## ABSTRACT

In this dissertation, various concepts for comparing fuzzy objects such as similarity, dissimilarity, symmetric similarity, relative similarity, multi-dimensional, and multi-attributes were studied. Some existing models such as Jaccard, Simple Matching Co-efficient, Vector, and Tversky were closely studied. The similarity measures introduced by Tversky(1977) and modified by (Dubois and Prade, 1980) using the cardinality of fuzzy sets and scalar evaluators and their operations such as T-norms (𝑇1,𝑇2,𝑇3) and T-conorms(𝑆1,𝑆2,𝑆3) were systematized. Finally,a new approach (Set Theoretic Measures) for comparing fuzzy objects by taking into consideration the degree of inclusion, partial matching and similarity was presented, some properties of the modified model elaborated and some areas of applications were identified.

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## TABLE OF CONTENTS

Cover page i

Fly leaf ii

Title page iii

Declaration iv

Certification v

Dedication vi

Acknowledgements vii

Abstract viii

Table of Contents ix

CHAPTER ONE

GENERAL INTRODUCTION

1.1Background of the Study 1

1.2Statement of the Research Problem 3

1.3Aim and Objectives of the Study 3

1.4Methodology 4

1.5Definition of Terms 4

1.6 Organization of Dissertation 27

CHAPTER TWO

LITERATURE REVIEW

2.0 literature Review 28

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CHAPTER THREE

APPLICATIONS OF COMPATIBILITY RELATION IN FUZZY SET CONTEXT

3.1 Concept of Similarity 34

3.2 Jaccard Ratio Model 36

3.3 Simple Matching Coefficient 38

3.4 Vector Ratio Model 38

3.5 Mean Character Difference Model 39

3.6 Canberra Metric Model 39

3.7 Tversky Ratio Model 40

CHAPTER FOUR

MODIFIED APPLICATIONS OF COMPATIBILITY RELATION IN FUZZY SET CONTEXT

4.1 Inclusion Indices 45

4.2 Partial Matching Indices 51

4.3 Similarity Indices 55

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATIONS

5.0 Summary 64

5.1 Conclusion 64

5.2Recommendations 64

xi

References 65

## CHAPTER ONE

GENERAL INTRODUCTION

1.1 Background of the Study

Fuzzy set was introduced by Zadeh in (1965) to represent or manipulate data and information possessing uncertainties.The theory of fuzzy set has advanced in a variety of ways and in many disciplines. Applications of this theory are found in artificial intelligence, computer science, operational research, pattern recognition, and robotics (Dubois, 1980). To handle the

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problems involving imprecise concepts, the conventional methods of set theory are found insufficient. In order to overcome shortcomings of the conventional approaches, development of fuzzy set theory has been found most successful in this direction.

One of the most fundamental notions in pure and applied science is the concept of relation. Science has been described as the discovery of relations between objects, states and events (Peterson, 1976).Fuzzy relations generalize the concept of relations in the same manner as fuzzy sets generalize the fundamental idea of sets.A relation is a mathematical description of a situation where certain elements of sets are related to one another in some way. Fuzzy relations are significant concepts in fuzzy theory and have been widely used in many fields such as fuzzy clustering, fuzzy control and uncertainty reasoning.

The theory of compatibility relation has been studied extensively in mathematics along with its applications in diverse fields. The term compatibility relation is used to encompass various types of comparisons frequently made between objects and concepts. The degree to which two objects are compatible is a fundamental component of human reasoning and consequently is critical in the development of automated diagnosis, information retrieval and decision systems. Assessment of compatibility relation has played an important role in diverse disciplines such as taxonomy, psychology, and the social sciences. Each discipline has proposed methods for quantifying compatibility judgments suitable for its particular applications. Applications of compatibility relation in various areas include expert systems, information retrieval, and intelligent database system etc., (Valerie, 2002).

Several measures of similarity among fuzzy sets have been proposed in literature. The motivation behind these measures is both geometric and set-theoretic. Geometric models dominate the theoretic analysis of similarity measures. Objects in these models are

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represented as points in a coordinate space, and the metric between the respective points is considered to be a measure for deciding the degree of similarity or dissimilarity among the objects. In most cases the Euclidean distance is used to define such a measure. In the set-theoretic approaches a different model is used, which is based on the concept of non dimensional and non metric similarity relation (Bashon, 2011).

