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.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
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.
In this paper we define some new operations in fuzzy soft multi set theory and show that the DeMorgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations in our way. Also some new results along with illustrating examples have been put
forward in our work.
This document discusses a fusion of soft expert set and matrix models. It begins by introducing soft sets, soft expert sets, fuzzy soft sets, and intuitionistic fuzzy soft sets. It then defines various types of matrices in the context of soft expert sets, including soft expert matrices, soft expert equal matrices, soft expert complement matrices, and operations on soft expert matrices like addition, subtraction, and multiplication. An example is provided to illustrate a soft expert matrix model for a manufacturing firm choosing a location based on expert opinions. The document aims to provide a new dimension to soft expert sets through the use of matrices to solve decision making problems.
A fusion of soft expert set and matrix modelseSAT Journals
Abstract
The purpose of this paper is to define different types of matrices in the light of soft expert sets. We then propose a decision making
model based on soft expert set.
Keywords: Soft set, soft expert set, Soft Expert matrix.
This document introduces the exponential Pareto distribution (EPD), a new probability distribution derived from the exponential and Pareto distributions. Some key properties of the EPD are derived, including its probability density function, cumulative distribution function, moments, coefficients of variation, skewness and kurtosis. Parameter estimation for the EPD is also discussed using maximum likelihood estimation. Plots of the density, distribution, and other functions of the EPD are provided for various parameter values.
Heuristic Algorithm for Finding Sensitivity Analysis in Interval Solid Transp...AM Publications
This paper develops a heuristic algorithm for finding the ranges of cost in the interval solid transportation
problem such that optimal basis is invariant. The procedure of the proposed approach is illustrated by numerical
example.
On optimization ofON OPTIMIZATION OF DOPING OF A HETEROSTRUCTURE DURING MANUF...ijcsitcejournal
We introduce an approach of manufacturing of a p-i-n-heterodiodes. The approach based on using a δ-
doped heterostructure, doping by diffusion or ion implantation of several areas of the heterostructure. After
the doping the dopant and/or radiation defects have been annealed. We introduce an approach to optimize
annealing of the dopant and/or radiation defects. We determine several conditions to manufacture more
compact p-i-n-heterodiodes
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
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.
In this paper we define some new operations in fuzzy soft multi set theory and show that the DeMorgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations in our way. Also some new results along with illustrating examples have been put
forward in our work.
This document discusses a fusion of soft expert set and matrix models. It begins by introducing soft sets, soft expert sets, fuzzy soft sets, and intuitionistic fuzzy soft sets. It then defines various types of matrices in the context of soft expert sets, including soft expert matrices, soft expert equal matrices, soft expert complement matrices, and operations on soft expert matrices like addition, subtraction, and multiplication. An example is provided to illustrate a soft expert matrix model for a manufacturing firm choosing a location based on expert opinions. The document aims to provide a new dimension to soft expert sets through the use of matrices to solve decision making problems.
A fusion of soft expert set and matrix modelseSAT Journals
Abstract
The purpose of this paper is to define different types of matrices in the light of soft expert sets. We then propose a decision making
model based on soft expert set.
Keywords: Soft set, soft expert set, Soft Expert matrix.
This document introduces the exponential Pareto distribution (EPD), a new probability distribution derived from the exponential and Pareto distributions. Some key properties of the EPD are derived, including its probability density function, cumulative distribution function, moments, coefficients of variation, skewness and kurtosis. Parameter estimation for the EPD is also discussed using maximum likelihood estimation. Plots of the density, distribution, and other functions of the EPD are provided for various parameter values.
Heuristic Algorithm for Finding Sensitivity Analysis in Interval Solid Transp...AM Publications
This paper develops a heuristic algorithm for finding the ranges of cost in the interval solid transportation
problem such that optimal basis is invariant. The procedure of the proposed approach is illustrated by numerical
example.
On optimization ofON OPTIMIZATION OF DOPING OF A HETEROSTRUCTURE DURING MANUF...ijcsitcejournal
We introduce an approach of manufacturing of a p-i-n-heterodiodes. The approach based on using a δ-
doped heterostructure, doping by diffusion or ion implantation of several areas of the heterostructure. After
the doping the dopant and/or radiation defects have been annealed. We introduce an approach to optimize
annealing of the dopant and/or radiation defects. We determine several conditions to manufacture more
compact p-i-n-heterodiodes
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
Neutrosophic Soft Topological Spaces on New OperationsIJSRED
The document summarizes a research paper on neutrosophic soft topological spaces based on new operations defined for neutrosophic soft sets. It introduces neutrosophic soft sets and operations such as union, intersection, complement, and subset. It then defines new operations for union, intersection, and difference of neutrosophic soft sets. Finally, it defines the union and intersection of a family of neutrosophic soft sets and explores properties of the new operations.
Fixed point and common fixed point theorems in complete metric spacesAlexander Decker
1) The document presents a fixed point and common fixed point theorem for four self-mappings (E, F, G, H) in a complete metric space that satisfy certain contractive conditions.
2) It establishes that if the mappings satisfy a contractive modulus condition and the pairs (G, E) and (H, F) are weakly compatible, then the mappings have a unique common fixed point.
3) The proof involves showing that a sequence defined using the mappings converges to a fixed point, and that this fixed point is unique.
Results from set-operations on Fuzzy soft setsIJMER
In this paper considering a class of Fuzzy-Soft Sets, seven set operations are defined and
several relations arising from these set-operations are established using matrix representation of fuzzy soft
sets.
Fixed point theorems for random variables in complete metric spacesAlexander Decker
This document presents two fixed point theorems for random variables in complete metric spaces. Theorem 1 proves that if a self-mapping E on a complete metric space satisfies certain rational inequalities involving distances between random variables, then E has a fixed point. Theorem 2 proves a similar result for a self-mapping E satisfying alternative rational inequalities, assuming E is onto. Both theorems use properties of complete metric spaces and rational inequalities to show the existence of fixed points for random variables under the given conditions.
This document proposes a new approach to multi-criteria decision making problems using neutrosophic bipolar vague sets. Neutrosophic bipolar vague sets are an extension of fuzzy sets that can handle imprecision, uncertainty, and bipolar membership degrees. The document defines operations and functions for neutrosophic bipolar vague sets, including weighted average and geometric aggregation operators. It then presents a multi-criteria decision making method based on these operators and functions to evaluate alternatives described by neutrosophic bipolar vague values and select the most desirable option. Numerical examples are provided to demonstrate the application of the proposed decision making approach.
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
This document presents an efficient soft set approach for mining association rules that uses an initial support constraint to filter out false frequent items and rarely occurring items. This improves the structure of the dataset and results in faster, more accurate results that use less memory than previous approaches. The improved dataset is transformed into a Boolean-valued information system, allowing it to be represented as a soft set. Association rules are then mined between sets of co-occurring parameters using soft set theory. Experimental results show this approach produces strong association rules faster with the same accuracy using less memory space.
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
The document describes a numerical method for solving the Helmholtz equation using multivariate Padé approximation. The Helmholtz equation arises in various fields and is difficult to solve using conventional methods. The document presents:
1) The Helmholtz equation and its application in different areas.
2) How two-dimensional differential transformation is used to convert the Helmholtz PDE to a power series.
3) How multivariate Padé approximation is then applied to the power series to accelerate its convergence and reduce computational time in obtaining the numerical solution.
An example demonstrating the method is provided.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
This document summarizes a research paper that proposes using Milne's predictor-corrector method to solve the dependency problem in fuzzy computations when numerically solving fuzzy differential equations (FDEs). The paper first provides background on fuzzy sets, fuzzy numbers, fuzzy processes, and fuzzy initial value problems (FIVPs). It then describes Milne's predictor-corrector method and illustrates how it can be applied to solve some example FIVPs. The goal is to address issues that arise from dependencies in fuzzy computations by using this numerical method to find solutions to FDEs.
Jurnal Study of Anisotropy Superconductor using Time-Dependent Ginzburg-Landa...Fuad Anwar
This document summarizes a study that used numerical simulations based on the time-dependent Ginzburg-Landau equation to study the properties of anisotropic superconductors. The study models an anisotropic superconductor with different effective masses for Cooper pairs along different crystal axes. It describes the theoretical formalism and numerical methods used to solve the time-dependent Ginzburg-Landau equations for an anisotropic superconductor. The results of the numerical simulations show that the anisotropic properties of superconductors can cause the critical field to be either lower or higher compared to isotropic superconductors.
This document discusses the formulation of fractional supersymmetric theories in one dimension. It begins by presenting fractional superspace and fractional supersymmetry of order F=3, including the fractional supersymmetry transformations. It then derives the fractional supercharges and Euler-Lagrange equations for F=3. Finally, it generalizes the formulation to arbitrary fractional order F ≥ 3 by introducing fractional superspace and supersymmetry transformations of order F, as well as an action invariant under such transformations.
The document describes a finite element analysis of a classroom seat. It includes:
1) Modeling of the seat structure using wood and steel materials and defining boundary conditions and load.
2) Generation of a finite element mesh and formulation of the stiffness matrix.
3) Computation of displacements, strains, and stresses using the stiffness matrix and material properties.
4) The maximum displacement and stresses in the seat structure are reported.
On Coincidence Points in Pseudocompact Tichonov Spaces and Common Fixed Point...inventionjournals
1) The document presents several theorems establishing the existence of coincidence points and common fixed points for self-maps on pseudocompact Tichonov spaces and pseudocompact topological spaces.
2) Theorem 1.1 proves that if three self-maps f, g, and h on a pseudocompact Tichonov space satisfy certain contractive conditions involving a metric-type function F, then f and h or g and h have a coincidence point.
3) Theorem 2.1 shows that if two weakly commuting self-maps f and g on a pseudocompact topological space satisfy a generalized contractive condition involving F, then f and g have a coincidence point and a unique common fixed point
On Decreasing of Dimensions of Field-Effect Heterotransistors in Logical CMOP...BRNSS Publication Hub
In this paper, we introduce an approach to decrease the dimensions of CMOP voltage differencing inverting buffered amplifier based on field-effect heterotransistors by increasing density of elements. Dimensions of the elements will be decreased due to manufacture heterostructure with a specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
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.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
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.
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The document summarizes research on complex fuzzy sets. Complex fuzzy sets extend traditional fuzzy sets by allowing membership functions to range over the unit circle in the complex plane rather than just [0,1]. The paper defines complex fuzzy sets and operations like union, intersection, and complement. It presents examples and properties of these operations. It also introduces a probabilistic interpretation of complex fuzzy sets to distinguish them from probability.
