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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
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.
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.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
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.
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.
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.
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
Abstract
In this study, we consider the numerical solution of the Helmholtz equation, arising from numerous physical phenomena and
engineering applications including energy systems. We make use of a Padé Approximation method for the solution of the
Helmholtz equation. Firstly, Helmholtz partial differential equation had been converted to power series by two-dimensional
differential transformation, then the numerical solution of the equation was put into Padé series form for accelerating
convergence and decreasing computational time. Hereby, we obtained numerical solution of Helmholtz type partial differential
equation.
Key Words: Helmholtz equation, Two-dimensional differential transformation, Multivariate Padé approximation,
power series
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
The study of this paper suggests on dependency problem in fuzzy computational method by using the numerical solution of Fuzzy differential equations(FDEs) in Milne’s predictor-corrector method. This method is adopted to solve the dependency problem in fuzzy computation. We solve some fuzzy initial value problems to illustrate the theory.
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 innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and 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.
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.
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.
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.
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.
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
Abstract
In this study, we consider the numerical solution of the Helmholtz equation, arising from numerous physical phenomena and
engineering applications including energy systems. We make use of a Padé Approximation method for the solution of the
Helmholtz equation. Firstly, Helmholtz partial differential equation had been converted to power series by two-dimensional
differential transformation, then the numerical solution of the equation was put into Padé series form for accelerating
convergence and decreasing computational time. Hereby, we obtained numerical solution of Helmholtz type partial differential
equation.
Key Words: Helmholtz equation, Two-dimensional differential transformation, Multivariate Padé approximation,
power series
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
The study of this paper suggests on dependency problem in fuzzy computational method by using the numerical solution of Fuzzy differential equations(FDEs) in Milne’s predictor-corrector method. This method is adopted to solve the dependency problem in fuzzy computation. We solve some fuzzy initial value problems to illustrate the theory.
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 innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate
the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function
ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the
member ship function in the form of complex number. The Compressive study of mathematical operation of
Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division
are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by
evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set
where we attempted to clarify the distinction between Fuzzy logic and 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.
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.
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.
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
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.
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
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
The degree of roughness characterizes the uncertainty contained in a rough set. The rough entropy was defined to measure the roughness of a rough set. Though, it was effective and useful, but not accurate
enough. Some authors use information measure in place of entropy for better understanding which measures the amount of uncertainty contained in fuzzy rough set .In this paper three new fuzzy rough information measures are proposed and their validity is verified. The application of these proposed information measures in decision making problems is studied and also compared with other existing
information measures
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
The degree of roughness characterizes the uncertainty contained in a rough set. The rough entropy was defined to measure the roughness of a rough set. Though, it was effective and useful, but not accurate enough. Some authors use information measure in place of entropy for better understanding which measures the amount of uncertainty contained in fuzzy rough set .In this paper three new fuzzy rough information measures are proposed and their validity is verified. The application of these proposed information measures in decision making problems is studied and also compared with other existing
information measures.
Similar to On Fuzzy Soft Multi Set and Its Application in Information Systems (20)
NEW ONTOLOGY RETRIEVAL IMAGE METHOD IN 5K COREL IMAGESijcax
Semantic annotation of images is an important research topic on both image understanding and database
or web image search. Image annotation is a technique to choosing appropriate labels for images with
extracting effective and hidden feature in pictures. In the feature extraction step of proposed method, we
present a model, which combined effective features of visual topics (global features over an image) and
regional contexts (relationship between the regions in Image and each other regions images) to automatic
image annotation.In the annotation step of proposed method, we create a new ontology (base on WordNet
ontology) for the semantic relationships between tags in the classification and improving semantic gap
exist in the automatic image annotation.Experiments result on the 5k Corel dataset show the proposed
method of image annotation in addition to reducing the complexity of the classification, increased accuracy
compared to the another 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
A Mobile Ad Hoc Network (MANET) is a collection of mobile nodes that want to communicate without any
pre-determined infrastructure and fixed organization of available links. Each node in MANET operates as
a router, forwarding information packets for other mobile nodes. There are many routing protocols that
possess different performance levels in different scenarios. The main task is to evaluate the existing routing
protocols and finding by comparing them the best one. In this article we compare AODV, DSR, DSDV,
OLSR and DYMO routing protocols in mobile ad hoc networks (MANETs) to specify the best operational
conditions for each MANETs protocol. We study these five MANETs routing protocols by different
simulations in NS-2 simulator. We describe that pause time parameter affect their performance. This
performance analysis is measured in terms of Packet Delivery Ratio, Average End-to-End Delay,
Normalized Routing Load and Average Throughput.
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
With the great development that, modern medical technology is witnessing today, medical devices and
equipment have become a basic pillar of any healthcare system in the world and cannot be dispensed with,
so we find competition between the major companies that manufacture medical devices and equipment
resulting in a huge variety of complex modern medical technologies. These medical devices and equipment
require high accuracy in manufacturing and packaging in addition to operation, maintenance, and followup, because any error in any of the previous stages will have bad consequences for the patients and the
health system, there are many accidents that have led to some deaths. Therefore, we find that many medical
device producers and medical companies in addition to health service providers seek to find systems and
protocols to reduce accidents resulting from medical devices. As a result, many systems have recently
appeared that seek to protect from the dangers of medical devices and equipment. This research aims to
conduct a study of the effects of international standards on the safety of medical devices and equipment
and reduce their risks. By counting the international standards in force in the Kingdom of Saudi Arabia
that are applied by the Saudi Food and Drug Authority, making questionnaires, and distributing them to
health service providers and regulatory bodies for medical devices and equipment, considering the data,
these data will be analysed and evaluated the effectiveness of quality systems and standards in maintaining
Effectiveness and quality of medical devices and equipment. The study will include governmental and
private health services sectors.
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
TOP 10 B TECH COLLEGES IN JAIPUR 2024.pptxnikitacareer3
Looking for the best engineering colleges in Jaipur for 2024?
Check out our list of the top 10 B.Tech colleges to help you make the right choice for your future career!
1) MNIT
2) MANIPAL UNIV
3) LNMIIT
4) NIMS UNIV
5) JECRC
6) VIVEKANANDA GLOBAL UNIV
7) BIT JAIPUR
8) APEX UNIV
9) AMITY UNIV.
10) JNU
TO KNOW MORE ABOUT COLLEGES, FEES AND PLACEMENT, WATCH THE FULL VIDEO GIVEN BELOW ON "TOP 10 B TECH COLLEGES IN JAIPUR"
https://www.youtube.com/watch?v=vSNje0MBh7g
VISIT CAREER MANTRA PORTAL TO KNOW MORE ABOUT COLLEGES/UNIVERSITITES in Jaipur:
https://careermantra.net/colleges/3378/Jaipur/b-tech
Get all the information you need to plan your next steps in your medical career with Career Mantra!
https://careermantra.net/
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Water billing management system project report.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]
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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
= ∩ = ∈ ∈
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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
= ∪ = ∈ ∈
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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∀ ∈
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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∀ ∈
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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.
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