Similarity measures are specific functions used to approximate the degree to which two compared objects are similar to one another. The functions are required to fulfill specific similarity conditions or axioms. The use of compatibility measures depends on the type of data characterizing the objects being compared. Data describing those objects could be categorical or numerical which can be presented in set representations such as fuzzy set.Based on literature reviews, various forms of similarity measures involving fuzzy sets have been proposed depending on the context in which they are to be applied. There is no unique way of determining the degree to which two such sets are compatible to one another(Suliaman and Mohamad, 2012). However, in this dissertation we narrow down our research to a special kind of relation called compatibility relation in fuzzy set context and modify the model proposed by Tverskyin order to develop a new model which shows the degree of inclusion, partial matching and similarity of fuzzy objects.

1.2 Statement of the Research Problem

We intend to investigate compatibility relation in fuzzy context and modify some of the applications of Tversky parameterized ratio model using the cardinality of fuzzy sets and other functions such as the scalar evaluators and their operators such asT-norms(𝑇1,𝑇2,𝑇3) and T-conorms(𝑆1,𝑆2,𝑆3). We propose a new approach for comparing fuzzy objects which

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include the degree of inclusion, partial matching and similarity and discuss some of the properties of the modified model.

1.3 Aim and Objectives of the Study

The aim of this research is to investigate compatibility relationin fuzzy context and modify Tversky model. To achieve this, the objectives are to:

i. investigate various concepts related to similarity, dissimilarity, symmetric similarity, relative similarity, multi-dimensional and multi-attribute of fuzzy objects,

ii. study existing ratio models, in particular Jaccardunparameterized ratio model and Tversky parameterized ratio model, and present a new model in terms of inclusion, partial matching, and similarity, and

iii. discuss some properties of the new model of similarity measures in relation to the degree of inclusion, partial matching and similarity of fuzzy objects.

1.4 Methodology

An up-to-date review of literatureson fuzzy relations, compatibility relation in fuzzy set context, and similarity measures introduced by Tversky which would be of help in modifying Tversky models to show the degree of inclusion, partial matching and similarity of fuzzy objects was conducted.

1.5 Definition of Terms

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These definitions are adopted from different sources [(Dubois and Prade, 1980), (Kaufmann, 1975), (Rosenfeld, 1975)].

Classical set

A classical set or crisp set is normally defined as the collection of elements or objects

that can be finite, countable, or uncountable.

Such a classical set 𝐴can be described as in different ways;

i. By stating the condition for membership (𝐴={𝑥∈𝑁∧𝑥≤5})

ii. Define the elements by listing the characteristic function, in which 1 indicates

membership and 0 non membership. That is 𝜇𝐴 𝑥 =1 if and only if 𝑥∈𝐴 and

𝜇𝐴 𝑥 =0 if and only if 𝑥∉𝐴.

Fuzzy set

Let 𝑋 be a collection of objects, the fuzzy set 𝐴 in 𝑋 is a set of ordered pairs. 𝐴={ 𝑥,𝜇𝐴 𝑥 │𝑥∈𝑋}

𝜇𝐴is called the membership function which maps each element in the universal set 𝑋 to the

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membership space [0,1].

That is, 𝜇𝐴:𝑋→[0,1]

For example, a house owner wants to classify the house he offers to his clients. One indicator of comfort of these houses is the number of bedrooms in it. Let 𝑋={1, 2, 3, …, 10} be the set of available type of houses described by𝑥∈𝑋 where 𝑥= number of bedrooms in the house. The fuzzy set “comfortable type of houses” for a four persons family may be described as

𝐴= {(1, 0.2),(2, 0.5), (3, 0.8), (4, 1), (5, 0.7), (6, 0.3)}

A fuzzy set is denoted by a set of ordered of pairs, the first element of which denotes the

element and the second the degree of membership.

Support of a fuzzy set

The support of a fuzzy set A, is the classical set of all 𝑥∈𝑋 such that 𝜇𝐴(𝑥)>0.

