Yeni Bir Tip Genişletilmiş Esnek Küme İşlemi: Tümleyenli Genişletilmiş Teta İşlemi

Year 2024, Volume: 7 Issue: 1, 62 - 88, 30.06.2024

Aslıhan Sezgin Ahmet Mücahit Demirci

https://doi.org/10.56728/dustad.1476447

Cited By: 1

Abstract

Molodtsov tarafından 1999'da öne sürülen esnek kümeler kavramı, belirsizlikle başa çıkmak için sağlam bir matematiksel temel sağlar. Klasik küme teorisinin aksine, esnek kümeler elemanların parametrelendirilmesine izin verir ve bu da belirsizliğin daha karmaşık bir temsilini sağlar. Esnek küme işlemleri, parametrik verileri içeren problemleri ele almak için yeni yaklaşımlar sunar ve bu nedenle esenek küme teorisinde önemli kavramlardır. Bu çalışmada, mevcut teoriye katkıda bulunmak amacıyla yeni bir esnek küme işlemi olan tümleyenli genişletilmiş teta işlemi tanıtılmıştır. İşlemin özelliklerini kapsamlı bir şekilde analiz edilmiş ve tümleyenli genişletilmiş teta işlemi ile diğer esnek küme işlemleri arasındaki ilişkiyi araştırarak dağılma kuralları elde edilmiştir. Bu, gelecekteki çalışmalarda esnek kümelerin cebirsel yapılarının bu yeni işlemle ilgili daha fazla incelenmesine olanak tanır. Esnek kümelerin cebirsel yapısını esnek küme işlemleri perspektifinden incelemek, uygulamalarının kapsamlı bir şekilde anlaşılmasını sağlamanın yanı sıra esnek kümelerin klasik ve klasik olmayan mantığa nasıl uygulanabileceğini anlama açısından da önemlidir. Bu makale bu kapsamda esnek kümeler konusundaki literatüre katkıda bulunmayı amaçlamaktadır.

Keywords

esnek kümeler, koşullu tümleyenler, esnek küme işlemleri, tümleyenli genişletilmiş esnek küme işlemleri

A New Type of Extended Soft Set Operation: Complementary Extended Theta Operation

Abstract

A thorough mathematical foundation for dealing with uncertainty is provided by the notion of soft sets introduced by Molodtsov in 1999. In contrast to classical set theory, soft sets allow elements to have parametrization, providing a more complex representation of uncertainty. Soft set operations are important concepts in soft set theory, as they provide new approaches to dealing with problems involving parametric data. In this paper, we introduce a new soft set operation which we call “complementary extended theta operation,” to contribute to the existing theory. We thoroughly analyze the properties of the operation and investigate the relationship between the complementary extended theta operation and other soft set operations by obtaining the distribution laws in order to further study the algebraic structures of soft sets with respect to this new operation in the future studies. Since studying the algebraic structure of soft sets from the perspective of soft set operations provides a thorough understanding of their application as well as an appreciation of how soft sets can be applied to classical and non-classical logic, this paper also aims to contribute to the literature of soft sets in this regard.

Keywords

soft sets, conditional complements, soft set operations, complementary extended soft set operations

