Research Article
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A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation

Year 2024, Volume: 8 Issue: 2, 90 - 114, 30.12.2024

Aslıhan Sezgin , Emre Akbulut , Hüseyin Demir

https://doi.org/10.47137/usufedbid.1473425

Abstract

Soft set theory has many theoretical and practical applications. It was first introduced by Molodtsov in 1999 as a way to represent specific situations including uncertainty. The fundamental building blocks of soft set theory are soft set operations. Since its debut, several types of soft set operations have been defined and utilized in diverse contexts. In order to further the theory, a new soft set operation known as the complementary extended difference operation is defined in this paper. Its properties are thoroughly discussed, with particular attention to how it differs from the difference operation in classical sets. Additionally, the distribution of this operation over other types of soft set operations is examined in order to determine how this operation relates to other soft set operations.

Keywords

Soft set Conditional complements Complementary extended soft set operations

Project Number

YOK

References

  • Zadeh LA. Fuzzy set theory, Inf. Control, 1965; 8(3): 338-353.
  • Molodtsov D. Soft set theory—first results, Comput Math Appl, 1999; 37(1): 19-31.
  • Maji PK, Biswas R and Roy AR. Soft set theory, Comput Math Appl, 2003; 45(1): 555-562.
  • Pei D and Miao D. From soft sets to information systems. In: Proceedings of Granular Computing IEEE, 2005; 2: 617-621.
  • Ali MI, Feng F, Liu X, Min WK, Shabir M. On some new operations in soft set theory, Comput Math Appl, 2009; 57(9): 1547-1553.
  • Sezgin A and Atagün AO. On operations of soft sets. Comput Math Appl, 2011; 61(5): 1457-1467.
  • Ali MI, Shabir M and Naz M. Algebraic structures of soft sets associated with new operations, Comput Math Appl 2011; 61: 2647–2654.
  • Sezgin A, Shahzad A and Mehmood A. New operation on soft sets: Extended difference of soft sets. J New Theory, 2019; (27): 33-42.
  • Stojanovic NS. A new operation on soft sets: Extended symmetric difference of soft sets. Military Technical Courier, 2021; 69(4): 779-791.
  • Eren ÖF and Çalışıcı H. On some operations of soft sets, The Fourth International Conference on Computational Mathematics and Engineering Sciences; 2019 Apr 20-22; Antalya, Türkiye.
  • Sezgin A and Çalışıcı H. A comprehensive study on soft binary piecewise difference operation, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler, 2024;12(1): 32-54.
  • Aybek FN. New restricted and extended soft set operations, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Çağman N. Conditional complements of sets and their application to group theory, J New Results Sci, 2021; 10(3): 67-74.
  • Sezgin A, Çağman N, Atagün AO and Aybek FN. Complemental binary operations of sets and their application to group theory, Matrix Science Mathematic, 2023; 7(2): 114-121.
  • Yavuz E. Soft binary piecewise operations and their properties, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Akbulut E. New type of extended operations of soft set: Complementary extended difference and lambda operation, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Demirci AM. A new type of extended operations of soft set: Complementary extended union, plus and theta operation, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Sarıalioğlu M. A new type of extended operations of soft set: Complementary extended intersection, gamma and star operation, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Sezgin A and Atagün AO. A new soft set operation: Complementary soft binary piecewise plus operation, Matrix Science Mathematic, 2023; 7(2): 125-142.
  • Sezgin A and Aybek FN. A new soft set operation: Complementary soft binary piecewise gamma operation, Matrix Science Mathematic, 2023; 7(1): 27-45.
  • Sezgin A, Aybek FN and Güngör NB. A new soft set operation: Complementary soft binary piecewise union operation, Acta Informatica Malaysia, 2023; 7(1): 38-53.
  • Sezgin A, Aybek FN and Atagün AO. A new soft set operation: Complementary soft binary piecewise intersection operation, BSJ Eng Sci, 2023; 6(4): 330-346.
  • Sezgin A and Çağman N. A new soft set operation: Complementary soft binary piecewise difference operation, Osmaniye Korkut Ata Üniv Fen Biliml Derg, 2024; 7(1): 58-94.
  • Sezgin A and Demirci AM. A new soft set operation: Complementary soft binary piecewise star operation, Ikonion Journal of Mathematics, 2023; 5(2): 24-52.
  • Sezgin A and Sarıalioğlu M. A new soft set operation: Complementary soft binary piecewise theta operation, Journal of Kadirli Faculty of Applied Sciences, 2024; 4(2): 325-357.
  • Sezgin A and Yavuz E. A new soft set operation: Complementary Soft Binary Piecewise Lambda Operation, Sinop University Journal of Natural Sciences, 2023; 8(2): 101-133.
  • Sezgin A and Yavuz E. A new soft set operation: Soft binary piecewise symmetric difference operation, Necmettin Erbakan University Journal of Science and Engineering, 2023; 5(2): 189-208.
  • Sezgin A and Dagtoros K. Complementary soft binary piecewise symmetric difference operation: a novel soft set operation, Scientific Journal of Mehmet Akif Ersoy University, 2023; 6(2): 31-45.
  • Çağman N, Çitak F and Aktaş H. Soft int-group and its applications to group theory, Neural Comput Appl, 2012; 2: 151–158.
  • Sezer AS, Çağman N, Atagün AO, Ali MI and Türkmen E. Soft intersection semigroups, ideals and bi-ideals; a new application on semigroup theory I, Filomat, 2015; 29(5): 917-946.
  • Sezer AS, Çağman N and Atagün AO. Soft intersection interior ideals, quasi-ideals and generalized bi-ideals; a new approach to semigroup theory II, J Mult.-Valued Log. Soft Comput, 2014; 23(1-2): 161-207.
  • Sezgin A and Orbay M. Analysis of semigroups with soft intersection ideals, Acta Univ Sapientiae Math, 2022; 14(1): 166-210.
  • Sezgin A. A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals, Algebra Lett, 2016; 2016(3): 1–46.
  • Jana C, Pal M, Karaaslan F and Sezgin A. (α, β)-soft intersectional rings and ideals with their applications, New Math Nat Comput, 2019; 15(2): 333–350.
  • Muştuoğlu E, Sezgin A and Türk ZK. Some characterizations on soft uni-groups and normal soft uni-groups, Int J Comput Appl, 2016; 155(10): 1-8.
  • Sezer AS, Çağman N and Atagün AO. Uni-soft substructures of groups, Ann Fuzzy Math Inform, 2015; 9(2): 235–246.
  • Sezer AS. Certain Characterizations of LA-semigroups by soft sets, J Intell Fuzzy Syst, 2014; 27(2): 1035-1046.
  • Özlü Ş and Sezgin A. Soft covered ideals in semigroups, Acta Univ Sapientiae Math, 2020: 12(2): 317-346.
  • Atagün AO and Sezgin A. Soft subnear-rings, soft ideals and soft n-subgroups of near-rings, Math Sci Letters, 2018; 7(1): 37-42.
  • Sezgin A. A new view on AG-groupoid theory via soft sets for uncertainty modeling, Filomat, 2018; 32(8): 2995–3030.
  • Sezgin A, Çağman N and Atagün AO. A completely new view to soft intersection rings via soft uni-int product, Appl Soft Comput, 2017; 54: 366-392.
  • Sezgin A, Atagün AO and Çağman N and Demir H. On near-rings with soft union ideals and applications, New Math Nat Comput, 2022; 18(2): 495-511.
  • Imai Y and Iseki K. On axiom systems of proposition calculi, Proc Jpn Acad, 1966; 42: 19–22.
  • Pant S, Dagtoros K, Kholil MI and Vivas A. Matrices: Peculiar determinant property, OPS Journal, 2024; 1: 1–7.