Hypersoft set is an extension of the soft set where there is more than one set of attributes occur and it is very much helpful in multi-criteria group decision making problem. In a hypersoft set, the function F is a multi-argument function. In this paper, we have used the notion of Fuzzy Hypersoft Set (FHSS), which is a combination of fuzzy set and hypersoft set. In earlier research works the concept of Fuzzy Soft Set (FSS) was introduced and it was applied successfully in various fields. The FHSS theory gives more flexibility as compared to FSS to tackle the parameterized problems of uncertainty. To overcome the issue where FSS failed to explain uncertainty and incompleteness there is a dire need for another environment which is known as FHSS. It works well when there is more complexity involved in the parametric data i.e the data that involves vague concepts. This work includes some basic set-theoretic operations on FHSSs and for the reliability and the authenticity of these operations, we have shown its application with the help of a suitable example. This example shows that how FHSS theory plays its role to solve real decision-making problems.
The document introduces the concept of generalized neutrosophic soft sets which incorporates properties of generalized neutrosophic sets and soft set theory. It defines generalized neutrosophic soft sets and provides an example. Generalized neutrosophic soft sets allow for modeling of vague, uncertain and inconsistent data and consist of a collection of parameterized generalized neutrosophic sets that can be used to approximate objects. Operations on generalized neutrosophic soft sets such as union and intersection are introduced to analyze and process such data.
International Journal of Computer Science, Engineering and Information Techno...ijcseit
In this paper we present a new concept called “generalized neutrosophic soft set”. This concept
incorporates the beneficial properties of both generalized neutrosophic set introduced by A.A.Salama [7]
and soft set techniques proposed by Molodtsov [4]. We also study some properties of this concept. Some
definitions and operations have been introduced on generalized neutrosophic soft set. Finally we present
an application of generalized neuutrosophic soft set in decision making problem.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex fuzzy relation are introduced.
Neutrosophic Soft Topological Spaces on New OperationsIJSRED
The document summarizes a research paper on neutrosophic soft topological spaces based on new operations defined for neutrosophic soft sets. It introduces neutrosophic soft sets and operations such as union, intersection, complement, and subset. It then defines new operations for union, intersection, and difference of neutrosophic soft sets. Finally, it defines the union and intersection of a family of neutrosophic soft sets and explores properties of the new operations.
Fixed point and common fixed point theorems in complete metric spacesAlexander Decker
1) The document presents a fixed point and common fixed point theorem for four self-mappings (E, F, G, H) in a complete metric space that satisfy certain contractive conditions.
2) It establishes that if the mappings satisfy a contractive modulus condition and the pairs (G, E) and (H, F) are weakly compatible, then the mappings have a unique common fixed point.
3) The proof involves showing that a sequence defined using the mappings converges to a fixed point, and that this fixed point is unique.
Results from set-operations on Fuzzy soft setsIJMER
In this paper considering a class of Fuzzy-Soft Sets, seven set operations are defined and
several relations arising from these set-operations are established using matrix representation of fuzzy soft
sets.
Fixed point theorems for random variables in complete metric spacesAlexander Decker
This document presents two fixed point theorems for random variables in complete metric spaces. Theorem 1 proves that if a self-mapping E on a complete metric space satisfies certain rational inequalities involving distances between random variables, then E has a fixed point. Theorem 2 proves a similar result for a self-mapping E satisfying alternative rational inequalities, assuming E is onto. Both theorems use properties of complete metric spaces and rational inequalities to show the existence of fixed points for random variables under the given conditions.
This document proposes a new approach to multi-criteria decision making problems using neutrosophic bipolar vague sets. Neutrosophic bipolar vague sets are an extension of fuzzy sets that can handle imprecision, uncertainty, and bipolar membership degrees. The document defines operations and functions for neutrosophic bipolar vague sets, including weighted average and geometric aggregation operators. It then presents a multi-criteria decision making method based on these operators and functions to evaluate alternatives described by neutrosophic bipolar vague values and select the most desirable option. Numerical examples are provided to demonstrate the application of the proposed decision making approach.
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
This document presents an efficient soft set approach for mining association rules that uses an initial support constraint to filter out false frequent items and rarely occurring items. This improves the structure of the dataset and results in faster, more accurate results that use less memory than previous approaches. The improved dataset is transformed into a Boolean-valued information system, allowing it to be represented as a soft set. Association rules are then mined between sets of co-occurring parameters using soft set theory. Experimental results show this approach produces strong association rules faster with the same accuracy using less memory space.
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
The document describes a numerical method for solving the Helmholtz equation using multivariate Padé approximation. The Helmholtz equation arises in various fields and is difficult to solve using conventional methods. The document presents:
1) The Helmholtz equation and its application in different areas.
2) How two-dimensional differential transformation is used to convert the Helmholtz PDE to a power series.
3) How multivariate Padé approximation is then applied to the power series to accelerate its convergence and reduce computational time in obtaining the numerical solution.
An example demonstrating the method is provided.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
This document summarizes a research paper that proposes using Milne's predictor-corrector method to solve the dependency problem in fuzzy computations when numerically solving fuzzy differential equations (FDEs). The paper first provides background on fuzzy sets, fuzzy numbers, fuzzy processes, and fuzzy initial value problems (FIVPs). It then describes Milne's predictor-corrector method and illustrates how it can be applied to solve some example FIVPs. The goal is to address issues that arise from dependencies in fuzzy computations by using this numerical method to find solutions to FDEs.
Jurnal Study of Anisotropy Superconductor using Time-Dependent Ginzburg-Landa...Fuad Anwar
This document summarizes a study that used numerical simulations based on the time-dependent Ginzburg-Landau equation to study the properties of anisotropic superconductors. The study models an anisotropic superconductor with different effective masses for Cooper pairs along different crystal axes. It describes the theoretical formalism and numerical methods used to solve the time-dependent Ginzburg-Landau equations for an anisotropic superconductor. The results of the numerical simulations show that the anisotropic properties of superconductors can cause the critical field to be either lower or higher compared to isotropic superconductors.
This document discusses the formulation of fractional supersymmetric theories in one dimension. It begins by presenting fractional superspace and fractional supersymmetry of order F=3, including the fractional supersymmetry transformations. It then derives the fractional supercharges and Euler-Lagrange equations for F=3. Finally, it generalizes the formulation to arbitrary fractional order F ≥ 3 by introducing fractional superspace and supersymmetry transformations of order F, as well as an action invariant under such transformations.
The document describes a finite element analysis of a classroom seat. It includes:
1) Modeling of the seat structure using wood and steel materials and defining boundary conditions and load.
2) Generation of a finite element mesh and formulation of the stiffness matrix.
3) Computation of displacements, strains, and stresses using the stiffness matrix and material properties.
4) The maximum displacement and stresses in the seat structure are reported.
On Coincidence Points in Pseudocompact Tichonov Spaces and Common Fixed Point...inventionjournals
1) The document presents several theorems establishing the existence of coincidence points and common fixed points for self-maps on pseudocompact Tichonov spaces and pseudocompact topological spaces.
2) Theorem 1.1 proves that if three self-maps f, g, and h on a pseudocompact Tichonov space satisfy certain contractive conditions involving a metric-type function F, then f and h or g and h have a coincidence point.
3) Theorem 2.1 shows that if two weakly commuting self-maps f and g on a pseudocompact topological space satisfy a generalized contractive condition involving F, then f and g have a coincidence point and a unique common fixed point
On Decreasing of Dimensions of Field-Effect Heterotransistors in Logical CMOP...BRNSS Publication Hub
In this paper, we introduce an approach to decrease the dimensions of CMOP voltage differencing inverting buffered amplifier based on field-effect heterotransistors by increasing density of elements. Dimensions of the elements will be decreased due to manufacture heterostructure with a specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
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.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
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.
PROBABILISTIC INTERPRETATION OF COMPLEX FUZZY SETIJCSEIT Journal
The document summarizes research on complex fuzzy sets. Complex fuzzy sets extend traditional fuzzy sets by allowing membership functions to range over the unit circle in the complex plane rather than just [0,1]. The paper defines complex fuzzy sets and operations like union, intersection, and complement. It presents examples and properties of these operations. It also introduces a probabilistic interpretation of complex fuzzy sets to distinguish them from probability.
Hypersoft set is an extension of the soft set where there is more than one set of attributes occur and it is very much helpful in multi-criteria group decision making problem. In a hypersoft set, the function F is a multi-argument function. In this paper, we have used the notion of Fuzzy Hypersoft Set (FHSS), which is a combination of fuzzy set and hypersoft set. In earlier research works the concept of Fuzzy Soft Set (FSS) was introduced and it was applied successfully in various fields. The FHSS theory gives more flexibility as compared to FSS to tackle the parameterized problems of uncertainty. To overcome the issue where FSS failed to explain uncertainty and incompleteness there is a dire need for another environment which is known as FHSS. It works well when there is more complexity involved in the parametric data i.e the data that involves vague concepts. This work includes some basic set-theoretic operations on FHSSs and for the reliability and the authenticity of these operations, we have shown its application with the help of a suitable example. This example shows that how FHSS theory plays its role to solve real decision-making problems.
The document introduces the concept of generalized neutrosophic soft sets which incorporates properties of generalized neutrosophic sets and soft set theory. It defines generalized neutrosophic soft sets and provides an example. Generalized neutrosophic soft sets allow for modeling of vague, uncertain and inconsistent data and consist of a collection of parameterized generalized neutrosophic sets that can be used to approximate objects. Operations on generalized neutrosophic soft sets such as union and intersection are introduced to analyze and process such data.
International Journal of Computer Science, Engineering and Information Techno...ijcseit
In this paper we present a new concept called “generalized neutrosophic soft set”. This concept
incorporates the beneficial properties of both generalized neutrosophic set introduced by A.A.Salama [7]
and soft set techniques proposed by Molodtsov [4]. We also study some properties of this concept. Some
definitions and operations have been introduced on generalized neutrosophic soft set. Finally we present
an application of generalized neuutrosophic soft set in decision making problem.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex fuzzy relation are introduced.
The aim of this paper is to investigate different definitions of soft points in the existing literature on soft set theory and its extensions in different directions. Then limitations of these definitions are illustrated with the help of examples. Moreover, the definition of soft point in the setup of fuzzy soft set, intervalvalued fuzzy soft set, hesitant fuzzy soft set and intuitionistic soft set are also discussed. We also suggest an approach to unify the definitions of soft point which is more applicable than the existing notions.