References

• Akbulut, E. (2024). New Type of Extended Operations of Soft Sets: Complementary Extended Lambda and Applied Sciences, Amasya.
• Ali, M. I., Feng, F., Liu, X., Min, W. K., & Shabir, M. (2009). On some new operations in soft set theory. Computers Mathematics with Applications, 57(9), 1547-1553.
• Ali, M. I., Shabir, M., & Naz, M. (2011). Algebraic structures of soft sets associated with new operations. Computers and Mathematics with Applications, 61(9), 2647–2654.
• Aybek, F. (2024). New Restricted and Extended Soft Set Operations. [Unpublished Master Thesis], Amasya University.
• Çağman, N. (2021). Conditional Complements of Sets and Their Application to Group Theory. Journal of New Results in Science, 10(3), 67-74.
• Çağman, N., Çıtak, F., & Aktaş, H. (2012). Soft int-group and its applications to group theory. Neural Computing and Applications, 2, 151–158.
• Çağman, N., & Enginoğlu, S. (2010). Soft set theory and uni-int decision making. European Journal of Operational Research, 20, 7(2), 848-855.
• Clifford, A. H. (1954). Bands of Semigroups. Proceedings of the American Mathematical Society, 5(3), 499-504. Eren, Ö. F., & Çalışıcı, H. (2019). On some operations of soft sets, The Fourth International Conference on Computational Mathematics and Engineering Sciences (CMES 2019), Antalya.
• Jana, C., Pal, M., Karaaslan, F., & Sezgin, A. (2019). (α, β)-soft intersectional rings and ideals with their applications. New Mathematics and Natural Computation, 15(2), 333–350.
• Mahmood, T., Rehman, Z. U., & Sezgin, A. (2018). Lattice ordered soft near rings. Korean Journal of Mathematics, 26(3), 503-517.
• Maji, P. K, Biswas, R. & Roy, A. R. (2003) Soft set theory. Computers and Mathematics with Applications, 45 (1), 555-562.
• Molodtsov, D. (1999.) Soft set theory-first results. Computers and Mathematics with Applications, 3 (4-5), 19–31.
• Maji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers and Mathematics with Applications, 45, 555–562.
• Muştuoğlu, E., Sezgin, A., & Türk, Z. K. (2016). Some characterizations on soft uni-groups and normal soft uni-groups. International Journal of Computer Applications, 155(10), 1-8.
• Özlü, Ş., & Sezgin, A. (2020). Soft covered ideals in semigroups. Acta Universitatis Sapientiae Mathematica, 12(2), 317-346.
• Pant, S., Dagtoros, K., Kholil, M. I., & Vivas, A. (2024). Matrices: Peculiar determinant property. Optimum Science Journal, 1, 1–7.
• Pei, D., & Miao, D. (2005). From soft sets to information systems. IEEE International Conference on Granular Computing, 2, 617-621.
• Sarıalioğlu, M. (2024). New Type of Extended Operations of Soft Sets: Complementary Extended Intersection, Gamma and Star Operation. [Unpublished Master's Thesis], Amasya University.
• Sezer, A. S. (2014). Certain Characterizations of LA-semigroups by soft sets. Journal of Intelligent and Fuzzy Systems, 27(2), 1035-1046.
• Sezer, A. S., Çağman, N., & Atagün, A. O. (2015). Uni-soft substructures of groups. Annals of Fuzzy Mathematics and Informatics, 9(2), 235–246.
• Sezgin, A. (2018). A new view on AG-groupoid theory via soft sets for uncertainty modeling. Filomat, 32(8), 2995–3030.
• Sezgin, A., Ahmad, S., & Mehmood, A. (2019). A new operation on soft sets: Extended difference of soft sets. Journal of New Theory, 27, 33-42.
• Sezgin, A., & Aybek, F. N. (2023). New soft set operation: Complementary soft binary piecewise gamma operation. Matrix Science Mathematic (1), 27-45.
• Sezgin, A., & Aybek, F. N. Atagün, A. O. (2023a). New soft set operation: Complementary soft binary piecewise intersection operation. Black Sea Journal of Engineering and Science, 6(4), 330-346.
• Sezgin, A., Aybek, F. N., & Güngör, N. B. (2023b). New soft set operation: Complementary soft binary piecewise union operation. Acta Informatica Malaysia, (7)1, 38-53.
• Sezgin, A., Atagün, A. O., Çağman, N., & Demir, H. (2022). On near-rings with soft union ideals and applications. New Mathematics and Natural Computation, 18(2), 495-511.
• Sezgin, A., & Atagün, A. O. (2011). On operations of soft sets. Computers and Mathematics with Applications, 61(5), 1457-1467.
• Sezgin, A., & Atagün, A. O. (2023). New soft set operation: Complementary soft binary piecewise plus operation. Matrix Science Mathematic, 7(2), 125-142.
• Sezgin, A., & Çağman, N. (2024). New soft set operation: Complementary soft binary piecewise difference operation. Osmaniye Korkut Ata University Journal of the Institute of Science and Technology, 7(1), 58-94.
• Sezgin, A., Çağman, N., & Atagün, A. O. (2017). A completely new view to soft intersection rings via soft uni-int product. Applied Soft Computing, 54, 366-392.
• Sezgin, A & Çalışıcı, H. (2024). A comprehensive study on soft binary piecewise difference operation. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler, 12(1), 32-54.
• Sezgin, A., & Dagtoros, K. (2023). Complementary soft binary piecewise symmetric difference operation: A novel soft set operation. Scientific Journal of Mehmet Akif Ersoy University, 6(2), 31-45. Sezgin, A., & Demirci, A. M. (2023). New soft set operation: complementary soft binary piecewise star operation. Ikonion Journal of Mathematics, 5(2), 24-52
• Sezgin, A., & Sarıalioğlu, M. (2024). A new soft set operation complementary soft binary piecewise theta operation. Journal of Kadirli Faculty of Applied Sciences, 4(1), 1-33.
• Sezgin, A., & Yavuz, E. (2023a). New soft set operation: Complementary soft binary piecewise lambda operation. Sinop University Journal of Natural Sciences, 8(2), 101-133.
• Sezgin, A., & Yavuz, E. (2023b). A new soft set operation: Soft binary piecewise symmetric difference operation. Necmettin Erbakan University Journal of Science and Engineering, 5(2), 189-208.
• Stojanovic, N. S. (2021). A new operation on soft sets: Extended symmetric difference of soft sets. Military Technical Courier, 69(4), 779-791.
• Yavuz E., (2024). Soft Binary Piecewise Operations and Their Properties, [Unpublished Master’s Thesis], Amasya University.