Yeni Bir Esnek Küme İşlemi: Tümleyenli Genişletilmiş Fark İşlemi

Year 2024, Volume: 8 Issue: 2, 90 - 114, 30.12.2024

Aslıhan Sezgin , Emre Akbulut , Hüseyin Demir

https://doi.org/10.47137/usufedbid.1473425

Abstract

Esnek küme teorisinin birçok teorik ve pratik uygulaması vardır. İlk kez 1999 yılında Molodtsov tarafından belirsizlik durumlarını temsil etmenin bir yolu olarak tanıtıldı. Esnek küme teorisinin temel yapı taşları esnek küme işlemleridir. İlk çıkışından bu yana, çeşitli bağlamlarda esnek küme işlemlerinin çeşitli türleri tanımlanmış ve kullanılmıştır. Teoriyi ilerletmek amacıyla bu çalışmada tümleyenli genişletilmiş fark işlemi olarak isimlendirilen yeni bir esnek küme işlemi tanımlanmıştır. Özellikleri, klasik kümelerdeki fark işlemi ile kıyaslanarak kapsamlı bir şekilde tartışılmıştır. Ayrıca, bu işlemin diğer esnek küme işlemleri ile nasıl bir ilişkisi olduğunu belirlemek amacıyla bu işlemin diğer esnek küme işlemlerine dağılımı da incelenmiştir.

