Abstract:In this paper we introduce the concept of restricted interval valued neutrosophic sets (... more Abstract:In this paper we introduce the concept of restricted interval valued neutrosophic sets (RIVNS in short). Some basic operations and properties of RIVNS are discussed. The concept of restricted interval valued neutrosophic topology is also introduced together with restricted interval valued neutrosophic finer and restricted interval valued neutrosophic coarser topology. We also define restricted interval valued neutrosophic interior and closer of a restricted interval valued neutrosophic set. Some theorems and examples are cites. Restricted interval valued neutrosophic subspace topology is also studied.
Molodtsov initiated the concept of fuzzy soft set theory in 1999. Maji et al. introduced the noti... more Molodtsov initiated the concept of fuzzy soft set theory in 1999. Maji et al. introduced the notion of fuzzy soft sets. By introducing the concept of intuitionistic fuzzy sets into the theory of soft sets, Maji et al. proposed the concept of intuitionistic fuzzy soft set theory. The notion of the interval-valued intuitionistic fuzzy sets was first introduced by Atanassov and Gargov. It is characterised by an interval-valued membership degree and an interval-valued non-membership degree. In 2010, Y. Jiang et al. introduced the concept of interval-valued intuitionistic fuzzy soft sets. In this chapter, the concept of generalised interval-valued intuitionistic fuzzy soft sets is introduced. The basic properties of these sets are presented. Also, an application of generalised interval-valued intuitionistic fuzzy soft sets in decision-making with respect to interval of degree of preference is investigated.
In our real-life problems, there are situations with the uncertain data that may not be successfu... more In our real-life problems, there are situations with the uncertain data that may not be successfully modelled by the classical mathematics. There are some mathematical tools for dealing with uncertainties—they are fuzzy set theory introduced by Zadeh [10], rough set theory introduced by Pawlak [7], and soft set theory initiated by Molodtsov [5]. In this chapter, we recall some basic notions relevant to our Chaps. 2– 10, such as fuzzy sets, intuitionistic fuzzy sets, interval fuzzy sets, soft set, fuzzy soft sets, rough sets, fuzzy rough sets, fuzzy rough soft set, and others.
In this paper, we procure the idea of pentapartitioned neutrosophic <em>Q</em>-ideal ... more In this paper, we procure the idea of pentapartitioned neutrosophic <em>Q</em>-ideal of <em>Q</em>-algebra.<br> Then, we formulate some definitions and results on it. Further, we furnish some suitable examples.<br>
International Journal of Neutrosophic Science, 2020
The purpose of this article is to study neutrosophic polynomials i.e. polynomials which are Neutr... more The purpose of this article is to study neutrosophic polynomials i.e. polynomials which are Neutrosophic in nature and study its properties with the help of neutrosophic numbers. Apart from this we discuss different types of neutrosophic polynomials with concrete examples and establish some theorems and results which will be useful for the further study. We also give a solution method to find the approximate roots of a neutrosophic polynomial equation.
In this chapter, we introduce the concept of intuitionistic fuzzy parameterised intuitionistic fu... more In this chapter, we introduce the concept of intuitionistic fuzzy parameterised intuitionistic fuzzy soft (ifpifs) sets and their operations with examples. We also define the approximate functions of ifpifs-set from the intuitionistic fuzzy parameterised set to the intuitionistic fuzzy subsets [1] of universal set. Lastly, we construct an ifpifs-set decision-making problem and try to solve the problem.
The purpose of this chapter was to study the concept of topological structure formed by soft mult... more The purpose of this chapter was to study the concept of topological structure formed by soft multi-sets. The notion of relative complement of soft multi-set, soft multi-point, soft multi-open set, soft multi-closed set, soft multi-basis, soft multi-sub-basis, neighbourhoods and neighbourhood system, interior and closure of a soft multi-set, etc., is to be introduced, and their basic properties are also to be investigated. It is seen that a soft multi-topological space gives a parameterised family of topological spaces. Lastly, the concept of soft multi-compact space is also introduced.
Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types... more Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.
In this paper, we introduce a new type of sequence of intuitionistic fuzzy soft multi sets and th... more In this paper, we introduce a new type of sequence of intuitionistic fuzzy soft multi sets and their basic properties are studied. The concepts of subsequence, increasing sequence, decreasing sequence and convergence sequence of intuitionistic fuzzy soft multi sets are proposed. The concepts of cluster intuitionistic fuzzy soft multi sets of sequences are also introduced. Some basic properties regarding the above concepts are explored.
Alkhazaleh and Salleh presented a fuzzy soft multi set theoretic approach to solve decision-makin... more Alkhazaleh and Salleh presented a fuzzy soft multi set theoretic approach to solve decision-making problems using the Roy-Maji Algorithm, which has some limitations. In this research work, we have proposed an algorithm to solve fuzzy soft multi set based decision making problems using the Feng’s algorithm, which is more stable and more feasible than the Alkhazaleh–Salleh Algorithm. The feasibility of our proposed algorithm in practical applications is illustrated by a numerical example.