In the above example, the support set of a fuzzy set 𝐴=(1,2,3,4,5,6)

α- level set

The crisp set of elements that belong to the fuzzy set A at least to the degree α is

𝐴𝛼={𝑥∈𝑋│(𝜇𝐴(𝑥) )≥𝛼}.

For example,

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i. 𝐴0.2={1,2,3,4,5,6}

ii. 𝐴0.5={2,3,4,5}

iii. 𝐴0.8= 3,4 .

Strong 𝜶-level set

The strong 𝛼-level set is known as strong 𝛼-cut and is defined by

Á𝛼={𝑥∈𝑋│(𝜇𝐴(𝑥) )>𝛼}.

For example, the strong α- level set for 𝛼=0.8 𝑖𝑠 𝐴0.8={4}

Convexity of a fuzzy set

A fuzzy set𝐴 is convex if 𝜇𝐴(𝑡)≥min{𝜇𝐴(𝑟),𝜇𝐴(𝑠)} where 𝑡=(𝜆𝑟+ 1−𝜆 𝑠),𝑟,𝑠,𝑡∈𝑋 and 𝜆∈ 0,1 .

Alternatively, a fuzzy set is convex if all α- level sets are convex.

For example

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Convex fuzzy set 𝜇𝐴(𝑡)≥𝜇𝐴(𝑟)

Cardinality

For a finite fuzzy set A, the cardinality denoted by |A| is defined as

𝐴 = 𝜇𝐴(𝑥)𝑥∈𝑋

𝐴 =|𝐴||𝑋|is called the relative cardinality of A.

For example, for a fuzzy set “comfortable type of houses” for a four person family, the cardinality is

𝐴 =0.2+0.5+0.8+1+0.7+0.3=3.5.

The relative cardinality is 𝐴 =3.510=0.35

Standard operations of fuzzy set

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Complement set 𝐴 , union 𝐴∪𝐵, and intersection 𝐴∩𝐵represent the standard operations of

fuzzy sets as follows 𝜇𝐴 𝑥 =1−𝜇𝐴(𝑥) 𝜇𝐴∪𝐵 𝑥 =max[𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 ] 𝜇𝐴∩𝐵 𝑥 =min[𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 ]

We describe these concepts in details as follows:

Fuzzy complement

The fuzzy complement of a fuzzy set 𝐴, is denoted as 𝐴 defined by 𝜇𝐴 𝑥 =1−𝜇𝐴(𝑥)

Complement function C is designed to map the membership function 𝜇𝐴(𝑥) of a fuzzy set A

to [0, 1] and the mapped value is written as 𝐶(𝜇𝐴 𝑥 )

Properties of fuzzy complement function;

i. 𝐶 0 =1,𝐶 1 =0 (Boundary conditions).

ii. 𝑎,𝑏∈ 0,1 𝑖𝑓 𝑎<𝑏,𝑡𝑒𝑛 𝐶(𝑎)≥𝐶 𝑏 (Monotonic non- increasing).

iii. 𝐶 is a continuous function.

iv. 𝐶 is involutive i.e. 𝐶 𝑎 =𝑎 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑎∈[0,1]

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Fuzzy union

∪ μA 𝑥 ,𝜇𝐵 𝑥 =max[𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 ]or 𝜇𝐴∪𝐵 𝑥 =max[𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 ]

The fuzzy union of two sets 𝐴and 𝐵 can be expressed by a function of the form ∪: 0,1 × 0,1 →[0,1]

The membership degree of union 𝐴∪𝐵 arises from the union function

Properties of fuzzy union function;

i. ∪ 0,0 =0,∪ 0,1 =1,∪ 1,0 =1,∪ 1,1 =1 (boundary conditions).

ii. ∪ 𝑎,𝑏 =∪ 𝑏,𝑎 commutativity.

iii. 𝑖𝑓 𝑎≤𝑎 𝑎𝑛𝑑 𝑏≤𝑏 𝑡𝑒𝑛∪ 𝑎,𝑏 ≤∪ 𝑎 ,𝑏 .

iv. ∪ ∪ 𝑎,𝑏 ,𝑐 =∪(𝑎,∪ 𝑏,𝑐 ).

v. ∪𝑖𝑠 𝑎 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛.

vi. ∪ 𝑎,𝑎 =𝑎 (Idempotency).

