Research article · 2024 · Full text
International Journal of Statistics and Applied MathematicsInter relations between algebraic boundary and other boundary
Volume 9(1) · Pages 39–43 · DOI: 10.22271/maths.2024.v9.i1a.1574 ↗
Research areasRough Sets and Fuzzy Logic
Abstract
Author-writtenIn this study, various closed algebras on fuzzy rough algebraic TM systems are defined. The relationships between closed algebras and between algebraic boundaries and other boundaries are also provided. In order to emphasize the results, examples are provided.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Fuzzy Set Theory was introduced by Zadeh in 1965 as a mathematical method to handle imprecise explanations. View evidence ↗
- Fuzzy logic is useful in various daily applications, especially in medicine, because it can flexibly handle varying data. View evidence ↗
- This research builds upon the features of fuzzy rough sets in TM boundary algebra previously established by Manimegalai and Tamilarasi. View evidence ↗
The question
- The study aims to define various closed algebras on fuzzy rough algebraic TM systems and determine the relationships between these closed algebras and different types of algebraic boundaries. View evidence ↗
What they did
- The authors define a fuzzy rough algebraic TM system as a non-empty set A in X that contains 0̃ and 1̃, and is closed under intersection and arbitrary union. View evidence ↗
- The fuzzy rough algebraic boundary of a set A is defined as the intersection of the fuzzy rough algebraic closure of A and the fuzzy rough algebraic closure of its complement A'. View evidence ↗
What they found
- The authors prove that the collection of all fuzzy rough algebraic alpha-open sets forms a fuzzy rough algebraic TM system on X. View evidence ↗
- The study finds that every fuzzy rough algebraic alpha-closed set is also a fuzzy rough algebraic beta-closed set. View evidence ↗
What it means
- The authors conclude that they have successfully defined closed algebras on fuzzy rough algebraic TM systems and established the inter-relations between various algebraic boundaries. View evidence ↗
Visual summary
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Hierarchy of Closed Sets
AI-generated conceptual illustration (Abstract geometric shapes representing mathematical sets). It explains the idea and does not show study data.
View source evidence ↗Figures & tables
From the original paper
Figure 2Introduction
In 1965, Zadeh [9] presented the concept of Fuzzy Set Theory. He presented a mathematical technique to create some imprecise explanations. A fuzzy set is one that has components of a particular degree but no clearly defined boundaries. The range of a fuzzy set is 0 to 1. Due to its flexibility with varying data, fuzzy logic has been shown to have applications in several aspects of our daily lives, particularly in the medical area. Some features of fuzzy rough sets in TM boundary algebra, initially established by K. Manimegalai and A. Tamilarasi [5] , are discussed in this study. In this study, various closed algebras on fuzzy rough algebraic TM systems are defined. The relationships between closed algebras and between algebraic boundaries and other boundaries are also provided. In order to emphasize the results, examples are provided. [2,7] Let X be the fuzzy topological space, and let 𝜆 be a fuzzy set. The boundary 𝐵𝑑 of 𝜆, is then determined as 𝐵𝑑(𝜆) = 𝑐𝑙 (𝜆) ∩ 𝑐𝑙 (𝜆)′. Obviously, 𝐵𝑑(𝜆) is a fuzzy closed set.
Preliminaries 2.1 Definition
Preliminaries 2.1 Definition
Definition [1]
A fuzzy set 𝐴 in a fuzzy topological system (𝑋, 𝜏) is called fuzzy 𝛼 open if 𝐴 ≤ 𝑖𝑛𝑡 (𝑐𝑙(𝑖𝑛𝑡(𝐴))). The compliment of fuzzy 𝛼 open is fuzzy 𝛼 closed. [3,6] A fuzzy set 𝐴 in a fuzzy topological system (𝑋, 𝜏) is called fuzzy 𝛽 open if 𝐴 ≤ 𝑐𝑙 (𝑖𝑛𝑡(𝑐𝑙(𝐴))). The compliment of fuzzy 𝛽 open is fuzzy 𝛽 closed.
Definition
Definition
Definition [7]
A fuzzy set 𝐴 in a fuzzy topological system (𝑋, 𝜏) is called fuzzy 𝑝𝑟𝑒 open if 𝐴 ≤ 𝑖𝑛𝑡(𝑐𝑙(𝐴)). The compliment of fuzzy pre open is fuzzy pre closed. [4] . A fuzzy subset 𝐴 of fuzzy topological space (𝑋, 𝜏) is said to be fuzzyopen set if 𝐴 ≤ (𝑐𝑙(𝑖𝑛𝑡(𝐴))) ∪ (𝑖𝑛𝑡(𝑐𝑙(𝐴))). The compliment of fuzzy 𝛾 open set is fuzzy closed set. [8] A fuzzy subset 𝐴 of fuzzy topological space (𝑋, 𝜏) is said to be fuzzy 𝛿 closed if 𝐴 = 𝑐𝑙(𝐴).
