HOMOMORPHISM AND ISOMORPHISM OF FUZZY SETS
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In this work, the concepts of the isomorphism and homomorphism of fuzzy sets are given. The sufficient and necessary conditions of isomorphism and homomorphism of fuzzy sets, and some properties of the new structural definition are discussed. The essence of fuzzy sets is point out and the operational properties of fuzzy sets based on isomorphism and homomorphism of fuzzy sets are discussed. The study also covers the theorem and mathematical definition of isomorphism and homomorphism of fuzzy relations.
TABLE OF CONTENTS
COVER PAGE
TITLE PAGE
APPROVAL PAGE
DEDICATION
ACKNOWLEDGEMENT
ABSTRACT
CHAPTER ONE
- INTRODUCTION
- Background of the project
- Aim and objective of the project
- Scope and limitation of the study
- Definition of terms
CHAPTER TWO
LITERATURE REVIEW
- Overview of fuzzy sets
- FuzzySetTheoryandFuzzyLogic
- Review Of Previous Studies
CHAPTER THREE
METHODOLOGY
- Mathematical definition of homomorphism of fuzzy set
CHAPTER FOUR
- Mathematical theorem and definition of isomorphic fuzzy set
CHAPTER FIVE
- Conclusion
- Recommendation
- References
CHAPTER ONE
1.0 INTRODUCTION
1.1 BACKGROUND OF THE STUDY
One of the best definitions of mathematics is given by Bourbaki school of thought as“Mathematics is a study of setswith structure”. The three basic structures are ‘Algebra, topology and order’. In algebraic structures, we deal with operations ofaddition and multiplication and their study, in topology the ideas of nearness, neighborhood etc. and in order structures, those ofgreater than, lessthanandsoon.
ModernAlgebrahasfoundanextensiveuseinprobability,statistics,informationtheory,relativity,quantummechanics and geometrics etc. Galois field and finite geometrics has been used in design of experiment in statistics. AbstractAlgebrahasalso beenfoundusefulinstudyofparticle physicsandmolecular structures.
A fuzzy set is a class of objects whose memberships arenot precisely defined. Fuzzy sets provide a betterrepresentation of reality than the classical mathematicalbinary representation. The membership in fuzzy sets isgradual,thatmakesthetheoryinvaluabletorepresentthe limited level of precision in mental representations (Dubois, 2010).
The fundamental concept of fuzzy sets was initiated by Zadeh in 1965 and opened a new path of thinking tomathematicians, engineers, physicists, chemists and many others due to its diverse applications in various fields. The fuzzyalgebraic structures play a prominent role in mathematics with wide applications in many other branches such as theoreticalphysics, computer science, control engineering, information science, coding theory, group theory, real analysis, measure theoryetc. In 1971, Rosenfeld first introduced the concept of fuzzy subgroups, which was the first fuzzification of any algebraicstructure.Thereafter the notion of different fuzzy algebraic structures such as fuzzy ideals in rings and emirings etc. have seriouslystudied by many mathematicians according to Priyadarshin (2018). In 1975, Zadeh introduced the concepts of interval-valued fuzzy sets (in short written by i-vfuzzy sets), where the values of the membership functions are intervals of real numbers instead of the real points (Yang, 2012). Thereafter in1994, Biswas define interval-valued fuzzy subgroups of the same nature of Rosenfeld’s fuzzy subgroups and discuss someimportantresults (Filep, 2019).
- Aim and Objectives
The purpose of this study is to provide an in-depth presentation of the homomorphism and isomorphism of fuzzy sets. The objectives of this study are:
- To study the concepts of the isomorphism and homomorphism of fuzzy sets
- To study different publications on fuzzy set theory
- To evaluate properties of fuzzy sets based on isomorphism and homomorphism of fuzzy sets
- scope / limitation of the study
The scope of the work covers the concept of the isomorphism and homomorphism of fuzzy sets. Also sufficient and necessary conditions of isomorphism and homomorphism of fuzzy sets. The essence of fuzzy sets is point out and the operational properties of fuzzy sets based on isomorphism and homomorphism of fuzzy sets are discussed. Fortunately, we studied the isomorphism and homomorphism of fuzzy relations, have given the concept of isomorphic classification of fuzzy similarity relation, and proved that if two fuzzy similarity relations are isomorphic, then they must be classified isomorphic.
- Definition of terms
Terms that are frequently used in this work are defined as below:
Fuzzification: Fuzzification is the process of assigning the numerical input of a system to fuzzy sets with some degree of membership.
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1.
Isomorphism: In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.
Homomorphism: In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type
Set theory:branch of mathematics which deals with the formal properties of sets as units (without regard to the nature of their individual constituents) and the expression of other branches of mathematics in terms of sets.
CHAPTER FIVE
CONCLUSION
In this study,the concept of the isomorphism and homomorphism of fuzzy sets has been known. The sufficient and necessary conditions of isomorphism and homomorphism of fuzzy sets, and also, the mathematical definition and theorem of isomorphism and homomorphism was also studied and known.
RECOMMENDATION
There many definitions and theorems of isomorphism and homomorphism of fuzzy set that was discussed in this work, I am recommending that more theorems shall be studied in the future work.
CHAPTER FIVE
CONCLUSION
In this study,the concept of the isomorphism and homomorphism of fuzzy sets has been known. The sufficient and necessary conditions of isomorphism and homomorphism of fuzzy sets, and also, the mathematical definition and theorem of isomorphism and homomorphism was also studied and known.
RECOMMENDATION
CHAPTER TWO: The chapter one of this work has been displayed above. The complete chapter two of "homomorphism and isomorphism of fuzzy sets" is also available. Order full work to download. Chapter two of "homomorphism and isomorphism of fuzzy sets" consists of the literature review. In this chapter all the related work on "homomorphism and isomorphism of fuzzy sets" was reviewed.
CHAPTER THREE: The complete chapter three of "homomorphism and isomorphism of fuzzy sets" is available. Order full work to download. Chapter three of "homomorphism and isomorphism of fuzzy sets" consists of the methodology. In this chapter all the method used in carrying out this work was discussed.
CHAPTER FOUR: The complete chapter four of "homomorphism and isomorphism of fuzzy sets" is available. Order full work to download. Chapter four of "homomorphism and isomorphism of fuzzy sets" consists of all the test conducted during the work and the result gotten after the whole work
CHAPTER FIVE: The complete chapter five of "homomorphism and isomorphism of fuzzy sets" is available. Order full work to download. Chapter five of "homomorphism and isomorphism of fuzzy sets"consist of conclusion, recommendation and references.
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