ABSTRACT
The theory of fuzzy set has been studied extensively in mathematics along with its application in diverse fields. Rosenfeld in 1971 used this concept to develop the theory of fuzzy groups. Gu in 1994 put forward the notion of M-fuzzy groups. In this research work, we review some of the fundamental works in M-fuzzy group theory and provide some new or alternative methods to proving some existing theorems in M-fuzzy group theory. We have given independent proof of several theorems on Level M- subgroups of M-fuzzy subgroups.
TABLE OF CONTENTS
Cover Page…………………………………………………………………………….……i
Fly Leaf………………………….…………………………………………………………ii
Title Page………………………………………………………………………………….iii
Declaration………………………………………………………………….……………..iv
Certifacation……………………………………………………………………………….v
Dedication…………………………………………………………………………………vi
Acknowledment…………………………………………………………………………..vii
Abstract…………………………………………………………………………………….ix
Table Of Contents……………………………………………… ………………………..x
CHAPTER ONE : INTRODUCTION…………………………………..…….…………..1
1.1 Background to the Study ………………………………………………………………………………. 1
1.2 Statement of the Problem ………………………………………………………………………………. 1
1.3 Justification ………………………………………………………………………………………………… 2
1.4 Aim and Objectives of the Thesis ………………………………………………………………….. 3
1.5 Research Methodology …………………………………………………………………………………. 3
1.6 Organization of the Thesis ……………………………………………………………………………. 4
1.7 Preliminaries ………………………………………………………………………………………………. 4
CHAPTER TWO : LITERATURE REVIEW…………………………………………….9
2.0 Introduction………………………………………………………………………….9
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CHAPTER THREE : FUNDAMENTALS OF FUZZY SUBGROUP……………….…11
3.1 Introduction ………………………………………………………………………………………………. 11
3.2 Concept of Fuzzy Subgroup ………………………………………………………………………… 11
3.3 Properties of Fuzzy Subgroup …………………………………………………………………….. 14
3.4 Composition and Inverse of Fuzzy Subgroup ………………………………………………… 13
3.5 Some Results on Algebraic Operations on Fuzzy Subgroups …………………………… 13
3.6 Level Subgroup of the Fuzzy Subgroup………………………………………………………… 14
3.7 Normal Fuzzy Subgroups ……………………………………………………………………………. 16
3.8 Order of Fuzzy Subgroup ……………………………………………………………………………. 16
3.9 Extended Theorems from Group Theory ………………………………………………………. 19
CHAPTER FOUR : -FUZZY SUBGROUP AND ITS LEVEL -SUBGROUP………21
4.1 Introduction………………………………………………………………………….21
4.2 -Fuzzy Subgroup of an -group………………………………………………….21
4.3 Properties of an -Fuzzy Subgroups of an -groups……….……………………..22
4.4 Level -Subgroup of an -Fuzzy Subgroup of an -group……………………….29
CHAPTER FIVE : SUMMARY, CONCLUSION AND RECOMMENDATIONS……..35
5.1 Summary………………………………………………………………………….. 36
5.2 Conclusion………………………………………………………………………… 35
5.3 Recommendations…………………………………………………………………..35
REFERENCES………………………………………………………………………….37
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CHAPTER ONE
INTRODUCTION
1.1 Background to the Study
The fundamental concept of fuzzy sets was introduced by Zadeh in 1965 to represents information possessing non-statistical uncertainties. 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.
The first publication in fuzzy set theory by Zadeh (1965) and then by Klaua (1965) showed the intention of the authors to generalize the classical set. In classical set theory, a subset of a set can be defined by its characteristic function which is defined by
The mapping may be represented as a set of ordered pairs with exactly one ordered pair present for each element of. The first element of the ordered pair is an element of the set and the second is its value in under. The value „0‟ is used to represent non-membership and the value „1‟ is used to represent membership of the element in. The truth or falsity of the statement “ is in” is determined by the ordered pair. The statement is true, if the second element of the ordered pair is „1‟, and the statement is false, if it is „0‟.
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Fuzzy set theory is an extension of classical set theory where elements have varying degrees of membership. A logic based on the two truth values, True and False, is sometimes inadequate when describing human reasoning. Fuzzy logic uses the whole interval between 0 (false) and 1(true) to describe human reasoning.
A fuzzy set in is characterized by a membership function which associates with each point in a real number in the interval, with the value of at representing the “grade of membership” of in. In 1971, Rosenfeld first introduced the concept of fuzzy subgroups, which was the first fuzzification of any algebraic structure and shows that many results in group theory can be extended in an elementary manner to develop the theory of fuzzy group. Gu et al (1994) studied the theory of fuzzy groups and developed the concept of M-fuzzy groups. This dissertation is an attempt to study M-fuzzy subgroups and its level M-subgroups.