Research on information and communication technologies have been developed rapidly since it can b... more Research on information and communication technologies have been developed rapidly since it can be applied easily to several areas like computer science, medical science, economics, environments, engineering, among other. Applications of soft set theory, especially in information systems have been found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft multi set theory and show that the De-Morgan's type of results hold in fuzzy soft multi set theory with respect to these newly defined operations. In this paper, we extend their work and study some more basic properties of their defined operations. Also, we define some basic supporting tools in information system also application of fuzzy soft multi sets in information system are presented and discussed. Here we define the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy soft multi set is a fuzzy multi valued information system.
F. Smarandache introduced the concept of neutrosophic set in 1995 and P. K. Maji introduced the n... more F. Smarandache introduced the concept of neutrosophic set in 1995 and P. K. Maji introduced the notion of neutrosophic soft set in 2013, which is a hybridization of neutrosophic set and soft set. Irfan Deli introduced the concept of interval valued neutrosophic soft sets.
In this paper we introduce the concept of interval valued neutrosophic soft topological space tog... more In this paper we introduce the concept of interval valued neutrosophic soft topological space together with interval valued neutrosophic soft finer and interval valued neutrosophic soft coarser topology. We also define interval valued neutrosophic interior and closer of an interval valued neutrosophic soft set. Some theorems and examples are cites. Interval valued neutrosophic soft sub-space topology are studied. Some examples and theorems regarding this concept are presented.. Keywords: Soft set, interval valued neutrosophic set, interval valued neutrosophic soft set, interval valued neutrosophic soft topo-logical space.
Interval valued neutrosophic soft set introduced by Irfan Deli in 2014[8] is a generalization of ... more Interval valued neutrosophic soft set introduced by Irfan Deli in 2014[8] is a generalization of neu-trosophic set introduced by F. Smarandache in 1995[19], which can be used in real scientific and engineering applications. In this paper the Hamming and Euclidean distances between two interval valued neutrosophic soft sets (IVNS sets) are defined and similarity measures based on distances between two interval valued neutrosophic soft sets are proposed. Similarity measure based on set theo-retic approach is also proposed. Some basic properties of similarity measures between two interval valued neutro-sophic soft sets is also studied. A decision making method is established for interval valued neutrosophic soft set setting using similarity measures between IVNS sets. Finally an example is given to demonstrate the possible application of similarity measures in pattern recognition problems.
Abstract:In this paper we introduce the concept of restricted interval valued neutrosophic sets (... more Abstract:In this paper we introduce the concept of restricted interval valued neutrosophic sets (RIVNS in short). Some basic operations and properties of RIVNS are discussed. The concept of restricted interval valued neutrosophic topology is also introduced together with restricted interval valued neutrosophic finer and restricted interval valued neutrosophic coarser topology. We also define restricted interval valued neutrosophic interior and closer of a restricted interval valued neutrosophic set. Some theorems and examples are cites. Restricted interval valued neutrosophic subspace topology is also studied.
Molodtsov initiated the concept of fuzzy soft set theory in 1999. Maji et al. introduced the noti... more Molodtsov initiated the concept of fuzzy soft set theory in 1999. Maji et al. introduced the notion of fuzzy soft sets. By introducing the concept of intuitionistic fuzzy sets into the theory of soft sets, Maji et al. proposed the concept of intuitionistic fuzzy soft set theory. The notion of the interval-valued intuitionistic fuzzy sets was first introduced by Atanassov and Gargov. It is characterised by an interval-valued membership degree and an interval-valued non-membership degree. In 2010, Y. Jiang et al. introduced the concept of interval-valued intuitionistic fuzzy soft sets. In this chapter, the concept of generalised interval-valued intuitionistic fuzzy soft sets is introduced. The basic properties of these sets are presented. Also, an application of generalised interval-valued intuitionistic fuzzy soft sets in decision-making with respect to interval of degree of preference is investigated.
In our real-life problems, there are situations with the uncertain data that may not be successfu... more In our real-life problems, there are situations with the uncertain data that may not be successfully modelled by the classical mathematics. There are some mathematical tools for dealing with uncertainties—they are fuzzy set theory introduced by Zadeh [10], rough set theory introduced by Pawlak [7], and soft set theory initiated by Molodtsov [5]. In this chapter, we recall some basic notions relevant to our Chaps. 2– 10, such as fuzzy sets, intuitionistic fuzzy sets, interval fuzzy sets, soft set, fuzzy soft sets, rough sets, fuzzy rough sets, fuzzy rough soft set, and others.