Definition
Definition
Definition
Definition
Inter relations between algebraic boundary and other Boundary 3.1 Definition
A non-empty set 𝐴 is a family of fuzzy rough algebraic in 𝑋 satisfies the following conditions:
Then 𝐴 is said to be a fuzzy rough algebraic TM system. Any member of fuzzy rough algebraic TM system is called fuzzy rough algebraic TM open. The complement of fuzzy rough open algebraic is fuzzy rough algebraic TM closed.
Definition
Let (𝑋, 𝑇𝑀) be a fuzzy rough algebraic TM system and 𝐴 be any fuzzy rough algebraic. Then the fuzzy rough algebraic boundary of 𝐴, is denoted and defined as ℱ ℛ𝒯ℳ 𝐵𝑑 (𝐴) = ℱ ℛ𝒯ℳ 𝑐𝑙(𝐴) ∩ ℱ ℛ𝒯ℳ 𝑐𝑙(𝐴′). Obviously ℱ ℛ𝒯ℳ 𝐵𝑑 (𝐴) is fuzzy rough algebraic TM closed.
Definition
Let (𝑋, 𝑇𝑀) be a fuzzy rough algebraic TM system. A fuzzy rough algebraic 𝐴 in 𝑋 is said to be fuzzy rough algebraic 𝑝𝑟𝑒 closed (ℱ ℛ𝒯ℳ 𝑝𝐶 shortly) if 𝐴 ⊇ (ℱ ℛ𝒯ℳ 𝑐𝑙(ℱ ℛ𝒯ℳ 𝑖𝑛𝑡(𝐴))). The complement of every fuzzy rough algebraic 𝑝𝑟𝑒 closed is fuzzy rough algebraic 𝑝𝑟𝑒 open (ℱ ℛ𝒯ℳ 𝑝𝑂 shortly).
Definition
Let (𝑋, 𝑇𝑀) be a fuzzy rough TM system and 𝐴 be any fuzzy rough algebraic. Then A is said to be a fuzzy rough algebraic 𝛼 closed (ℱ ℛ𝒯ℳ 𝛼𝐶 shortly) if 𝐴 ⊇ ℱ ℛ𝒯ℳ 𝑐𝑙 (ℱ ℛ𝒯ℳ 𝑖𝑛𝑡(ℱ ℛ𝒯ℳ 𝑐𝑙(𝐴))). The complement of fuzzy rough algebraic 𝛼 closed is fuzzy rough algebraic 𝛼 open (ℱ ℛ𝒯ℳ 𝛼𝑂 shortly).
Definition
Let (𝑋, 𝑇𝑀) be a fuzzy rough TM system and 𝐴 be any fuzzy rough algebraic. Then A is said to be a fuzzy rough algebraic 𝛽 closed (ℱ ℛ𝒯ℳ 𝛽𝐶 shortly) if 𝐴 ⊇ ℱ ℛ𝒯ℳ 𝑖𝑛𝑡 (ℱ ℛ𝒯ℳ 𝑐𝑙(ℱ ℛ𝒯ℳ 𝑖𝑛𝑡(𝐴))). The complement of fuzzy rough algebraic 𝛽 closed is fuzzy rough algebraic 𝛽 open (ℱ ℛ𝒯ℳ 𝛽𝑂 shortly) .
Definition
Let (𝑋, 𝑇𝑀) be a fuzzy rough TM system and 𝐴 be any fuzzy rough algebraic. Then A is said to be a fuzzy rough algebraic closed (ℱ ℛ𝒯ℳ 𝛾𝐶 shortly) if 𝐴 ⊇ (ℱ ℛ𝒯ℳ 𝑖𝑛𝑡(ℱ ℛ𝒯ℳ 𝑐𝑙(𝐴))) ∩ (ℱ ℛ𝒯ℳ 𝑐𝑙(ℱ ℛ𝒯ℳ 𝑖𝑛𝑡(𝐴))). The complement of fuzzy rough algebraic 𝛾 closed is fuzzy rough algebraic open (ℱ ℛ𝒯ℳ 𝛾𝑂 shortly).
Definition
Let (𝑋, 𝑇𝑀) be a fuzzy rough TM system and 𝐴 be any fuzzy rough algebraic. Then A is said to be a fuzzy rough algebraic 𝛿 closed (ℱ ℛ𝒯ℳ 𝛿𝐶 shortly) if 𝐴 = ℱ ℛ𝒯ℳ 𝑐𝑙(𝐴). The complement of fuzzy rough algebraic 𝛿 closed is fuzzy rough algebraic 𝛿 open (ℱ ℛ𝒯ℳ 𝛿𝑂 shortly).
Definition
Let (𝑋, 𝑇𝑀) be a fuzzy rough algebraic TM system and 𝐴 be a fuzzy rough algebraic in 𝑋.
Definition
Let A be any fuzzy rough algebraic of fuzzy rough algebraic TM system(𝑋, 𝑇𝑀). Then A is said to be
Figures and tables from the paper
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