A common random fixed point theorem for rational inequality in hilbert spaceAlexander Decker
This document presents a common random fixed point theorem for four continuous random operators defined on a non-empty closed subset of a separable Hilbert space. It begins with introducing basic concepts such as separable Hilbert spaces, random operators, and common random fixed points. It then defines a condition (A) that the four mappings must satisfy. The main result is Theorem 2.1, which proves the existence of a unique common random fixed point for the four operators under condition (A) and a rational inequality condition. The proof constructs a sequence of measurable functions and shows it converges to the common random fixed point. This establishes the common random fixed point theorem for these operators.
In real life situations, there are many issues in which we face uncertainties, vagueness, complexities and unpredictability. Neutrosophic sets are a mathematical tool to address some issues which cannot be met using the existing methods. Neutrosophic soft matrices play a crucial role in handling indeterminant and inconsistent information during decision making process. The main focus of this article is to discuss the concept of neutrosophic sets, neutrosophic soft sets and neutrosophic soft matrices theory which are very useful and applicable in various situations involving uncertainties and imprecisions. Thereafter our intention is to find a new method for constructing a decision matrix using neutrosophic soft matrices as an application of the theory. A neutrosophic soft matrix based algorithm is considered to solve some problems in the diagnosis of a disease from the occurrence of various symptoms in patients. This article deals with patient-symptoms and symptoms-disease neutrosophic soft matrices. To come to a decision, a score matrix is defined where multiplication based on max-min operation and complementation of neutrosophic soft matrices are taken into considerations.
An Application of Interval Valued Fuzzy Soft Matrix In Medical Diagnosisiosrjce
: Today is a world of uncertainty with its associated problems, which can be well handled by soft set
theory. In this paper, we extend sanchez`s approach for medical diagnosis using the representation of an
interval valued fuzzy soft matrices. we introduce the definition of union and intersection of Interval valued fuzzy
soft matrices with examples.Finally, we extend our approach in application of these matrices in medical
Diagnosis.
A COUNTEREXAMPLE TO THE FORWARD RECURSION IN FUZZY CRITICAL PATH ANALYSIS UND...Wireilla
Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy version of the backward recursion has been shown to produce results that incorrectly amplify the level of uncertainty. However, the fuzzy version of the forward recursion has been widely proposed as an approach for determining the fuzzy set of critical path lengths. In this paper, the direct application of the extension principle leads to a proposition that must be satisfied in fuzzy critical path analysis. Using a counterexample it is demonstrated that the fuzzy forward recursion when discrete fuzzy sets are used to represent activity durations produces results that are not consistent with the theory presented. The problem is shown to be the application of the fuzzy maximum. Several methods presented in the literature are described and shown to provide results that are consistent with the extension principle.
A Counterexample to the Forward Recursion in Fuzzy Critical Path Analysis Und...ijfls
This document presents a counterexample demonstrating that the fuzzy forward recursion method for determining critical paths does not always produce results consistent with the extension principle when discrete fuzzy sets are used to represent activity durations.
The document first provides background on fuzzy sets and critical path analysis. It then presents a proposition stating that the membership function for fuzzy critical path lengths can be determined by taking the maximum of the minimum membership values across all activity durations in each configuration.
The document goes on to present a counterexample using a simple series-parallel network with 18 configurations. It shows that applying the fuzzy forward recursion produces a different membership value for one critical path length compared to directly applying the extension principle. This difference proves the fuzzy forward
Fundamentals of Parameterised Covering Approximation SpaceYogeshIJTSRD
Combination of theories has not only advanced the research, but also helped in handling the issues of impreciseness in real life problems. The soft rough set has been defined by many authors by combining the theories of soft set and rough set. The concept Soft Covering Based Rough Set be given by J.Zhan et al 2008 , Feng Feng et al 2011 , S.Yuksel et al 2015 by taking full soft set instead of Covering. In this note We first consider the covering soft set and then covering based soft rough set. Again it defines a mapping from the coverings of element of universal set U to the parameters attributes . The new model “Parameterised Soft Rough Set on Covering Approximation Space is conceptualised to capture the issues of vagueness, and impreciseness of information. Also dependency on this new model and some properties be studied. Kedar Chandra Parida | Debadutta Mohanty "Fundamentals of Parameterised Covering Approximation Space" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38761.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/38761/fundamentals-of-parameterised-covering-approximation-space/kedar-chandra-parida
This document presents a new multivariate fuzzy time series forecasting method to predict car road accidents. The method uses four secondary factors (number killed, mortally wounded, died 30 days after accident, severely wounded, and lightly casualties) along with the main factor of total annual car accidents in Belgium from 1974 to 2004. The new method establishes fuzzy logical relationships between the factors to generate forecasts. Experimental results show the proposed method performs better than existing fuzzy time series forecasting approaches at predicting car accidents. Actuaries can use this kind of multivariate fuzzy time series analysis to help define insurance premiums and underwriting.
In this paper, we study the cartesian product of intuitionistic fuzzy soft normal subgroup structure over snorm. By using s-norm of S, we characterize some basic results of intuitionistic S-fuzzy soft subgroup of normal subgroup. Also, we define the relationship between intuitionistic S-fuzzy soft subgroup and intuitionistic S-fuzzy soft normal subgroup. Finally we prove some basic properties
A Real Time Application of Soft Set in Parameterization Reduction for Decisio...IJECEIAES
In each and every field of science and technology Information science plays an important role. Sometimes information science is facing different types of problems to handle the data and information. Data Uncertainty is one of the challenging difficulties to handle. In past, there are several theories like fuzzy set, Rough set, Probability etc.to dealing with uncertainty. Soft set theory is the youngest theory to deal with uncertainty. In this paper we discussed how to find reducts. This paper focuses on how we can transform a sample data set to binary valued information system. We are also going to reduce the dimension of data set by using the binary valued information that results a better decision.
An Interval Based Fuzzy Multiple Expert System to Analyze the Impacts of Clim...ijdmtaiir
Indian agriculture is completely dependent on the
environment and any undesirable change in the environment
has an adverse impact on agriculture. Climate change and
pollution in India have caused great damage to the
environment. In this paper we analyse the impact of climate
change on Indian agriculture. The first section gives an
introduction to the problem. In section two we introduce a new
fuzzy tool called interval based fuzzy multiple expert system.
Section three adapt the new fuzzy tool to analyse the problem
of agricultural impacts of climate change and in section four
we give the results and suggestions based on our analysis
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
This document summarizes a research paper that proposes three new information measures for fuzzy rough sets and evaluates their applications. It begins with background on fuzzy rough sets and existing information measures. It then defines a new logarithmic information measure for fuzzy rough values and proves its validity. Another proposed information measure for fuzzy rough sets is described along with an illustration of its application. A weighted information measure for fuzzy rough sets is also discussed. The paper compares the proposed information measures to other existing ones and concludes with references.
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
This document summarizes a research paper that proposes three new information measures for fuzzy rough sets and evaluates their applications. It begins with background on fuzzy rough sets and existing information measures. It then defines a new logarithmic information measure for fuzzy rough values and proves its validity. Another proposed information measure for fuzzy rough sets is described along with an illustration of its application. A weighted information measure for fuzzy rough sets is also discussed. The paper compares the proposed information measures to other existing ones and concludes with references.
Similar to On Fuzzy Soft Multi Set and Its Application in Information Systems (20)
NEW ONTOLOGY RETRIEVAL IMAGE METHOD IN 5K COREL IMAGESijcax
This document summarizes a research paper that proposes a new ontology-based method for automatic image retrieval and annotation using 5,000 images from the Corel dataset. The method combines global and regional visual features with contextual relationships defined in an ontology. It creates a new ontology based on WordNet to semantically relate tags and reduce gaps between low-level features and high-level concepts. Experimental results show the proposed method increases annotation accuracy compared to other methods.
THE STUDY OF CUCKOO OPTIMIZATION ALGORITHM FOR PRODUCTION PLANNING PROBLEMijcax
Constrained Nonlinear programming problems are hard problems, and one of the most widely used and
common problems for production planning problem to optimize. In this study, one of the mathematical
models of production planning is survey and the problem solved by cuckoo algorithm. Cuckoo Algorithm is
efficient method to solve continues non linear problem. Moreover, mentioned models of production
planning solved with Genetic algorithm and Lingo software and the results will compared. The Cuckoo
Algorithm is suitable choice for optimization in convergence of solution
COMPARATIVE ANALYSIS OF ROUTING PROTOCOLS IN MOBILE AD HOC NETWORKSijcax
This document compares the performance of five routing protocols (AODV, DSR, DSDV, OLSR, DYMO) in mobile ad hoc networks through simulations. It summarizes each protocol and discusses the simulation setup. The protocols are categorized as reactive, proactive, or hybrid. Key performance metrics like packet delivery ratio, end-to-end delay, and routing load are evaluated under varying pause times using the NS-2 simulator. The analysis seeks to determine the best operational conditions for each protocol in mobile ad hoc networks.
PREDICTING ACADEMIC MAJOR OF STUDENTS USING BAYESIAN NETWORKS TO THE CASE OF ...ijcax
In this study, which took place current year in the city of Maragheh in IRAN. Number of high school
students in the fields of study: mathematics, Experimental Sciences, humanities, vocational, business and
science were studied and compared. The purpose of this research is to predict the academic major of high
school students using Bayesian networks. The effective factors have been used in academic major selection
for the first time as an effective indicator of Bayesian networks. Evaluation of Impacts of indicators on
each other, discretization data and processing them was performed by GeNIe. The proper course would be
advised for students to continue their education.
A Multi Criteria Decision Making Based Approach for Semantic Image Annotation ijcax
Automatic image annotation has emerged as an important research topic due to its potential application on
both image understanding and web image search. This paper presents a model, which integrates visual
topics and regional contexts to automatic image annotation. Regional contexts model the relationship
between the regions, while visual topics provide the global distribution of topics over an image. Previous
image annotation methods neglected the relationship between the regions in an image, while these regions
are exactly explanation of the image semantics, therefore considering the relationship between them are
helpful to annotate the images. Regional contexts and visual topics are learned by PLSA (Probability
Latent Semantic Analysis) from the training data. The proposed model incorporates these two types of
information by MCDM (Multi Criteria Decision Making) approach based on WSM (Weighted Sum
Method). Experiments conducted on the 5k Corel dataset demonstrate the effectiveness of the proposed
model.
