(3, 2)-Fuzzy Sets and Their Applications to Topology and Optimal Choices

Area of Science:

• Fuzzy Set Theory
• Topology
• Decision Making

Background:

• Existing fuzzy sets like Pythagorean and intuitionistic fuzzy sets have limitations in handling uncertainty.
• A need exists for more robust mathematical tools to describe complex uncertain situations.

Purpose of the Study:

• To define and explore the properties of (3, 2)-fuzzy sets.
• To introduce and analyze (3, 2)-fuzzy topological spaces and related concepts.
• To develop a decision-making approach for student placement using (3, 2)-fuzzy relations.

Main Methods:

• Definition and exploration of basic set operations for (3, 2)-fuzzy sets.
• Introduction of (3, 2)-fuzzy topological spaces, continuous maps, and fuzzy points.
• Study of separation axioms within the (3, 2)-fuzzy topological framework.
• Formulation of (3, 2)-fuzzy relations and their properties.

Main Results:

• (3, 2)-Fuzzy sets provide a broader range for membership grades, enhancing uncertainty description.
• The paper establishes fundamental properties of (3, 2)-fuzzy topological spaces and continuous maps.
• A novel (3, 2)-fuzzy relation-based decision-making model is presented for student-college suitability assessment.

Conclusions:

• (3, 2)-Fuzzy sets offer a significant advancement in modeling uncertainty.
• The developed framework for (3, 2)-fuzzy topology and relations has practical applications.
• The proposed decision-making approach aids in optimizing student placement based on academic performance.