaching, Neutrosophic Sets and Systems, Vol. 87, 2025, pp. 665-677, DOI: 10.5281/zenodo.15708533,  https://fs.unm.edu/NSS/41RefinedNeutrosophic.pdf  

[9] Takaaki Fujita & Florentin Smarandache (2025). Some Types of HyperNeutrosophic Set: Cubic, Trapozoidal, q-Rung Orthopair, Overset, Underset, and Offset, pp. 193-208; in their book: “Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Fourth volume: HyperUncertain Set (Collected Papers).NSIA Publishing House, Gallup, University of New Mexico, United States of America - and Universidad de Guayaquil, Ecuador, 314 p., 2025. https://fs.unm.edu/ScArt/SomeTypesHyperNeutrosophic4.pdf