Web is a collection of inter-related files on one or more web servers while web mining means extracting
valuable information from web databases. Web mining is one of the data mining domains where data
mining techniques are used for extracting information from the web servers. The web data includes web
pages, web links, objects on the web and web logs. Web mining is used to understand the customer
behaviour, evaluate a particular website based on the information which is stored in web log files. Web
mining is evaluated by using data mining techniques, namely classification, clustering, and association
rules. It has some beneficial areas or applications such as Electronic commerce, E-learning, Egovernment, E-policies, E-democracy, Electronic business, security, crime investigation and digital library.
Retrieving the required web page from the web efficiently and effectively becomes a challenging task
because web is made up of unstructured data, which delivers the large amount of information and increase
the complexity of dealing information from different web service providers. The collection of information
becomes very hard to find, extract, filter or evaluate the relevant information for the users. In this paper,
we have studied the basic concepts of web mining, classification, processes and issues. In addition to this,
this paper also analyzed the web mining research challenges.
SPAM FILTERING SECURITY EVALUATION FRAMEWORK USING SVM, LR AND MILR ijcax
The Pattern classification system classifies the pattern into feature space within a boundary. In case
adversarial applications use, for example Spam Filtering, the Network Intrusion Detection System (NIDS),
Biometric Authentication, the pattern classification systems are used. Spam filtering is an adversary
application in which data can be employed by humans to attenuate perspective operations. To appraise the
security issue related Spam Filtering voluminous machine learning systems. We presented a framework for
the experimental evaluation of the classifier security in an adversarial environments, that combines and
constructs on the arms race and security by design, Adversary modelling and Data distribution under
attack. Furthermore, we presented a SVM, LR and MILR classifier for classification to categorize email as
legitimate (ham) or spam emails on the basis of thee text samples.
Visually impaired people face many problems in their day to day lives. Among them, outdoor navigation is
one of the major concerns. The existing solutions based on Wireless Sensor Networks(WSN) and Global
Positioning System (GPS) track ZigBee units or RFID (Radio Frequency Identification) tags fixed on the
navigation system. The issues pertaining to these solutions are as follows: (1) It is suitable only when the
visually impaired person is commuting in a familiar environment; (2) The device provides only a one way
communication; (3) Most of these instruments are heavy and sometimes costly. Preferable solution would
be to make a system which is easy to carry and cheap.
The objective of this paper is to break down the technological barriers, and to propose a system by
developing an Android App which would help a visually impaired person while traveling via the public
transport system like Bus. The proposed system uses an inbuilt feature of smart phone such as GPS
location tracker to track the location of the user and Text to Speech converter. The system also integrates
Google Speech to Text converter for capturing the voice input and converts them to text. This system
recommends the requirement of installing a GPS module in buses for real time tracking. With minor
modification, this App can also help older people for independent navigation.
INTELLIGENT AGENT FOR PUBLICATION AND SUBSCRIPTION PATTERN ANALYSIS OF NEWS W...ijcax
The rapid growth of Internet has revolutionized online news reporting. Many users tend to use online news
websites to obtain news information. When considering Sri Lanka, there are numerous news websites,
which are subscribed on a daily basis. With the rise in this number of news websites, the Sri Lankan
authorities of media face the issue of lacking a proper methodology or a tool which is capable of tracking
and regulating publications made by different disseminators of news.
This paper proposes a News Agent toolbox which periodically extracts news articles and associated
comments with the aid of a concept called Mapping Rules; to classify them into Personalized Categories
defined in terms of keywords based Category Profiles. The proposed tool also analyzes comments made by
the readers with the aid of simple statistical techniques to discover the most popular news articles and
fluctuations in popularity of news stories.
ADVANCED E-VOTING APPLICATION USING ANDROID PLATFORMijcax
The advancement in the mobile devices, wireless and web technologies given rise to the new application
that will make the voting process very easy and efficient. The E-voting promises the possibility of
convenient, easy and safe way to capture and count the votes in an election[1]. This research project
provides the specification and requirements for E-Voting using an Android platform. The e-voting means
the voting process in election by using electronic device. The android platform is used to develop an evoting application. At first, an introduction about the system is presented. Sections II and III describe all
the concepts (survey, design and implementation) that would be used in this work. Finally, the proposed evoting system will be presented. This technology helps the user to cast the vote without visiting the polling
booth. The application follows proper authentication measures in order to avoid fraud voters using the
system. Once the voting session is completed the results can be available within a fraction of seconds. All
the candidates vote count is encrypted and stored in the database in order to avoid any attacks and
disclosure of results by third person other than the administrator. Once the session is completed the admin
can decrypt the vote count and publish results and can complete the voting process.
The design of silicon chips in every semiconductor industry involves the testing of these chips with other
components on the board. The platform developed acts as power on vehicle for the silicon chips. This
Printed Circuit Board design that serves as a validation platform is foundational to the semiconductor
industry.
The manual/repetitive design activities that accompany the development of this board must be minimized to
achieve high quality, improve design efficiency, and eliminate human-errors. One of the time consuming
tasks in the board design is the Trace Length matching. The paper aims to reduce the length matching time
by automating it using SKILL scripts.
RESEARCH TRENDS İN EDUCATIONAL TECHNOLOGY İN TURKEY: 2010-2018 YEAR THESIS AN...ijcax
The purpose of this research is the analysis using meta-analysis of studies in the field of Educational
Technology in Turkey and in the field is to demonstrate how to get to that trend. For this purpose, a total of
263 studies were analyzed including 98 theses and 165 articles published between 2010-2018. Purpose
sampling method was used when selecting publications. In the research, while selecting articles and theses;
Turkey addressed; YOK Tez Tarama Database, Journal of Hacettepe University Faculty of Education,
Educational Sciences : Theory & Practice Journal, Education and Science Journal, Elementary Education
Online Journal, The Turkish Online Journal of Education and The Turkish Online Journal of Educational
Technology used in journals. Publications have been reviewed under 11 criteria. Index, year of
publication, research scope, method, education level, sample, number of samples, data collection methods,
analysis techniques, and research tendency, research topics in Educational Technology Research in Turkey
has revealed. The data is interpreted based on percentage and frequency and the results are shown using
the table.
RESEARCH TRENDS İN EDUCATIONAL TECHNOLOGY İN TURKEY: 2010-2018 YEAR THESIS AN...ijcax
The purpose of this research is the analysis using meta-analysis of studies in the field of Educational
Technology in Turkey and in the field is to demonstrate how to get to that trend. For this purpose, a total of
263 studies were analyzed including 98 theses and 165 articles published between 2010-2018. Purpose
sampling method was used when selecting publications. In the research, while selecting articles and theses;
Turkey addressed; YOK Tez Tarama Database, Journal of Hacettepe University Faculty of Education,
Educational Sciences : Theory & Practice Journal, Education and Science Journal, Elementary Education
Online Journal, The Turkish Online Journal of Education and The Turkish Online Journal of Educational
Technology used in journals. Publications have been reviewed under 11 criteria. Index, year of
publication, research scope, method, education level, sample, number of samples, data collection methods,
analysis techniques, and research tendency, research topics in Educational Technology Research in Turkey
has revealed. The data is interpreted based on percentage and frequency and the results are shown using
the table
IMPACT OF APPLYING INTERNATIONAL QUALITY STANDARDS ON MEDICAL EQUIPMENT IN SA...ijcax
This document summarizes a study on the impact of applying international quality standards on medical equipment in Saudi Arabia. A questionnaire was distributed to 300 healthcare professionals in public and private hospitals to collect data. The results showed that around 80% of respondents strongly agreed that international standards like ISO have significantly improved safety, testing and calibration of medical devices, use of consistent terminology, and compatibility/interoperability. Adopting global quality standards appears to have reduced risks associated with medical equipment and enhanced patient safety in Saudi Arabia according to the healthcare workers surveyed.
SPAM FILTERING SECURITY EVALUATION FRAMEWORK USING SVM, LR AND MILR ijcax
The Pattern classification system classifies the pattern into feature space within a boundary. In case adversarial applications use, for example Spam Filtering, the Network Intrusion Detection System (NIDS), Biometric Authentication, the pattern classification systems are used. Spam filtering is an adversary
application in which data can be employed by humans to attenuate perspective operations. To appraise the
security issue related Spam Filtering voluminous machine learning systems. We presented a framework for the experimental evaluation of the classifier security in an adversarial environments, that combines and constructs on the arms race and security by design, Adversary modelling and Data distribution under
attack. Furthermore, we presented a SVM, LR and MILR classifier for classification to categorize email as legitimate (ham) or spam emails on the basis of thee text samples
Developing Product Configurator Tool Using CADs’ API with the help of Paramet...ijcax
Order placingis a crucial phase of lifecycle of a Mass-customizable product and seeks improvement in
Mechanical industry. ‘Product Configurator’ is a good solution to bring in data transparency and speed
up the process. Configuration tools arebeing used on a very small scale,reasons being lack of awareness
and dearer costs of existing tools. In this research work a product configurator is developedfor
Hydraulic Actuator (HA).This method uses Applicable Programing Interface (API) of a CAD tool coupled
with Visual Basics (VB) and MS Excel.Itis a standaloneapplication of VB and its integration into web
portal can be the future scope. The final aim was to reduce time delay at CRM phase,bring more
transparency in the ordering system and to establish a method which, small and medium scale enterprises
canafford. Trails on the tool developed generated Part-Assembly drawings, BOM and JT files in
moments.
DESIGN AND DEVELOPMENT OF CUSTOM CHANGE MANAGEMENT WORKFLOW TEMPLATES AND HAN...ijcax
A large no. of automobile companies finding a convinient way to manage design changes with the use of
various PLM techniques. Change in any product is something that should occur on timely basis to match
up with customer requirement and cost reduction. The change made in the vehicle designs directly affects
various concerned agencies. Automobile Vehicle structures contains thousands of parts and if there is any
change is occurring in child parts then it becomes important to track that impacted part, propose a solution
on that part and release a new assembly structure with feasible changes such that all efforts need to be
done for cost reduction.
