Pythagorean Fuzzy Soft Nano Topological Space and it’s Applications

Abstract

Our primary goal is to introduce the conceptual framework of Pythagorean fuzzy soft nano topological space (PFSNTS) from the fuzzy topological space (FTS), nano topological space (NTS), and fuzzy soft topological space (FSTS). We also look at the Pythagorean distinctive characteristics. This article has discussed several concepts related to intuitionistic, nano, and Pythagorean. The primary intention of this work is to use programming languages to make decisions in the contemporary world.

Introduction

Overview of Fuzzy and Soft Set Theories

• Fuzzy Sets: Introduced by Lotfi A. Zadeh (1965), they allow elements to have varying degrees of membership.

• Intuitionistic Fuzzy Sets (IFS): Introduced by Atanassov, they extend fuzzy sets by also considering non-membership with the condition: μ(x) + ν(x) ≤ 1.

• Pythagorean Fuzzy Sets (PFS): Further extend IFS by allowing μ²(x) + ν²(x) ≤ 1 (Yager and Abbasov).

• Soft Sets: Introduced by Molodtsov (1999), soft sets are parameterized families of subsets.

• Nano Topology (NT): A form of topology based on equivalence relations and used for finer approximations.

• Pythagorean Fuzzy Soft Nano Topological Space (PFSNTS): Integrates PFS, soft sets, and NT for enhanced uncertainty modeling.

II. Key Definitions and Structures

1. Topology: A collection of subsets on a set X satisfying union, intersection, and inclusion of empty and full sets.

2. Approximations:

   • Lower (L) and Upper (U) approximations define set boundaries based on equivalence classes.

   • Boundary (B): Difference between upper and lower approximations.

3. PFSNTS:

   • Defined using fuzzy soft approximations under PFS conditions.

   • Requires τ to contain lower, upper, and boundary sets and be closed under union and intersection.

III. Theorems and Properties of PFSNTS

• Theorem 1: Outlines the structure of PFSNTS under different conditions of lower and upper approximations (e.g., empty, universal set).

• Theorem 2: Shows that intuitionistic and Pythagorean fuzzy soft sets are interrelated, and conditions for set inclusion and membership/non-membership hold under fuzzy logic.

IV. Applications

A. House Painting Color Selection:

• Evaluates combinations like (Blue, White), (Red, Yellow) using Pythagorean fuzzy logic.

• Checks if α² + β² ≤ 1 and selects the best color combination based on computed values.

• Result: (Blue, White) is the most suitable pair.

B. Best Skin Care Combinations:

• Compares combinations of ingredients (e.g., Rice Flour, Green Gram) for different skin types.

• Uses membership (α) and non-membership (β) degrees.

• Each table confirms conditions for being Pythagorean fuzzy soft subsets.

• Results:

  • Normal skin: Rice Flour + Green Gram

  • Oily skin: Orange Peel + Chickpea Flour

  • Dry skin: Rose Petal + Rice Flour

  • Sensitive skin: Vetiver + Rose Petal Powder

V. Algorithm for Decision-Making

Steps include:

1. Input and validate values.

2. Calculate hypotenuse (for fuzzy parameter verification).

3. Apply fuzzy conditions (e.g., α² + β² ≤ 1).

4. Identify optimal solutions (colors or ingredients).

Conclusion

In this paper, we introduced and explored the concept of Pythagorean Fuzzy Soft Nano Topological Space (PFSNTS) by integrating fuzzy topology, nano topology, and soft topology. We established key definitions, theorems, and properties of this novel framework. Furthermore, we demonstrated its practical applicability in decision-making scenarios, particularly in selecting optimal choices such as color combinations for house painting and ingredient blends for skin care. Our findings highlight the effectiveness of PFSNTS in handling uncertainty and imprecision in real-world problems. Future research can extend this work by applying PFSNTS to other domains, such as medical diagnosis, risk assessment, and artificial intelligence.

References

[1] D. Ajay, J. Joseline Charisma. Pythagorean Nano Topological Space. International Journal of Recent Technology and Engineering 2020, DOI:10.35940/ijrte. E6477.018520. [2] K. Atanassov. Intuitionistic fuzzy sets, Fuzzy Sets and Systems 1986, 87–96. [3] N Rajesh, P Sudha, On weakly-?-continuous functions, International Journal of Pure and Applied Mathematics 2013, vol 86, 325 – 332. [4] N Rajesh, P Sudha, Properties of ?-continuous functions, Fasciculi Mathematici 2015, Vol – 55, 170 -195 [5] Saeid Jafari, M. Rishwanth, Parimala, Ibtesam Alshmmari. On Pythagorean Fuzzy Soft Topological Spaces. Journal of Intelligent & Fuzzy Systems 2021, DOI:10.3233/JIFS-21085. [6] L. A. Zadeh. Fuzzy Sets Information and Control. 1965, 83,38-–53. [7] Coker D. An Introduction to IF Topological Spaces. Fuzzy Sets and Systems. 1997; 88: 81–9. https://doi.org/10.1016/S0165-0114(96)00076-0. [8] Coker, D., An introduction to intuitionistic topological spaces, Busefal, 81(2000), 51-56. [9] Kim, J. H., P. K. Lim, J. G. Lee, and K. Hur, Intuitionistic topological spaces, Infinite Study, 2018. [10] X. Peng and Y. Yang (2015), Some results for Pythagorean fuzzy sets, Int. J Intell Syst. 30, 1133-1160. [11] N. Turanli and D. Coker, “Fuzzy connectedness in intuitionistic fuzzy topological spaces,” Fuzzy Sets and Systems, vol. 116, no. 3, pp. 369–375, 2000. [12] Olgun M., Unver M. and Yard S., Pythagorean fuzzy topological spaces, Complex & Intelligent Systems (2019), 1–7. [13] S. Naz, S. Ashraf, and M. Akram, A novel approach to decision-making with Pythagorean fuzzy information, Mathematics, vol. 6, no. 6, p. 95, 2018. [14] Ramachandran, M.; Stephan Antony Raj, A., Intuitionistic fuzzy nano topological space: Theory and Application, Sciexplore, International Journal of Research in Science, 2017, 4(1), pp. 1–6.

Copyright

Copyright © 2025 P. Sudha, R. Sayeelakshmi, A. Shameena Yasmin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.