Neutrosophic Sets and Logic

2017 ◽  
pp. 18-63
Author(s):  
Mumtaz Ali ◽  
Florentin Smarandache ◽  
Luige Vladareanu
Keyword(s):  
Fuzzy Sets ◽  
Approximation Theory ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Hybrid Structures ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Basic Concepts ◽  
Interval Valued

Neutrosophic sets and Logic plays a significant role in approximation theory. It is a generalization of fuzzy sets and intuitionistic fuzzy set. Neutrosophic set is based on the neutrosophic philosophy in which every idea Z, has opposite denoted as anti(Z) and its neutral which is denoted as neut(Z). This is the main feature of neutrosophic sets and logic. This chapter is about the basic concepts of neutrosophic sets as well as some of their hybrid structures. This chapter starts with the introduction of fuzzy sets and intuitionistic fuzzy sets respectively. The notions of neutrosophic set are defined and studied their basic properties in this chapter. Then we studied neutrosophic crisp sets and their associated properties and notions. Moreover, interval valued neutrosophic sets are studied with some of their properties. Finally, we presented some applications of neutrosophic sets in the real world problems.

scholarly journals Certain Notions of Neutrosophic Topological K-Algebras

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 234 ◽  
Author(s):  
Muhammad Akram ◽  
Hina Gulzar ◽  
Florentin Smarandache ◽  
Said Broumi
Keyword(s):  
Fuzzy Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Point Of View ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Research Article

The concept of neutrosophic set from philosophical point of view was first considered by Smarandache. A single-valued neutrosophic set is a subclass of the neutrosophic set from a scientific and engineering point of view and an extension of intuitionistic fuzzy sets. In this research article, we apply the notion of single-valued neutrosophic sets to K-algebras. We introduce the notion of single-valued neutrosophic topological K-algebras and investigate some of their properties. Further, we study certain properties, including C 5 -connected, super connected, compact and Hausdorff, of single-valued neutrosophic topological K-algebras. We also investigate the image and pre-image of single-valued neutrosophic topological K-algebras under homomorphism.

scholarly journals MBJ-neutrosophic subalgebras and filters in $ BE $-algebras

2022 ◽  
Vol 7 (4) ◽  
pp. 6016-6033
Author(s):  
Rajab Ali Borzooei ◽  
◽  
Hee Sik Kim ◽  
Young Bae Jun ◽  
Sun Shin Ahn ◽  
...  
Keyword(s):  
Membership Function ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Neutrosophic Set ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
The Relationship ◽  
Interval Valued

<abstract><p>The concept of a neutrosophic set, which is a generalization of an intuitionistic fuzzy set and a para consistent set etc., was introduced by F. Smarandache. Since then, it has been studied in various applications. In considering a generalization of the neutrosophic set, Mohseni Takallo et al. used the interval valued fuzzy set as the indeterminate membership function because interval valued fuzzy set is a generalization of a fuzzy set, and introduced the notion of MBJ-neutrosophic sets, and then they applied it to BCK/BCI-algebras. The aim of this paper is to apply the concept of MBJ-neutrosophic sets to a $ BE $-algebra, which is a generalization of a BCK-algebra. The notions of MBJ-neutrosophic subalgebras and MBJ-neutrosophic filters of $ BE $-algebras are introduced and related properties are investigated. The conditions under which the MBJ-neutrosophic set can be a MBJ-neutrosophic subalgebra/filter are searched. Characterizations of MBJ-neutrosophic subalgebras and MBJ-neutrosophic filters are considered. The relationship between an MBJ-neutrosophic subalgebra and an MBJ-neutrosophic filter is established.</p></abstract>

Interval-Valued Intuitionistic Fuzzy Subnear Rings

2020 ◽  
pp. 213-224
Author(s):  
Amal Kumar Adak
Keyword(s):  
Fuzzy Sets ◽  
Level Set ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy ◽  
The Family ◽  
Interval Valued

The theory of interval-valued intuitionistic fuzzy sets is a generalization of both interval-valued fuzzy sets and intuitionistic fuzzy sets. In this chapter, the notion of interval-valued intuitionistic fuzzy subnear-ring is introduced, and some interesting properties are discussed. Some relations on the family of all interval-valued intuitionistic fuzzy subnear-ring are presented, and some related properties are investigated. Also, the authors represent upper and lower level set of interval-valued intuitionistic fuzzy set.

scholarly journals A Comparative Analysis on Geometric Measure in Intuitionistic Fuzzy Set and Interval-Valued Intuitionistic Fuzzy Set

2019 ◽  
pp. 485-491
Author(s):  
A. Manonmani ◽  
M. Suganya
Keyword(s):  
Comparative Analysis ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Distance Measure ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy ◽  
Wide Range ◽  
And Algebra ◽  
Interval Valued

