Description
DEDICATION
This project is dedicated to Almighty God for his protection, kindness, strength over my life throughout the period and also to my — for his financial support and moral care towards me.Also to my mentor — for her academic advice she often gives to me. May Almighty God shield them from the peril of this world and bless their entire endeavour Amen.
ACKNOWLEDGEMENT
The successful completion of this project work could not have been a reality without the encouragement of my — and other people. My immensely appreciation goes to my humble and able supervisor Mr. — for his kindness in supervising this project. My warmest gratitude goes to my parents for their moral, spiritual and financial support throughout my study in this institution.
My appreciation goes to some of my lecturers among whom are Mr. —, and Dr. —. I also recognize the support of some of the staff of — among whom are: The General Manager, Deputy General manager, the internal Auditor Mr. — and the —. Finally, my appreciation goes to my elder sister —, my lovely friends mercy —, —, — and many others who were quite helpful.
ABSTRACT
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
- Fuzzy Set Theory and Fuzzy Logic
- 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 sets with structure”. The three basic structures are ‘Algebra, topology and order’. In algebraic structures, we deal with operations of addition and multiplication and their study, in topology the ideas of nearness, neighborhood etc. and in order structures, those of greater than, less than and so on.
Modern Algebra has found an extensive use in probability, statistics, information theory, relativity, quantum mechanics and geometrics etc. Galois field and finite geometrics has been used in design of experiment in statistics. Abstract Algebra has also been found useful in study of particle physics and molecular structures.
A fuzzy set is a class of objects whose memberships are not precisely defined. Fuzzy sets provide a better representation of reality than the classical mathematical binary representation. The membership in fuzzy sets is gradual,that makes the theory invaluable to represent the 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 to mathematicians, engineers, physicists, chemists and many others due to its diverse applications in various fields. The fuzzy algebraic structures play a prominent role in mathematics with wide applications in many other branches such as theoretical physics, computer science, control engineering, information science, coding theory, group theory, real analysis, measure theory etc. In 1971, Rosenfeld first introduced the concept of fuzzy subgroups, which was the first fuzzification of any algebraic structure. Thereafter the notion of different fuzzy algebraic structures such as fuzzy ideals in rings and emirings etc. have seriously studied by many mathematicians according to Priyadarshin (2018). In 1975, Zadeh introduced the concepts of interval-valued fuzzy sets (in short written by i-v fuzzy sets), where the values of the membership functions are intervals of real numbers instead of the real points (Yang, 2012). Thereafter in 1994, Biswas define interval-valued fuzzy subgroups of the same nature of Rosenfeld’s fuzzy subgroups and discuss some important results (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.
PROJECT DESCRIPTION
Format = Microsoft word
Chapters = 1-5 chapters
Price: N3,000.
For more information contact us through any of the following means:
Mobile No: +2348146561114 or +2347015391124
Email address: engr4project@gmail.com
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