Research Article
Year 2024,
Volume: 8 Issue: 1, 23 - 30, 30.06.2024
Abstract
In this article, we reminded the properties of crisp sets, the definition of fuzzy sets and some properties of fuzzy sets. We gave a new Minkowski difference definition for fuzzy sets by adapting the geometry of the Minkowski difference concept for crisp sets. We proved some algebraic properties of this new fuzzy Minkowski difference. At the end of the article, we presented some illustrative geometric examples on the calculation of fuzzy Minkowski difference.
References
- Zadeh LA. "Fuzzy sets." Information and control, 1965,8(3): 338-353.
- Dubois D, Ostasiewicz W and Prade H. "Fuzzy sets: history and basic notions." Fundamentals of fuzzy sets. Boston, MA: Springer US, 2000;21-124.
- Demirhan A, Kılıç YA & İnan G. Tıpta yapay zeka uygulamaları. (2010).
- Dehghan OR. "Convex Fuzzy Cones of Hypervector Spaces.", 2021.
- Kon M. "A scalarization method for fuzzy set optimization problems." Fuzzy Optimization and Decision Making, 2020;(19):135-152.
- Takeda A, Mitsugi H and Kanamori T. "A unified classification model based on robust optimization." Neural Computation 25(3); 2013: 759-804.
- Barki H. et al. "Contributing vertices-based Minkowski difference (CVMD) of polyhedra and applications." 3D Research 4(4);2013:1-16.
- Cameron S. "Enhancing GJK: Computing minimum and penetration distances between convex polyhedra." Proceedings of international conference on robotics and automation. Vol. 4. IEEE, 1997.
- Ericson C. Real-time collision detection. CRC Press, 2004.
- Ghosh PK. "A unified computational framework for Minkowski operations." Computers & Graphics 1993,17(4):357-378.
- KARAMAN E. "Gömme Fonksiyonu Kullanılarak Küme Optimizasyonuna Göre Verilen Küme Değerli Optimizasyon Problemlerinin Optimallik Koşulları." Süleyman Demirel University Faculty of Arts and Science Journal of Science 2019;14(1):105-111.
- Karaman E. Küme değerli dönüşümlerin optimizasyon yöntemleri. Diss. Anadolu University, Turkey, 2018.
- KLIR, George J and Bo Y. "Fuzzy sets and fuzzy logic: theory and applications." Possibility Theory versus Probab. Theory, 1996;32(2):207-208.
- Kon M. "Operation and ordering of fuzzy sets, and fuzzy set-valued convex mappings." Journal of fuzzy set valued Analysis, 2014;(2014):1-17.
- Chalco-Cano Y, Silva GN and Rufián-Lizana A. "On the Newton method for solving fuzzy optimization problems." Fuzzy Sets and Systems, 2015(272):60-69.
- Brown JG. "A note on fuzzy sets." Information and control, 1971;18(1):32-39.
- Şeker S. Kümelerin cebirsel toplamı, Minkowski farkı ve uygulamaları üzerine (2014). (Master's thesis, Anadolu University (Turkey).
Year 2024,
Volume: 8 Issue: 1, 23 - 30, 30.06.2024
Abstract
Bu çalışmada, katı(crisp) kümelerin özellikleri, bulanık(fuzzy) küme tanımı ve bulanık küme özelliklerini hatırlattık. Katı kümeler için Minkowski(Pontryagin) fark tanımını göz önüne alarak bulanık kümeler için yeni bir Minkowski fark tanımı verdik. Tanımlanan bu yeni bulanık Minkowski farkın bazı özelliklerinin sağladığını gösterdik. Daha sonra ise tanımlamış olduğumuz bulanık Minkowski farkın hesaplamaları üzerine bazı açıklayıcı geometrik örnekler sunduk.
References
- Zadeh LA. "Fuzzy sets." Information and control, 1965,8(3): 338-353.
- Dubois D, Ostasiewicz W and Prade H. "Fuzzy sets: history and basic notions." Fundamentals of fuzzy sets. Boston, MA: Springer US, 2000;21-124.
- Demirhan A, Kılıç YA & İnan G. Tıpta yapay zeka uygulamaları. (2010).
- Dehghan OR. "Convex Fuzzy Cones of Hypervector Spaces.", 2021.
- Kon M. "A scalarization method for fuzzy set optimization problems." Fuzzy Optimization and Decision Making, 2020;(19):135-152.
- Takeda A, Mitsugi H and Kanamori T. "A unified classification model based on robust optimization." Neural Computation 25(3); 2013: 759-804.
- Barki H. et al. "Contributing vertices-based Minkowski difference (CVMD) of polyhedra and applications." 3D Research 4(4);2013:1-16.
- Cameron S. "Enhancing GJK: Computing minimum and penetration distances between convex polyhedra." Proceedings of international conference on robotics and automation. Vol. 4. IEEE, 1997.
- Ericson C. Real-time collision detection. CRC Press, 2004.
- Ghosh PK. "A unified computational framework for Minkowski operations." Computers & Graphics 1993,17(4):357-378.
- KARAMAN E. "Gömme Fonksiyonu Kullanılarak Küme Optimizasyonuna Göre Verilen Küme Değerli Optimizasyon Problemlerinin Optimallik Koşulları." Süleyman Demirel University Faculty of Arts and Science Journal of Science 2019;14(1):105-111.
- Karaman E. Küme değerli dönüşümlerin optimizasyon yöntemleri. Diss. Anadolu University, Turkey, 2018.
- KLIR, George J and Bo Y. "Fuzzy sets and fuzzy logic: theory and applications." Possibility Theory versus Probab. Theory, 1996;32(2):207-208.
- Kon M. "Operation and ordering of fuzzy sets, and fuzzy set-valued convex mappings." Journal of fuzzy set valued Analysis, 2014;(2014):1-17.
- Chalco-Cano Y, Silva GN and Rufián-Lizana A. "On the Newton method for solving fuzzy optimization problems." Fuzzy Sets and Systems, 2015(272):60-69.
- Brown JG. "A note on fuzzy sets." Information and control, 1971;18(1):32-39.
- Şeker S. Kümelerin cebirsel toplamı, Minkowski farkı ve uygulamaları üzerine (2014). (Master's thesis, Anadolu University (Turkey).
There are 17 citations in total.
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Pure Mathematics (Other) |
| Journal Section | Research Article |
| Authors | |
| Publication Date | June 30, 2024 |
| Submission Date | March 21, 2024 |
| Acceptance Date | May 17, 2024 |
| Published in Issue | Year 2024 Volume: 8 Issue: 1 |