Intuitionistic Fuzzy set (IFS) was proposed in early 80’s. It is a well known theory. As a developer in Fuzzy Mathematics, interval – valued Intuitionistic Fuzzy sets (IVIFS) were developed afterwards by Gargo and Atanssov. It has a wide range of applications in the field of Optimization and algebra. There are many distance measure in Fuzzy such as Hamming, Normalized Hamming, Euclidean, Normalized Euclidean, Geometric, Normalized Geometric etc… to calculate the distance between two fuzzy numbers. In this paper, the comparison between Geometric distance measure in Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets is explored. The step-wise conservation of Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets is also proposed. This type of comparative analysis shows that the distance between Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets varies slightly due to boundaries of interval – valued Intuitionistic Fuzzy sets.

scholarly journals On Neutrosophic Offuninorms

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1136 ◽  
Author(s):  
Erick González Caballero ◽  
Florentin Smarandache ◽  
Maikel Leyva Vázquez
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Theory ◽  
Intuitionistic Fuzzy Sets ◽  
Membership Functions ◽  
Neutrosophic Sets ◽  
Intuitionistic Fuzzy ◽  
Wide Range ◽  
The Relationship ◽  
Interval Valued ◽  
Important Kind

Uninorms comprise an important kind of operator in fuzzy theory. They are obtained from the generalization of the t-norm and t-conorm axiomatic. Uninorms are theoretically remarkable, and furthermore, they have a wide range of applications. For that reason, when fuzzy sets have been generalized to others—e.g., intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets, or neutrosophic sets—then uninorm generalizations have emerged in those novel frameworks. Neutrosophic sets contain the notion of indeterminacy—which is caused by unknown, contradictory, and paradoxical information—and thus, it includes, aside from the membership and non-membership functions, an indeterminate-membership function. Also, the relationship among them does not satisfy any restriction. Along this line of generalizations, this paper aims to extend uninorms to the framework of neutrosophic offsets, which are called neutrosophic offuninorms. Offsets are neutrosophic sets such that their domains exceed the scope of the interval [0,1]. In the present paper, the definition, properties, and application areas of this new concept are provided. It is necessary to emphasize that the neutrosophic offuninorms are feasible for application in several fields, as we illustrate in this paper.

scholarly journals An Improved Correlation Coefficient of Intuitionistic Fuzzy Sets

2019 ◽  
Vol 28 (2) ◽  
pp. 231-243 ◽  
Author(s):  
Han-Liang Huang ◽  
Yuting Guo
Keyword(s):  
Fuzzy Sets ◽  
Correlation Coefficient ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Medical Diagnosis ◽  
Intuitionistic Fuzzy Set ◽  
Intuitionistic Fuzzy Sets ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

Abstract The intuitionistic fuzzy set is a useful tool to deal with vagueness and uncertainty. Correlation coefficient of the intuitionistic fuzzy sets is an important measure in intuitionistic fuzzy set theory and has great practical potential in a variety of areas, such as decision making, medical diagnosis, pattern recognition, etc. In this paper, an improved correlation coefficient of the intuitionistic fuzzy sets is defined, and it can overcome some drawbacks of the existing ones. The properties of this correlation coefficient are discussed. Then, the generalization of the coefficient of interval-valued intuitionistic fuzzy sets is also introduced. Finally, two examples about the application of the proposed correlation coefficient of the intuitionistic fuzzy sets in medical diagnosis and clustering are shown to illustrate the advantages over the existing methods.

Width-based distance measures on interval-valued intuitionistic fuzzy sets

2021 ◽  
pp. 1-13
Author(s):  
Xi Li ◽  
Chunfeng Suo ◽  
Yongming Li
Keyword(s):  
Pattern Recognition ◽  
Fuzzy Sets ◽  
Medical Diagnosis ◽  
Intuitionistic Fuzzy Sets ◽  
Distance Measures ◽  
Intuitionistic Fuzzy ◽  
Interval Valued ◽  
Dissimilarity Function ◽  
Better Than

An essential topic of interval-valued intuitionistic fuzzy sets(IVIFSs) is distance measures. In this paper, we introduce a new kind of distance measures on IVIFSs. The novelty of our method lies in that we consider the width of intervals so that the uncertainty of outputs is strongly associated with the uncertainty of inputs. In addition, better than the distance measures given by predecessors, we define a new quaternary function on IVIFSs to construct the above-mentioned distance measures, which called interval-valued intuitionistic fuzzy dissimilarity function. Two specific methods for building the quaternary functions are proposed. Moreover, we also analyzed the degradation of the distance measures in this paper, and show that our measures can perfectly cover the measures on a simpler set. Finally, we provide illustrative examples in pattern recognition and medical diagnosis problems to confirm the effectiveness and advantages of the proposed distance measures.

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