1.2 Statement of the Research problem
After the introduction of the notion of M-fuzzy subgroup by Gu (1994), several researches were conducted using this notion. Many algebraic structures of fuzzy subgroup with operators have been developed so far; however, there is no author that uses this alternative method of proving theorems in fuzzy subgroup with operators (i.e. M-fuzzy subgroup). Since there is no author that uses this alternative method, this gives us a room to provide new proofs of some existing theorems in M-fuzzy subgroup and to obtained independent proof of several theorems on Level M- subgroups of M-fuzzy subgroups.
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1.3 Justification
Many algebraic structures such as Groups, Monoids, Semigroups, Quasigroups, Ring, Semirings, Lattice, Semilattice, Boolean algebra etc. were developed using set as their underlying structure. Since fuzzy set is a generalization of classical set, various algebras based on fuzzy set could be developed such as Rosenfeld in 1971 used this concept to develop the theory of fuzzy groups and Gu in 1994 also used the concept to put forward the notion of M-fuzzy groups. In this research work, we reviewed the concepts of -fuzzy group theory and come up with a new method to develop the concept M-fuzzy subgroup and their Level -subgroups.
1.4 Aim and Objectives of the Dissertation
The aim of this research work is to study an algebraic structure called -fuzzy subgroup and its level -subgroup. The objectives are to:
1. Review some of the fundamental works done in M-fuzzy subgroup theory;
2. Provide some new or alternative methods of proving some existing theorems in M-fuzzy subgroup theory;
3. Provide proofs of some theorems on M-fuzzy subgroup theory which, to the very best of our knowledge, do not exist in the literature;
4. Obtain independent proof of several theorems on Level M- subgroups of M-fuzzy subgroups.
1.5 Research Methodology
The method of research adopted in this dissertation is by consulting necessary and relevant literature (papers) on the theory of M-fuzzy subgroups. These consultations make it
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possible to develop new or alternative methods of proving some existing theorems in M-fuzzy subgroup theory and provide some independent proof of several theorems on Level M- subgroups of M-fuzzy subgroups..
1.6 Organization of the Dissertation
This dissertation contains four chapters after this introductory chapter. The organizations of the remaining chapters are as follows: Chapter two: In this chapter, we present a survey of the necessary and relevant literature for the fuzzy sets and fuzzy subgroups. Chapter three: In this chapter, Basic definitions required for the dissertation and fundamentals of fuzzy subgroups are presented.
Chapter four: In this chapter, we focus our attention on the study of -fuzzy subgroups and its level -subgroups. Chapter five: In this chapter, we present our concluding remarks. In the end, a list of references cited in the dissertation is presented.
1.7 Preliminaries
We start by presenting the basic concepts of fuzzy sets theory needed for understanding the result of this dissertation. All definitions and results are typical and can be found in any introductory text on fuzzy sets theory
Definition 1.7.1: (Fuzzy sets and membership function)
If is a collection of objects denoted generically bythen a fuzzy set in is defined as a set of ordered pairs where is called the membership function (or MF for short) for the fuzzy set. A fuzzy set expresses the
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degree to which an element belongs to a set. Hence the MF maps each element of to a membership grade (or membership value) in the interval [0, 1].
Example 1.7.2: Let be a set. Then is a fuzzy set in.
Definition 1.7.3: (subset of fuzzy set)
Let be a fuzzy sets. Ifthenis said to be contained in (or contains), and we write If andthen is said to be properly contained in (or properly contains) and we write (or ).
Example 1.7.4: let be a universal set.
are fuzzy sets in , . Then Definition 1.7.5: (equal fuzzy sets)
Two fuzzy sets and are said to be equal if and it is denoted by
Example 1.7.6: let be a universal set.
and are fuzzy sets of Then
Definition 1.7.7: (
Let be a fuzzy sets. For define as follows:
is called the of
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Example 1.7.8: Let be a set. Then
is a fuzzy set in
The possible Definition 1.7.9: (Support of a fuzzy set)
Let be a fuzzy set. The set is called the support In particular, is called a finite fuzzy subset if is a finite set, and an infinite fuzzy subset otherwise.
Example 1.7.10: Let be the set. Then the fuzzy set may be described as
The support of is:
The elements are not part of the support of Definition 1.7.11: (complement)
Let be a fuzzy sets. Then the complement is defined by the equation
Example 1.7.12: Let be the set. Then the fuzzy set may be described as
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The complement Definition 1.7.13 (Union)
Let be a fuzzy sets. Then the union is defined by the equation.
Where,
Example 1.7.14: Let be the set. Then the fuzzy sets may be described as
Then, the union is given by: Definition 1.7.15: (Fuzzy Intersection)
Let be a fuzzy sets. Then the intersection defined by the equation.
Where,
Example 1.7.16: Let be the set. Then the fuzzy sets may be described as
The intersection
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Remark 1.7.16: These definitions can be generated for countable number of fuzzy sets. For any collection, of fuzzy subsets ofwhere is a nonempty index set with membership functions, then the membership functions of and of the are given by and respectively.
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