Visually impaired people face many problems in their day to day lives. Among them, outdoor navigation is
one of the major concerns. The existing solutions based on Wireless Sensor Networks(WSN) and Global
Positioning System (GPS) track ZigBee units or RFID (Radio Frequency Identification) tags fixed on the
navigation system. The issues pertaining to these solutions are as follows: (1) It is suitable only when the
visually impaired person is commuting in a familiar environment; (2) The device provides only a one way
communication; (3) Most of these instruments are heavy and sometimes costly. Preferable solution would
be to make a system which is easy to carry and cheap.
The objective of this paper is to break down the technological barriers, and to propose a system by
developing an Android App which would help a visually impaired person while traveling via the public
transport system like Bus. The proposed system uses an inbuilt feature of smart phone such as GPS
location tracker to track the location of the user and Text to Speech converter. The system also integrates
Google Speech to Text converter for capturing the voice input and converts them to text. This system
recommends the requirement of installing a GPS module in buses for real time tracking. With minor
modification, this App can also help older people for independent navigation.
TEACHER’S ATTITUDE TOWARDS UTILISING FUTURE GADGETS IN EDUCATION ijcax
Today’s era is an era of modernization and globalization. Everything is happening at a very fast rate
whether it is politics, societal reforms, commercialization, transportation, or educational innovations. In
every few second, technology grows either in the form of arrival of the new devices/gadgets with millions of
apps and these latest technological objects may be in the form of hardware/software devices. We are the
educationists, teachers, students and stakeholders of present Indian educational system. These
gadgets/devices are partly being used by us or most of them are still unaware of these innovative
technologies due to the mass media or economical factor. So, there is a need to improvise ourselves
towards utilizing the future gadgets in order to explore the educational uses, barriers and preparatoryneeds of these available devices for educational purposes. This paper aims to study the opinion of the
teacher-educators about the usage of future gadgets in higher education. It will also contribute towards
establishing the list of latest technological devices, and how it can enhances the process of teachinglearning system.
IMPROVING REAL TIME TASK AND HARNESSING ENERGY USING CSBTS IN VIRTUALIZED CLOUDijcax
Cloud computing provides the facility for the business customers to scale up and down their resource usage
based on needs. This is because of the virtualization technology. The scheduling objectives are to improve
the system’s schedule ability for the real-time tasks and to save energy. To achieve the objectives, we
employed the virtualization technique and rolling-horizon optimization with vertical scheduling operation.
The project considers Cluster Scoring Based Task Scheduling (CSBTS) algorithm which aims to decrease
task’s completion time and the policies for VM’s creation, migration and cancellation are to dynamically
adjust the scale of cloud in a while meets the real-time requirements and to save energy
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Walmart Business+ and Spark Good for Nonprofits.pdf
On Fuzzy Soft Multi Set and Its Application in Information Systems
1. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
DOI:10.5121/ijcax.2015.2303 29
On Fuzzy Soft Multi Set and Its Application in
Information Systems
Anjan Mukherjee1
and Ajoy Kanti Das2
1
Department of Mathematics,Tripura University, Agartala-799022,Tripura, India
2
Department of Mathematics, ICV-College, Belonia -799155, Tripura, India
ABSTRACT
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.
KEYWORDS
Soft set, fuzzy set, soft multi set, fuzzy soft multi set, information system.
1. INTRODUCTION
In recent years vague concepts have been used in different areas such as information and
communication technologies, medical applications, pharmacology, economics and engineering
since; these kinds of problems have their own uncertainties. There are many mathematical tools
for dealing with uncertainties; some of them are fuzzy set theory [18] and soft set theory [12]. In
soft set theory there is no limited condition to the description of objects; so researchers can
choose the form of parameters they need, which greatly simplifies the decision making process
and make the process more efficient in the absence of partial information. Although many
mathematical tools are available for modelling uncertainties such as probability theory, fuzzy set
theory, rough set theory, interval valued mathematics etc, but there are inherent difficulties
associated with each of these techniques. Soft set theory is standing in a unique way in the sense
that it is free from the above difficulties. Soft set theory has a rich potential for application in
many directions, some of which are reported by Molodtsov [12] in his work. Later on Maji et
al.[10, 11] presented some new definitions on soft sets and discussed in details the application of
soft set in decision making problem. Based on the analysis of several operations on soft sets
introduced in [12], Ali et al. [2] presented some new algebraic operations for soft sets and proved
that certain De Morgan’s law holds in soft set theory with respect to these new definitions.
Combining soft sets [12] with fuzzy sets [18], Maji et al. [9] defined fuzzy soft sets, which are
rich potential for solving decision making problems. Alkhazaleh and others [[1], [4], [5], [7],
[16], [17]] as a generalization of Molodtsov’s soft set, presented the definition of a soft multi set
2. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
30
and its basic operations such as complement, union, and intersection etc. In 2012 Alkhazaleh and
Salleh [3] introduced the concept of fuzzy soft multi set theory and studied the application of
these sets and recently, Mukherjee and Das[14] defined some new algebraic operations for fuzzy
soft multi sets and proved that certain De Morgan's law holds in fuzzy soft multi set theory with
respect to these new definitions. They also presented an application of fuzzy soft multi set based
decision making problems in [13].
Molodtsov [12] presented some applications of soft set theory in several directions, which
includes: the study of smoothness of functions, game theory, operations research, Riemann-
integration, probability, theory of measurement, etc. It has been shown that there is compact
connection between soft sets and information system. From the concept and the example of fuzzy
soft multisets given in the section 4.3, it can be seen that a fuzzy soft multi set is a multi-valued
information system. In this paper we define the notion of fuzzy multi-valued information system
in fuzzy soft multi set theory and application of fuzzy soft multi sets in information system were
presented and discussed.
2. PRELIMINARY NOTES
In this section, we recall some basic notions in soft set theory and fuzzy soft multi set theory. Let
U be an initial universe and E be a set of parameters. Let P(U) denotes the power set of U and
A⊆E.
A pair (F, A) is called a soft set over U, where F is a mapping given by F: A→ P(U). In other
words, soft set over U is a parameterized family of subsets of the universe U.
2.1. Definition [12]
Let { }:iU i I∈ be a collection of universes such that i I iU φ∈ =I and let { }:iUE i I∈ be a
collection of sets of parameters. Let ( )i I iU FS U∈= ∏ where ( )iFS U denotes the set of all fuzzy
subsets of iU , ii I UE E∈= ∏ and A E⊆ . A pair (F, A) is called a fuzzy soft multi set over U,
where :F A U→ is a mapping given by ,e A∀ ∈
( )
( ) : :
( )
i
F e
u
F e u U i I
uµ
= ∈ ∈
For illustration, we consider the following example.
2.2. Example
Let us consider three universes { }1 1 2 3 4 5, , , ,U h h h h h= , { }2 1 2 3 4, , ,U c c c c= and { }3 1 2 3, ,U v v v= be the
sets of “houses,” “cars,” and “hotels”, respectively. Suppose Mr. X has a budget to buy a house, a
car and rent a venue to hold a wedding celebration. Let us consider a intuitionistic fuzzy soft
multi set (F, A) which describes “houses,” “cars,” and “hotels” that Mr. X is considering for
accommodation purchase, transportation purchase, and a venue to hold a wedding celebration,
respectively. Let { }1 2 3
, ,U U UE E E be a collection of sets of decision parameters related to the above
universes, where
{ }1 1 1 1,1 ,2 ,3expensive, cheap, woodenU U U UE e e e= = = =
3. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
31
{ }2 2 2 2,1 ,2 ,3beautiful, cheap, sporty ,U U U UE e e e= = = =
{ }3 3 3 3,1 ,2 ,3expensive, model, beautiful .U U U UE e e e= = = =
Let ( )3
1i iU FS U== ∏ , 3
1 ii UE E== ∏ and A E⊆ , such that
{ }1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 31 ,1 ,1 ,1 2 ,1 ,2 ,1 3 ,2 ,3 ,1 4 ,3 ,3 ,1 5 ,1 ,1 ,2 6 ,1 ,2 ,2( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ) .U U U U U U U U U U U U U U U U U UA a e e e a e e e a e e e a e e e a e e e a e e e= = = = = = =
Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of
choice parameters. Then fuzzy soft multi set (F, A) can be represent as in Table 1.
Table 1. The tabular representation of the fuzzy soft multi-set (F, A)
Ui a1 a2 a3 a4 a5 a6
U1
h1
h2
h3
h4
h5
0.8
0.4
0.9
0.4
0.1
0.6
0.5
0
0.4
1
0.5
0.7
1
0.4
0.8
0.4
0.5
0.1
0
0.8
0.7
0.5
1
0.4
1
0.6
0.4
0.8
1
0.1
U2
c1
c2
c3
c4
0.8
1
0.8
1
0.9
0.6
1
0
0.6
0.1
0
1
0.8
0
0.8
1
0
0.1
0.7
1
0.7
0.1
0.9
0.1
U3
v1
v2
v3
0.8
0.7
0.6
0.6
0.6
0
0
0.7
0.7
1
0.7
0
1
0.8
0
0.1
0.9
0.1
2.4. Definition [14]
The restricted union of two fuzzy soft multi sets (F, A) and (G, B) over U is a fuzzy soft multi set
(H,C)whereC A B= ∩ and ,e C∀ ∈
( )( ) ( ), ( )H e F e G e= U
{ }( ) ( )
: :
max ( ), ( )
i
F e G e
u
u U i I
u uµ µ
= ∈ ∈
and is written as ( , ) ( , ) ( , ).RF A G B H C=∪%
2.5 Definition [14]
The extended union of two fuzzy soft multi sets (F, A) and (G, B) over U is a fuzzy soft multi set
(H, D), where D A B= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
H e G e if e B A
F e G e if e A B
∈ −
= ∈ −
∈ ∩U
where ( )( ), ( )F e G eU
{ }( ) ( )
: :
max ( ), ( )
i
F e G e
u
u U i I
u uµ µ
= ∈ ∈
4. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
32
and is written as ( , ) ( , ) ( , ).EF A G B H D=∪%
2.6 Definition [14]
The restricted intersection of two intuitionistic fuzzy soft multi sets (F, A) and (G, B) over U is a
fuzzy soft multi set (H, D) where D A B= ∩ and ,e D∀ ∈
( )( ) ( ), ( )H e F e G e= I
{ }( ) ( )
: :
min ( ), ( )
i
F e G e
u
u U i I
u uµ µ
= ∈ ∈
and is written as ( , ) ( , ) ( , ).RF A G B H D=∩%
2.7 Definition [14]
The extended intersection of two fuzzy soft multi sets (F, A) and (G, B) over U is a fuzzy soft
multi set (H, D), where D A B= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) ,
F e if e A B
H e G e if e B A
F e G e if e A B
∈ −
= ∈ −
∈ ∩I
Where
( )( ), ( )F e G eI
{ }( ) ( )
: :
min ( ), ( )
i
F e G e
u
u U i I
u uµ µ
= ∈ ∈
and is written as ( , ) ( , ) ( , ).EF A G B H D=∩%
2.8. Definition [14]
If (F, A) and (G, B) be two fuzzy soft multi sets over U, then ( ) ( )" , AND , "F A G B is a fuzzy soft
multi set denoted by ( ) ( ), ,F A G B∧ and is defined by ( ) ( ) ( ), , , ,F A G B H A B∧ = × where H is
mapping given by H: A×B→U and
( )
{ }( ) ( )
, , ( , ) : : .