In this paper, we procure the idea of pentapartitioned neutrosophic <em>Q</em>-ideal ... more In this paper, we procure the idea of pentapartitioned neutrosophic <em>Q</em>-ideal of <em>Q</em>-algebra.<br> Then, we formulate some definitions and results on it. Further, we furnish some suitable examples.<br>
International Journal of Neutrosophic Science, 2020
The purpose of this article is to study neutrosophic polynomials i.e. polynomials which are Neutr... more The purpose of this article is to study neutrosophic polynomials i.e. polynomials which are Neutrosophic in nature and study its properties with the help of neutrosophic numbers. Apart from this we discuss different types of neutrosophic polynomials with concrete examples and establish some theorems and results which will be useful for the further study. We also give a solution method to find the approximate roots of a neutrosophic polynomial equation.
In this chapter, we introduce the concept of intuitionistic fuzzy parameterised intuitionistic fu... more In this chapter, we introduce the concept of intuitionistic fuzzy parameterised intuitionistic fuzzy soft (ifpifs) sets and their operations with examples. We also define the approximate functions of ifpifs-set from the intuitionistic fuzzy parameterised set to the intuitionistic fuzzy subsets [1] of universal set. Lastly, we construct an ifpifs-set decision-making problem and try to solve the problem.
The purpose of this chapter was to study the concept of topological structure formed by soft mult... more The purpose of this chapter was to study the concept of topological structure formed by soft multi-sets. The notion of relative complement of soft multi-set, soft multi-point, soft multi-open set, soft multi-closed set, soft multi-basis, soft multi-sub-basis, neighbourhoods and neighbourhood system, interior and closure of a soft multi-set, etc., is to be introduced, and their basic properties are also to be investigated. It is seen that a soft multi-topological space gives a parameterised family of topological spaces. Lastly, the concept of soft multi-compact space is also introduced.
Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types... more Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.
In this paper, we introduce a new type of sequence of intuitionistic fuzzy soft multi sets and th... more In this paper, we introduce a new type of sequence of intuitionistic fuzzy soft multi sets and their basic properties are studied. The concepts of subsequence, increasing sequence, decreasing sequence and convergence sequence of intuitionistic fuzzy soft multi sets are proposed. The concepts of cluster intuitionistic fuzzy soft multi sets of sequences are also introduced. Some basic properties regarding the above concepts are explored.
Alkhazaleh and Salleh presented a fuzzy soft multi set theoretic approach to solve decision-makin... more Alkhazaleh and Salleh presented a fuzzy soft multi set theoretic approach to solve decision-making problems using the Roy-Maji Algorithm, which has some limitations. In this research work, we have proposed an algorithm to solve fuzzy soft multi set based decision making problems using the Feng’s algorithm, which is more stable and more feasible than the Alkhazaleh–Salleh Algorithm. The feasibility of our proposed algorithm in practical applications is illustrated by a numerical example.
Research on information and communication technologies have been developed rapidly since it can b... more Research on information and communication technologies have been developed rapidly since it can be applied easily to several areas like computer science, medical science, economics, environments, engineering, among other. Applications of soft set theory, especially in information systems have been found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft multi set theory and show that the De-Morgan's type of results hold in fuzzy soft multi set theory with respect to these newly defined operations. In this paper, we extend their work and study some more basic properties of their defined operations. Also, we define some basic supporting tools in information system also application of fuzzy soft multi sets in information system are presented and discussed. Here we define the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy soft multi set is a fuzzy multi valued information system.
F. Smarandache introduced the concept of neutrosophic set in 1995 and P. K. Maji introduced the n... more F. Smarandache introduced the concept of neutrosophic set in 1995 and P. K. Maji introduced the notion of neutrosophic soft set in 2013, which is a hybridization of neutrosophic set and soft set. Irfan Deli introduced the concept of interval valued neutrosophic soft sets.
In this paper we introduce the concept of interval valued neutrosophic soft topological space tog... more In this paper we introduce the concept of interval valued neutrosophic soft topological space together with interval valued neutrosophic soft finer and interval valued neutrosophic soft coarser topology. We also define interval valued neutrosophic interior and closer of an interval valued neutrosophic soft set. Some theorems and examples are cites. Interval valued neutrosophic soft sub-space topology are studied. Some examples and theorems regarding this concept are presented.. Keywords: Soft set, interval valued neutrosophic set, interval valued neutrosophic soft set, interval valued neutrosophic soft topo-logical space.
Interval valued neutrosophic soft set introduced by Irfan Deli in 2014[8] is a generalization of ... more Interval valued neutrosophic soft set introduced by Irfan Deli in 2014[8] is a generalization of neu-trosophic set introduced by F. Smarandache in 1995[19], which can be used in real scientific and engineering applications. In this paper the Hamming and Euclidean distances between two interval valued neutrosophic soft sets (IVNS sets) are defined and similarity measures based on distances between two interval valued neutrosophic soft sets are proposed. Similarity measure based on set theo-retic approach is also proposed. Some basic properties of similarity measures between two interval valued neutro-sophic soft sets is also studied. A decision making method is established for interval valued neutrosophic soft set setting using similarity measures between IVNS sets. Finally an example is given to demonstrate the possible application of similarity measures in pattern recognition problems.
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