Keywords

Esnek küme Koşullu tümleyenler Tümleyenli genişletilmiş işlemler

Project Number

YOK

References

  • Zadeh LA. Fuzzy set theory, Inf. Control, 1965; 8(3): 338-353.
  • Molodtsov D. Soft set theory—first results, Comput Math Appl, 1999; 37(1): 19-31.
  • Maji PK, Biswas R and Roy AR. Soft set theory, Comput Math Appl, 2003; 45(1): 555-562.
  • Pei D and Miao D. From soft sets to information systems. In: Proceedings of Granular Computing IEEE, 2005; 2: 617-621.
  • Ali MI, Feng F, Liu X, Min WK, Shabir M. On some new operations in soft set theory, Comput Math Appl, 2009; 57(9): 1547-1553.
  • Sezgin A and Atagün AO. On operations of soft sets. Comput Math Appl, 2011; 61(5): 1457-1467.
  • Ali MI, Shabir M and Naz M. Algebraic structures of soft sets associated with new operations, Comput Math Appl 2011; 61: 2647–2654.
  • Sezgin A, Shahzad A and Mehmood A. New operation on soft sets: Extended difference of soft sets. J New Theory, 2019; (27): 33-42.
  • Stojanovic NS. A new operation on soft sets: Extended symmetric difference of soft sets. Military Technical Courier, 2021; 69(4): 779-791.
  • Eren ÖF and Çalışıcı H. On some operations of soft sets, The Fourth International Conference on Computational Mathematics and Engineering Sciences; 2019 Apr 20-22; Antalya, Türkiye.
  • Sezgin A and Çalışıcı H. A comprehensive study on soft binary piecewise difference operation, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler, 2024;12(1): 32-54.
  • Aybek FN. New restricted and extended soft set operations, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Çağman N. Conditional complements of sets and their application to group theory, J New Results Sci, 2021; 10(3): 67-74.
  • Sezgin A, Çağman N, Atagün AO and Aybek FN. Complemental binary operations of sets and their application to group theory, Matrix Science Mathematic, 2023; 7(2): 114-121.
  • Yavuz E. Soft binary piecewise operations and their properties, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Akbulut E. New type of extended operations of soft set: Complementary extended difference and lambda operation, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Demirci AM. A new type of extended operations of soft set: Complementary extended union, plus and theta operation, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Sarıalioğlu M. A new type of extended operations of soft set: Complementary extended intersection, gamma and star operation, MSc Thesis, Amasya University, Amasya, Türkiye, 2024.
  • Sezgin A and Atagün AO. A new soft set operation: Complementary soft binary piecewise plus operation, Matrix Science Mathematic, 2023; 7(2): 125-142.
  • Sezgin A and Aybek FN. A new soft set operation: Complementary soft binary piecewise gamma operation, Matrix Science Mathematic, 2023; 7(1): 27-45.
  • Sezgin A, Aybek FN and Güngör NB. A new soft set operation: Complementary soft binary piecewise union operation, Acta Informatica Malaysia, 2023; 7(1): 38-53.
  • Sezgin A, Aybek FN and Atagün AO. A new soft set operation: Complementary soft binary piecewise intersection operation, BSJ Eng Sci, 2023; 6(4): 330-346.
  • Sezgin A and Çağman N. A new soft set operation: Complementary soft binary piecewise difference operation, Osmaniye Korkut Ata Üniv Fen Biliml Derg, 2024; 7(1): 58-94.
  • Sezgin A and Demirci AM. A new soft set operation: Complementary soft binary piecewise star operation, Ikonion Journal of Mathematics, 2023; 5(2): 24-52.
  • Sezgin A and Sarıalioğlu M. A new soft set operation: Complementary soft binary piecewise theta operation, Journal of Kadirli Faculty of Applied Sciences, 2024; 4(2): 325-357.
  • Sezgin A and Yavuz E. A new soft set operation: Complementary Soft Binary Piecewise Lambda Operation, Sinop University Journal of Natural Sciences, 2023; 8(2): 101-133.
  • Sezgin A and Yavuz E. A new soft set operation: Soft binary piecewise symmetric difference operation, Necmettin Erbakan University Journal of Science and Engineering, 2023; 5(2): 189-208.
  • Sezgin A and Dagtoros K. Complementary soft binary piecewise symmetric difference operation: a novel soft set operation, Scientific Journal of Mehmet Akif Ersoy University, 2023; 6(2): 31-45.
  • Çağman N, Çitak F and Aktaş H. Soft int-group and its applications to group theory, Neural Comput Appl, 2012; 2: 151–158.
  • Sezer AS, Çağman N, Atagün AO, Ali MI and Türkmen E. Soft intersection semigroups, ideals and bi-ideals; a new application on semigroup theory I, Filomat, 2015; 29(5): 917-946.
  • Sezer AS, Çağman N and Atagün AO. Soft intersection interior ideals, quasi-ideals and generalized bi-ideals; a new approach to semigroup theory II, J Mult.-Valued Log. Soft Comput, 2014; 23(1-2): 161-207.
  • Sezgin A and Orbay M. Analysis of semigroups with soft intersection ideals, Acta Univ Sapientiae Math, 2022; 14(1): 166-210.
  • Sezgin A. A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals, Algebra Lett, 2016; 2016(3): 1–46.
  • Jana C, Pal M, Karaaslan F and Sezgin A. (α, β)-soft intersectional rings and ideals with their applications, New Math Nat Comput, 2019; 15(2): 333–350.
  • Muştuoğlu E, Sezgin A and Türk ZK. Some characterizations on soft uni-groups and normal soft uni-groups, Int J Comput Appl, 2016; 155(10): 1-8.
  • Sezer AS, Çağman N and Atagün AO. Uni-soft substructures of groups, Ann Fuzzy Math Inform, 2015; 9(2): 235–246.
  • Sezer AS. Certain Characterizations of LA-semigroups by soft sets, J Intell Fuzzy Syst, 2014; 27(2): 1035-1046.
  • Özlü Ş and Sezgin A. Soft covered ideals in semigroups, Acta Univ Sapientiae Math, 2020: 12(2): 317-346.
  • Atagün AO and Sezgin A. Soft subnear-rings, soft ideals and soft n-subgroups of near-rings, Math Sci Letters, 2018; 7(1): 37-42.
  • Sezgin A. A new view on AG-groupoid theory via soft sets for uncertainty modeling, Filomat, 2018; 32(8): 2995–3030.
  • Sezgin A, Çağman N and Atagün AO. A completely new view to soft intersection rings via soft uni-int product, Appl Soft Comput, 2017; 54: 366-392.
  • Sezgin A, Atagün AO and Çağman N and Demir H. On near-rings with soft union ideals and applications, New Math Nat Comput, 2022; 18(2): 495-511.
  • Imai Y and Iseki K. On axiom systems of proposition calculi, Proc Jpn Acad, 1966; 42: 19–22.
  • Pant S, Dagtoros K, Kholil MI and Vivas A. Matrices: Peculiar determinant property, OPS Journal, 2024; 1: 1–7.
There are 44 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Research Article
Authors