min ( ), ( )
i
F a G b
u
a b A B H a b u U i I
u uµ µ
∀ ∈ × = ∈ ∈
2.9. Definition [14]
If (F, A) and (G, B) be two fuzzy soft multi sets over U, then ( ) ( )" , OR , "F A G B is a fuzzy soft
multi set denoted by ( ) ( ), ,F A G B∨ and is defined by ( ) ( ) ( ), , , ,F A G B K A B∨ = × where K is
mapping given by K: A×B→U and
( )
{ }( ) ( )
, , ( , ) : :
max ( ), ( )
i
F a G b
u
a b A B K a b u U i I
u uµ µ
∀ ∈ × = ∈ ∈
2.10. Definition [8]
5. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
33
An information system is a 4-tuple S = (U, A, V, f ), where U = {u1, u2, u3, …., un} is a non-empty
finite set of objects, A = {a1, a2, a3,…., am} is a non-empty finite set of attributes, V =∪a∈A Va, Va
is the domain of attribute a, f : U × A→V is an information function, such that f (u, a) ∈ Va for
every (u, a) ∈ U × A, called information(knowledge) function.
3. MAIN RESULTS
Mukherjee and Das [14] defined some new operations in fuzzy soft multi set theory and show that
the De Morgan’s types of results hold in fuzzy soft multi set theory with respect to these newly
defined operations. In this section, we extend their work and study some more basic properties of
their defined operations.
3.1. Proposition (Associative Laws)
Let (F, A), (G, B) and (H, C) are three fuzzy soft multi sets over U, then we have the following
properties:
1. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % %
2. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % %
Proof. 1. Assume that ( , ) ( , ) ( , )RG B H C I D=∩% , where D B C= ∩ and ,e D∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
G e H e
u
I e G e H e u U i I
u uµ µ
= ∩ = ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∩ ∩ ∩% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∩% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
= ∩ = ∈ ∈
{ } { }{ } { }( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )min min min min( ) ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( )K e F e I e F e G e H e F e G e H e
where
u u u u u u u u uµ µ µ µ µ µ µ µ µ= = =
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∩% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
= ∩ = ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C J N H C=∩ ∩ ∩% % % , we suppose that ( , ) ( , ) ( , )RJ N H C O N C= ∩∩% where
6. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
34
( )
, ( ) ( ) ( ) : :
( )
i
O e
u
e N C A B C O e J e H e u U i I
uµ
∀ ∈ ∩ = ∩ ∩ = ∩ = ∈ ∈
{ } { }{ }
{ }
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
min min min
min
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O e J e H e F e G e H e
F e G e H e K e
u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =
Consequently, K and O are the same operators. Thus
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).R R R RF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % %
2. Assume that ( , ) ( , ) ( , )RG B H C I D=∪% , where D B C= ∩ and ,e D∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
G e H e
u
I e G e H e u U i I
u uµ µ
= ∪ = ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∪ ∪ ∪% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∪% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
= ∪ = ∈ ∈
{ } { }{ } { }( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )max max max max( ) ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( )K e F e I e F e G e H e F e G e H e
where
u u u u u u u u uµ µ µ µ µ µ µ µ µ= = =
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∪% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
= ∪ = ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C J N H C=∪ ∪ ∪% % % , we suppose that ( , ) ( , ) ( , )RJ N H C O N C= ∩∪%
where
( )
, ( ) ( ) ( ) : :
( )
i
O e
u
e N C A B C O e J e H e u U i I
uµ
∀ ∈ ∩ = ∩ ∩ = ∪ = ∈ ∈
{ } { }{ }
{ }
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
max max max
max
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O e J e H e F e G e H e
F e G e H e K e
u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =
Consequently, K and O are the same operators. Thus
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).R R R RF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % %
7. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
35
3.2. Proposition (Distributive Laws)
Let (F, A), (G, B) and (H, C) are three fuzzy soft multi sets over U, then we have the following
properties:
( ) ( ) ( )(1). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % %
( ) ( ) ( )(2). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % %
Proof. (1). Assume that ( , ) ( , ) ( , )RG B H C I D=∩% , where D B C= ∩ and ,e D∀ ∈
( ) ( ) ( )I e G e H e= ∩ =
{ }( ) ( )min
: :
( ), ( )
i
G e H e
u
u U i I
u uµ µ
∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∪ ∩ ∪% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∪% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
= ∪ = ∈ ∈
{ }
{ }{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
max
max min
min
( ) ( ), ( )
( ), ( ), ( )
max ( ), ( ) ,max ( ), ( )
K e F e I e
F e G e H e
F e G e F e H e
where u u u
u u u
u u u u
µ µ µ
µ µ µ
µ µ µ µ
=
=
=
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∪% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
= ∪ = ∈ ∈
Again, let ( , ) ( , ) ( , ),RF A H C S T=∪% where T A C= ∩ and ,e T∀ ∈
{ }( ) ( )max
( ) ( ) ( ) : :
( ), ( )
i
F e H e
u
S e F e H e u U i I
u uµ µ
= ∪ = ∈ ∈
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B F A H C J N S T=∪ ∩ ∪ ∩% % % % , we suppose that
( , ) ( , ) ( , )RJ N S T O N T= ∩∩% where ,e N T A B C∀ ∈ ∩ = ∩ ∩
( )
( ) ( ) ( ) : :
( )
i
O e
u
O e J e S e u U i I
uµ
= ∩ = ∈ ∈
8. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
36
{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
min
min max max
( ) ( ), ( )
( ), ( ) , ( ), ( ) ( )
O e J e S e
F e G e F e H e K e
u u u
u u u u u
µ µ µ
µ µ µ µ µ
=
= =
Consequently, K and O are the same operators. Thus
( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % %
(2). Assume that ( , ) ( , ) ( , )RG B H C I D=∪% , where D B C= ∩ and ,e D∀ ∈ ( ) ( ) ( )I e G e H e= ∪
{ }( ) ( )max
: :
( ), ( )
i
G e H e
u
u U i I
u uµ µ
= ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )R R RF A G B H C F A I D=∩ ∪ ∩% % % , we suppose that ( )( , ) ( , ) ,RF A I D K M=∩% where
M A D A B C= ∩ = ∩ ∩ and ,e M∀ ∈
( )
( ) ( ) ( ) : :
( )
i
K e
u
K e F e I e u U i I
uµ
= ∩ = ∈ ∈
{ }
{ }{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
min
min max
max
( ) ( ), ( )
( ), ( ), ( )
min ( ), ( ) ,min ( ), ( )
K e F e I e
F e G e H e
F e G e F e H e
where u u u
u u u
u u u u
µ µ µ
µ µ µ
µ µ µ µ
=
=
=
Suppose that ( , ) ( , ) ( , ),RF A G B J N=∩% where N A B= ∩ and ,e N∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
F e G e
u
J e F e G e u U i I
u uµ µ
= ∩ = ∈ ∈
Again, let ( , ) ( , ) ( , ),RF A H C S T=∩% where T A C= ∩ and ,e T∀ ∈
{ }( ) ( )min
( ) ( ) ( ) : :
( ), ( )
i
F e H e
u
S e F e H e u U i I
u uµ µ
= ∩ = ∈ ∈
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R RF A G B F A H C J N S T=∩ ∪ ∩ ∪% % % % , we suppose that
( , ) ( , ) ( , )RJ N S T O N T= ∩∪% where ,e N T A B C∀ ∈ ∩ = ∩ ∩
( )
( ) ( ) ( ) : :
( )
i
O e
u
O e J e S e u U i I
uµ
= ∪ = ∈ ∈
9. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
37
{ }
{ } { }{ }
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
max
max min min
( ) ( ), ( )
( ), ( ) , ( ), ( ) ( )
O e J e S e
F e G e F e H e K e
u u u
u u u u u
µ µ µ
µ µ µ µ µ
=
= =
Consequently, K and O are the same operators. Thus
( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )R R R R RF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % %
3.3. Proposition (Associative Laws)
Let (F, A), (G, B) and (H, C) are three fuzzy soft multi sets over U, then we have the following
properties:
1. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % %
2. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % %
Proof. 1. Suppose that ( , ) ( , ) ( , )EG B H C I D=∪% , where D B C= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
G e if e B C
I e H e if e C B
G e H e if e B C
∈ −
= ∈ −
∈ ∩U
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∪ ∪ ∪% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∪% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
J e G e H e if e B C A
F e H e if e A C B
G e F e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
U
U
U
U ,C
Assume that ( , ) ( , ) ( , )EF A G B K S=∪% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
∈ −
= ∈ −
∈ ∩U
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C K S H C=∪ ∪ ∪% % % , we suppose that ( )( , ) ( , ) ,EK S H C L T=∪% where
T S C A B C= ∪ = ∪ ∪ and ,e T∀ ∈
10. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
38
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
L e G e H e if e B C A
F e H e if e A C B
G e F e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
U
U
U
U ,C
Therefore it is clear that M T= and , ( ) ( )e M J e L e∀ ∈ = , that is J and L are the same operators.
Thus ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∪ ∪ ∪ ∪% % % % .
2. Suppose that ( , ) ( , ) ( , )EG B H C I D=∩% , where D B C= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
G e if e B C
I e H e if e C B
G e H e if e B C
∈ −
= ∈ −
∈ ∩I
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∩ ∩ ∩% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∩% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
J e G e H e if e B C A
F e H e if e A C B
G e F e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
I
I
I
I ,C
Assume that ( , ) ( , ) ( , )EF A G B K S=∩% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
∈ −
= ∈ −
∈ ∩I
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C K S H C=∩ ∩ ∩% % % , we suppose that ( )( , ) ( , ) ,EK S H C L T=∩% where
T S C A B C= ∪ = ∪ ∪ and ,e T∀ ∈
11. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
39
( )
( )
( )
( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
L e G e H e if e B C A
F e H e if e A C B
G e F e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩ ∩
I
I
I
I ,C
Therefore it is clear that M T= and , ( ) ( )e M J e L e∀ ∈ = , that is J and L are the same operators.
