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

Öz

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.

Anahtar Kelimeler

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

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

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