Mathematical Techniques To Transform (3, 2)-Fuzzy Bags In Uncertainty Environment

The concept of fuzzy sets was proposed by Zadeh [1]. The theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. After the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. /e integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [2–4]. The idea of intuitionistic fuzzy sets suggested by Atanassov [5] is one of the extensions of fuzzy sets with better applicability. Applications of intuitionistic fuzzy sets appear in various fields, including medical diagnosis, optimization problems, and multi criteria decision making [6–8]. Yager [9] offered a new fuzzy set called a Pythagorean fuzzy set, which is the generalization of intuitionistic fuzzy sets. Fermatean fuzzy sets were introduced by Senapati and Yager [10], and they also defined basic operations over the Fermatean fuzzy sets. /e concept of fuzzy topological spaces was introduced by Chang [11]. He studied the topological concepts like continuity and compactness via fuzzy topological spaces. Then, Lowen [12] presented a new type of fuzzy topological spaces. Çoker [13] subsequently initiated a study of intuitionistic fuzzy topological spaces. Recently, Olgun et al. [14] presented the concept of Pythagorean fuzzy topological spaces and Ibrahim [15] defined the concept of Fermatean fuzzy topological spaces. In this paper, we introduce the concept of (3,2)-fuzzy multi-groups from (3,2)-fuzzy bags as on extension of intuitionistic fuzzy multi-group, Pythagorean fuzzy multi-group addressing limitation in uncertainty presentation.  We develop the theoretical framework for the algebraic structures deriving fundamental properties including closure under group operations and characterizing union and intersection ideas. Key results includes necessary and sufficient conditions for (3,2)-fuzzy uncertainty multi-group properties and comprehensive analysis of their algebraic structures. The theory which helps enhanced tools for decision making under certainty with multiple membership degrees.

2. Preliminaries:

In this section, we establish the fundamental criteria’s and definitions that from theoretical foundation for our investigation. We start with the fundamental concept of multi sets and the multi groups,  then proceed to uncertainty extension, ultimately leading to the framework necessary for defining (3,2)-uncertainty multi groups.

Example-3.2: Let ‘A

 such that

Definition-3.3: The cardinality of 

 or that of 

 in a (3, 2)-fuzzy bag

’ is called length of the elements

, and is denoted by

(i.e) 

When 

denotes the cardinality of the membership sequence

 and

 that of non-membership sequence

Definition-3.4: Two (3, 2)-fuzzy bags

 and

 drawn from a non-empty set

’ are said to be equivalent, written 

 if and only if

The following fundamental operations on (3, 2)-fuzzy bags are explored from (Ejegwa) and provide the algebraic frame work for subsequent development.

CONCLUSION

(3, 2)-fuzzy multi groups is introduced from (3, 2)-fuzzy bags and studied some characterizations of this. The length of the (3, 2)-fuzzy bags defined with suitable example. The necessary and sufficient conditions of (3, 2)-fuzzy multi groups is proved with on existence. Many key results and comprehensive analysis is explained.

FUTURE WORK:

One can obtain the similar result in the domain of Hesitant uncertainty collection and Bipolar uncertainty set.

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