Thus ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B H C F A G B H C=∩ ∩ ∩ ∩% % % % .
3.4. Proposition (Distributive Laws)
Let (F, A), (G, B) and (H, C) are three fuzzy soft multi sets over U, then we have the following
properties:
( ) ( ) ( )(1). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % %
( ) ( ) ( )(2). ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % %
Proof. (1). Suppose that ( , ) ( , ) ( , )EG B H C I D=∪% , where D B C= ∪ and ,e D∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
G e if e B C
I e H e if e C B
G e H e if e B C
∈ −
= ∈ −
∈ ∩U
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∩ ∪ ∩% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∩% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
J e G e H e if e B C A
F e H e if e A C B
F e G e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
U
I
I
I U ,C
∩
Assume that ( , ) ( , ) ( , )EF A G B K S=∩% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
∈ −
= ∈ −
∈ ∩I
and ( , ) ( , ) ( , )EF A H C N T=∩% , where T A C= ∪ and ,e T∀ ∈
12. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
40
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A C
N e H e if e C A
F e H e if e A C
∈ −
= ∈ −
∈ ∩I
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B F A H C K S N T=∩ ∪ ∩ ∪% % % % , we suppose that ( )( , ) ( , ) ,EK S N T O P=∪%
where ( ) ( )P S T A B A C A B C= ∪ = ∪ ∪ ∪ = ∪ ∪ and ,e P∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
O e G e H e if e B C A
F e H e if e A C B
F e G e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
U
I
I
I U ,C
∩
Therefore it is clear that M P= and , ( ) ( )e M J e O e∀ ∈ = , that is “J” and “O” are the same
operators. Thus ( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∩ ∪ ∩ ∪ ∩% % % % % .
(2). Suppose that ( , ) ( , ) ( , )EG B H C I D=∩% , where D B C= ∪ and ,e D∀ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )E E EF A G B H C F A I D=∪ ∩ ∪% % % , we suppose that ( )( , ) ( , ) ,EF A I D J M=∪% where
M A D A B C= ∪ = ∪ ∪ and ,e M∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
J e G e H e if e B C A
F e H e if e A C B
F e G e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
I
U
U
U I ,C
∩
Assume that ( , ) ( , ) ( , )EF A G B K S=∪% , where S A B= ∪ and ,e S∀ ∈
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A B
K e G e if e B A
F e G e if e A B
∈ −
= ∈ −
∈ ∩U
and ( , ) ( , ) ( , )EF A H C N T=∪% , where T A C= ∪ and ,e T∀ ∈
13. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
41
( )
( ),
( ) ( ),
( ), ( ) , ,
F e if e A C
N e H e if e C A
F e H e if e A C
∈ −
= ∈ −
∈ ∩U
Since ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E EF A G B F A H C K S N T=∪ ∩ ∪ ∩% % % % , we suppose that ( )( , ) ( , ) ,EK S N T O P=∩%
where ( ) ( )P S T A B A C A B C= ∪ = ∪ ∪ ∪ = ∪ ∪ and ,e P∀ ∈
( )
( )
( )
( )( )
( ),
( ),
( ),
( ) ( ), ( ) , ,
( ), ( ) , ,
( ), ( ) , ,
( ), ( ), ( ) ,
G e if e B C A
H e if e C B A
F e if e A B C
O e G e H e if e B C A
F e H e if e A C B
F e G e if e A B C
F e G e H e if e A B
∈ − −
∈ − −
∈ − −
= ∈ ∩ −
∈ ∩ −
∈ ∩ −
∈ ∩
I
U
U
U I ,C
∩
Therefore it is clear that M P= and , ( ) ( )e M J e O e∀ ∈ = , that is “J” and “O” are the same
operators. Thus ( ) ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )E E E E EF A G B H C F A G B F A H C=∪ ∩ ∪ ∩ ∪% % % % % .
3.5. Proposition (Associative Laws)
Let (F, A), (G, B) and (H, C) are three fuzzy soft multi sets over U, then we have the following
properties:
1. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A G B H C∧ ∧ = ∧ ∧
2. ( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A G B H C∨ ∨ = ∨ ∨
Proof. 1. Assume that ( , ) ( , ) ( , )G B H C I B C∧ = × , where ( ), ,b c B C∀ ∈ ×
{ }( ) ( )min
( , ) ( ) ( ) : :
( ), ( )
i
G b H c
u
I b c G b H c u U i I
u uµ µ
= ∩ = ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A I B C∧ ∧ = ∧ × , we suppose that ( )( , ) ( , ) , ( )F A I B C K A B C∧ × = × ×
where ( ) ( ), , ,a b c A B C A B C∀ ∈ × × = × ×
( , , )
( , , ) ( ) ( , ) : :
( )
i
K a b c
u
K a b c F a I b c u U i I
uµ
= ∩ = ∈ ∈
{ } { }{ }
{ }
( , , ) ( ) ( , ) ( ) ( ) ( )
( ) ( ) ( )
min min min
min
( ) ( ), ( ) ( ), ( ), ( )
( ), ( ), ( )
K a b c F e I b c F e G b H c
F e G b H c
where u u u u u u
u u u
µ µ µ µ µ µ
µ µ µ
= =
=
We take ( , ) .a b A B∈ × Suppose that ( , ) ( , ) ( , )F A G B J A B∧ = × where ( ), ,a b A B∀ ∈ ×
{ }( ) ( )min
( , ) ( ) ( ) : :
( ), ( )
i
F a G b
u
J a b F a G b u U i I
u uµ µ
= ∩ = ∈ ∈
14. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
42
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C J A B H C∧ ∧ = × ∧ , we suppose that ( , ) ( , ) ( ,( ) )J A B H C O A B C× ∧ = × ×
where ( ) ( ), , ,a b c A B C A B C∀ ∈ × × = × ×
( , , )
( , , ) ( , ) ( ) : :
( )
i
O a b c
u
O a b c J a b H c u U i I
uµ
= ∩ = ∈ ∈
{ } { }{ }
{ }
( , , ) ( , ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( , , )
min min min
min
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O a b c J a b H c F e G b H c
F e G b H c K a b c
u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =
Consequently, K and O are the same operators. Thus
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).F A G B H C F A G B H C∧ ∧ = ∧ ∧
2. Assume that ( , ) ( , ) ( , )G B H C I B C∨ = × , where ( ), ,b c B C∀ ∈ ×
{ }( ) ( )max
( , ) ( ) ( ) : :
( ), ( )
i
G b H c
u
I b c G b H c u U i I
u uµ µ
= ∪ = ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C F A I B C∨ ∨ = ∨ × , we suppose that ( )( , ) ( , ) , ( )F A I B C K A B C∨ × = × ×
( ) ( )
( , , )
where , , , ( , , ) ( ) ( , ) : :
( )
i
K a b c
u
a b c A B C A B C K a b c F a I b c u U i I
uµ
∀ ∈ × × = × × = ∪ = ∈ ∈
{ } { }{ }
{ }
( , , ) ( ) ( , ) ( ) ( ) ( )
( ) ( ) ( )
max max max
max
( ) ( ), ( ) ( ), ( ), ( )
( ), ( ), ( )
K a b c F e I b c F e G b H c
F e G b H c
where u u u u u u
u u u
µ µ µ µ µ µ
µ µ µ
= =
=
We take ( , ) .a b A B∈ × Suppose that ( , ) ( , ) ( , )F A G B J A B∨ = × where ( ), ,a b A B∀ ∈ ×
{ }( ) ( )max
( , ) ( ) ( ) : :
( ), ( )
i
F a G b
u
J a b F a G b u U i I
u uµ µ
= ∪ = ∈ ∈
Since ( )( , ) ( , ) ( , ) ( , ) ( , )F A G B H C J A B H C∨ ∨ = × ∨ , we suppose that ( , ) ( , ) ( ,( ) )J A B H C O A B C× ∨ = × ×
where
( ) ( )
( , , )
, , , ( , , ) ( , ) ( ) : :
( )
i
O a b c
u
a b c A B C A B C O a b c J a b H c u U i I
uµ
∀ ∈ × × = × × = ∪ = ∈ ∈
{ } { }{ }
{ }
( , , ) ( , ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( , , )
max max max
max
( ) ( ), ( ) ( ), ( ) , ( )
( ), ( ), ( ) ( )
O a b c J a b H c F e G b H c
F e G b H c K a b c
where u u u u u u
u u u u
µ µ µ µ µ µ
µ µ µ µ
= =
= =
Consequently, K and O are the same operators. Thus
15. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
43
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , ).F A G B H C F A G B H C∨ ∨ = ∨ ∨
4. An Application Of Fuzzy Soft Multi Set Theory In Information
System
In this section we define some basic supporting tools in information system and also application
of fuzzy soft multi sets in information system are presented and discussed.
4.1. Definition
A fuzzy multi-valued information system is a quadruple ( ), , ,systemInf X A f V= where X is
a non empty finite set of objects, A is a non empty finite set of attribute, a
a A
V V
∈
= ∪ , where V is
the domain (a fuzzy set,) set of attribute, which has multi value and :f X A V× → is a total
function such that ( ), af x a V∈ for every ( ), .x a X A∈ ×
4.1. Proposition
If ( ),F A is a fuzzy soft multi set over universe U, then ( ),F A is a fuzzy multi-valued
information system.
Proof. Let { }:iU i I∈ be a collection of universes such that i I iU φ∈ =I and let { }:iE i I∈
be a collection of sets of parameters. Let ( )i I iU IFS U∈= ∏ where ( )iIFS U denotes the set of
all fuzzy subsets of iU , ii I UE E∈= ∏ and A E⊆ . Let ( ),F A be an fuzzy soft multi set over
U and i
i I
X U
∈
= ∪ . We define a mapping f where :f X A V× → , defined as
( ) ( ) ( ), / F a
f x a x xµ= .
Hence a
a A
V V
∈
= ∪ where aV is the set of all counts of in ( )F a and ∪ represent the classical set
union. Then the fuzzy multi-valued information system ( ), , ,X A f V represents the fuzzy soft
multi set ( ),F A .