Aslıhan Sezgin 0000-0002-1519-7294

Emre Akbulut 0009-0004-2855-0140

Hüseyin Demir 0000-0002-2289-8662

Project Number YOK
Publication Date December 30, 2024
Submission Date April 25, 2024
Acceptance Date August 5, 2024
Published in Issue Year 2024 Volume: 8 Issue: 2

Cite

APA Sezgin, A., Akbulut, E., & Demir, H. (2024). A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation. Uşak Üniversitesi Fen Ve Doğa Bilimleri Dergisi, 8(2), 90-114. https://doi.org/10.47137/usufedbid.1473425
AMA Sezgin A, Akbulut E, Demir H. A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation. Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi. December 2024;8(2):90-114. doi:10.47137/usufedbid.1473425
Chicago Sezgin, Aslıhan, Emre Akbulut, and Hüseyin Demir. “A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation”. Uşak Üniversitesi Fen Ve Doğa Bilimleri Dergisi 8, no. 2 (December 2024): 90-114. https://doi.org/10.47137/usufedbid.1473425.
EndNote Sezgin A, Akbulut E, Demir H (December 1, 2024) A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation. Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi 8 2 90–114.
IEEE A. Sezgin, E. Akbulut, and H. Demir, “A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation”, Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi, vol. 8, no. 2, pp. 90–114, 2024, doi: 10.47137/usufedbid.1473425.
ISNAD Sezgin, Aslıhan et al. “A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation”. Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi 8/2 (December 2024), 90-114. https://doi.org/10.47137/usufedbid.1473425.
JAMA Sezgin A, Akbulut E, Demir H. A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation. Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi. 2024;8:90–114.
MLA Sezgin, Aslıhan et al. “A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation”. Uşak Üniversitesi Fen Ve Doğa Bilimleri Dergisi, vol. 8, no. 2, 2024, pp. 90-114, doi:10.47137/usufedbid.1473425.
Vancouver Sezgin A, Akbulut E, Demir H. A New Type of Extended Soft Set Operations: Complementary Extended Difference Operation. Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi. 2024;8(2):90-114.