4.1. Application in information system
Let us consider three universes { }1 1 2 3 4 5, , , ,U h h h h h= , { }2 1 2 3 4, , ,U c c c c= and
{ }3 1 2 3, ,U v v v= be the sets of “houses,” “cars,” and “hotels”, respectively. Suppose Mr. X has a
budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider a
intuitionistic fuzzy soft multi set (F, A) which describes “houses,” “cars,” and “hotels” that Mr. X
is considering for accommodation purchase, transportation purchase, and a venue to hold a
16. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
44
wedding celebration, respectively. Let { }1 2 3
, ,U U UE E E be a collection of sets of decision
parameters related to the above universes, where
{ }
{ }
{ }
1 1 1 1
2 2 2 2
3 3 3 3
,1 ,2 ,3
,1 ,2 ,3
,1 ,2 ,3
expensive, cheap, wooden
beautiful, cheap, sporty ,
expensive, model, beautiful .
U U U U
U U U U
U U U U
E e e e
E e e e
E e e e
= = = =
= = = =
= = = =
Let ( )3
1i iU FS U== ∏ , 3
1 ii UE E== ∏ and A E⊆ , such that
{
}
1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
1 ,1 ,1 ,1 2 ,1 ,2 ,1 3 ,2 ,3 ,1
4 ,3 ,3 ,1 5 ,1 ,1 ,2 6 ,1 ,2 ,2
( , , ), ( , , ), ( , , ),
( , , ), ( , , ), ( , , ) .
U U U U U U U U U
U U U U U U U U U
A a e e e a e e e a e e e
a e e e a e e e a e e e
= = = =
= = =
Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of
choice parameters. Let
( ) { } { } { }( )
( ) { } { } { }( )
( ) { }
1 1 2 3 4 5 1 2 3 4 1 2 3
2 1 2 3 4 5 1 2 3 4 1 2 3
3 1 2 3 4 5 1 2
/ 0.2, / 0.4, / 0, / 0.4, / 0 , / 0.8, / 0.1, / 0, /1 , / 0.8, / 0.7, / 0 ,
/ 0.3, / 0.5, / 0, / 0.4, /1 , / 0.9, / 0.6, /1, / 0 , / 0.6, / 0.6, / 0 ,
/ 0.5, / 0.7, /1, / 0.4, / 0 , / 0.6,
F a h h h h h c c c c v v v
F a h h h h h c c c c v v v
F a h h h h h c c
=
=
= { } { }( )
( ) { } { } { }( )
( ) { } { } { }( )
( )
3 4 1 2 3
4 1 2 3 4 5 1 2 3 4 1 2 3
5 1 2 3 4 5 1 2 3 4 1 2 3
6 1 2
/ 0.1, / 0, /1 , / 0, / 0.7, / 0.7 ,
/ 0.4, / 0.5, / 0.1, / 0, /1 , / 0.8, / 0, / 0.8, /1 , /1, / 0.7, / 0 ,
/ 0.7, / 0.5, /1, / 0.4, /1 , / 0, / 0.1, / 0.7, /1 , /1, / 0.8, / 0 ,
/ 0.6, /
c c v v v
F a h h h h h c c c c v v v
F a h h h h h c c c c v v v
F a h h
=
=
= { } { } { }( )3 4 5 1 2 3 4 1 2 30.4, / 0.8, /1, / 0.1 , / 0, / 0.1, /1, /1 , /1, / 0, /1 ,h h h c c c c v v v
Then the fuzzy soft multi set ( ),F A defined above describes the conditions of some “house”,
“car” and “hotel” in a state. Then the quadruple ( ), , ,X A f V corresponding to the fuzzy soft
multi set given above is a fuzzy multi-valued information system.
Where
3
1
i
i
X U
=
= ∪ and A is the set of parameters in the fuzzy soft multi set and
{ }
{ }
1
2
3
1 2 3 4 5 1 2 3 4 1 2 3
1 2 3 4 5 1 2 3 4 1 2 3
1 2 3 4 5 1 2
/ 0.2, / 0.4, / 0, / 0.4, / 0, / 0.8, / 0.1, / 0, /1, / 0.8, / 0.7, / 0 ,
/ 0.3, / 0.5, / 0, / 0.4, /1, / 0.9, / 0.6, /1, / 0, / 0.6, / 0.6, / 0 ,
/ 0.5, / 0.7, /1, / 0.4, / 0, / 0.6,
a
a
a
V h h h h h c c c c v v v
V h h h h h c c c c v v v
V h h h h h c c
=
=
= { }
{ }
{ }
4
5
6
3 4 1 2 3
1 2 3 4 5 1 2 3 4 1 2 3
1 2 3 4 5 1 2 3 4 1 2 3
1 2
/ 0.1, / 0, /1, / 0, / 0.7, / 0.7 ,
/ 0.4, / 0.5, / 0.1, / 0, /1, / 0.8, / 0, / 0.8, /1, /1, / 0.7, / 0 ,
/ 0.7, / 0.5, /1, / 0.4, /1, / 0, / 0.1, / 0.7, /1, /1, / 0.8, / 0 ,
/ 0.6, /
a
a
a
c c v v v
V h h h h h c c c c v v v
V h h h h h c c c c v v v
V h h
=
=
= { }3 4 5 1 2 3 4 1 2 30.4, / 0.8, /1, / 0.1, / 0, / 0.1, /1, /1, /1, / 0, /1h h h c c c c v v v
,
17. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
45
For the pair ( )1 1,h a we have ( )1 1,f h a = 0.2, for ( )2 1,h a , we have ( )2 1,f h a = 0.4. Continuing
in this way we obtain the values of other pairs. Therefore, according to the result above, it is seen
that fuzzy soft multi sets are fuzzy soft multi-valued information systems. Nevertheless, it is
obvious that fuzzy soft multi-valued information systems are not necessarily fuzzy soft multi sets.
We can construct an information table representing fuzzy soft multi set ( ),F A defined above as
follows.
4.1. An Information Table
Table 2. An information table
X a1 a2 a3 a4 a5 a6
h1
h2
h3
h4
h5
0.2
0.4
0
0.4
0
0.3
0.5
0
0.4
1
0.5
0.7
1
0.4
0
0.4
0.5
0.1
0
1
0.7
0.5
1
0.4
1
0.6
0.4
0.8
1
0.1
c1
c2
c3
c4
0.8
0.1
0
1
0.9
0.6
1
0
0.6
0.1
0
1
0.8
0
0.8
1
0
0.1
0.7
1
0
0.1
1
1
v1
v2
v3
0.8
0.7
0
0.6
0.6
0
0
0.7
0.7
1
0.7
0
1
0.8
0
1
0
1
5. Conclusion
The theory of soft set, fuzzy soft set and multi set are vital mathematical tools used in handling
uncertainties about vague concepts. In this paper, we have introduced the notion of fuzzy multi-
valued information system in fuzzy soft multi set theory. Here we shall present the application of
fuzzy soft multi set in information system and show that every fuzzy soft multi set is a fuzzy
multi valued information system.
References
[1] K.Alhazaymeh & N. Hassan, (2014) “Vague Soft Multiset Theory”, Int. J. Pure and Applied Math.,
Vol. 93, pp511-523.
[2] M.I.Ali, F. Feng, X. Liu, W.K. Minc & M. Shabir, (2009) “On some new operations in soft set
theory”, Comp. Math. Appl., vol. 57, pp1547-1553.
[3] S.Alkhazaleh & A.R. Salleh, (2012) “Fuzzy Soft Multi sets Theory”, Hindawi Publishing
Corporation, Abstract and Applied Analysis, Article ID 350603, 20 pages, doi: 10.1155/2012/350603.
[4] S.Alkhazaleh, A.R. Salleh & N. Hassan, (2011) Soft Multi sets Theory, Appl. Math. Sci., Vol. 5,
pp3561– 3573.
[5] K.V Babitha & S.J. John, (2013) “On Soft Multi sets”, Ann. Fuzzy Math. Inform., Vol. 5 pp35-44.
[6] T. Herewan & M.M. Deris, (2011) “A soft set approach for association rules mining”, Knowl.-Based
Syst. Vol. 24, pp186-195.
[7] A.M. Ibrahim & H.M. Balami, (2013) “Application of soft multiset in decision making problems”, J.
Nig. Ass. of mathematical physics, Vol. 25, pp307- 311.
[8] H.M. Balami & A. M. Ibrahim, (2013) “Soft Multiset and its Application in Information System”,
International Journal of scientific research and management, Vol. 1, pp471-482.
[9] P.K. Maji, R. Biswas & A.R. Roy, (2001) “Fuzzy soft sets”, J. Fuzzy Math., Vol. 9, pp589-602.
18. International Journal of Computer-Aided Technologies (IJCAx) Vol.2, No.3, July 2015
46
[10] P.K. Maji, R. Biswas & A.R. Roy, (2003) “Soft set theory”, Comp. Math. Appl., Vol. 45 pp555-562.
[11] P.K. Maji, R. Biswas & A.R. Roy, (2002) “An application of soft sets in a decision making problem”,
Comput. Math. Appl., Vol. 44, pp1077-1083.
[12] D.Molodtsov, (1999) “Soft set theory-first results”, Comp. Math. Appl., Vol. 37, pp19-31.
[13] A.Mukherjee & A.K. Das, (2015) “Application of fuzzy soft multi sets in decision making problems”,
Smart Innovation Systems and Technologies, Springer Verlag, (Accepted).
[14] A.Mukherjee & A.K. Das, (2015) “Some results on fuzzy soft multi sets. Int. J. Cybernetics &
Informatics. Vol. 4, pp 51-65.
[15] A.Mukherjee & A.K. Das, (2013) “Topological structure formed by fuzzy soft multisets”, Rev. Bull.
Cal.Math.Soc., Vol. 21, No.2, pp193-212.
[16] A.Mukherjee & A.K. Das, (2014) “Topological structure formed by soft multi sets and soft multi
compact space”, Ann. Fuzzy Math. Inform. Vol. 7, pp919-933.
[17] D.Tokat & I. Osmanoglu, (2011) “Soft multi set and soft multi topology”, Nevsehir Universitesi Fen
Bilimleri Enstitusu Dergisi Cilt., Vol. 2, pp109-118.
[18] L.A. Zadeh, (1965) “Fuzzy sets”, Inform. Control., Vol. 8, pp338-353.