Fuzzy intersection

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I 𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 =min 𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 𝑜𝑟 μA∩B 𝑥 =min[𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 ]

The intersection of two fuzzy sets A and B is defined by the function; 𝐼: 0,1 × 0,1 →[0,1]

Properties of fuzzy set intersection function;

i. 𝐼 1,1 =1,𝐼 1,0 =0,𝐼 0,1 =0 𝑎𝑛𝑑 𝐼 0,0 =0 (Boundary conditions).

ii. 𝐼 𝑎,𝑏 =𝐼 𝑏,𝑎 . Commutativity.

iii. 𝑖𝑓 𝑎≤𝑎 𝑎𝑛𝑑 𝑏≤𝑏 𝑡𝑒𝑛 𝐼(𝑎,𝑏)≤𝐼(𝑎 ,𝑏 ). 𝐼is a monotonic non decreasing function.

iv. 𝐼 𝐼 𝑎,𝑏 ,𝑐 =𝐼 𝑎,𝐼 𝑏,𝑐 . Associativity.

v. 𝐼 is a continuous function

vi. 𝐼 𝑎,𝑎 =𝑎. (Idempotency).

Properties of complement, union and intersection

Let 𝐴 and 𝐵 be two fuzzy sets with membership function 𝐴 𝑥 𝑎𝑛𝑑 𝐵 𝑦 respectively, then

the following properties hold:

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(1) Commutativity

i. 𝐴∪𝐵=𝐵∪𝐴

ii. 𝐴∩𝐵=𝐵∩𝐴

Proof

i. max{𝐴 𝑥 ,𝐵 𝑦 }=max{ 𝐵 𝑦 ,𝐴 𝑥 =𝐴 𝑥 ∨𝐵 𝑦 =𝐵 𝑦 ∨𝐴 𝑥 .

This can be verified by considering the two possibilities as follows;

𝐴 𝑥 <𝐵 𝑦 𝑜𝑟 𝐴 𝑥 >𝐵 𝑦 .

That is, if 𝐴 𝑥 <𝐵 𝑦 , we have 𝐴 𝑥 ∨𝐵 𝑦 =𝐵 𝑦 ∨𝐴 𝑥 =𝐵 𝑦 .

Also if 𝐴 𝑥 >𝐵 𝑦 , we have 𝐴 𝑥 ∨𝐵 𝑦 =𝐵 𝑦 ∨𝐴 𝑥 =𝐴 𝑥 .

ii. min 𝐴 𝑥 ,𝐵 𝑦 =min 𝐵 𝑦 ,𝐴 𝑥 =𝐴 𝑥 ∧𝐵 𝑦 =𝐵 𝑦 ∧𝐴 𝑥

𝐴 𝑥 <𝐵 𝑦 𝑜𝑟 𝐴 𝑥 >𝐵 𝑦 .

If 𝐴 𝑥 <𝐵 𝑦 , then we have 𝐴 𝑥 ∧𝐵 𝑦 =𝐵 𝑦 ∧𝐴 𝑥 =𝐴 𝑥 .

Also if 𝐴 𝑥 >𝐵 𝑦 , then 𝐴 𝑥 ∧𝐵 𝑦 =𝐵 𝑦 ∧𝐴 𝑥 =𝐵 𝑦 .

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(2) Associativity :

𝐴∪ 𝐵∪𝐶 = 𝐴∪𝐵 ∪𝐶, 𝐴∩ 𝐵∩𝐶 = 𝐴∩𝐵 ∩𝐶.

Proof; the proof is almost the same as above, this also apply to properties (3) to (7).

(3) Idempotency:

𝐴∪𝐴=𝐴, 𝐴∩𝐴=𝐴.

(4) Distributivity:

𝐴∪ 𝐵∩𝐶 = 𝐴∪𝐵 ∩ 𝐴∪𝐶 , 𝐴∩ 𝐵∪𝐶 = 𝐴∩𝐵 ∪ 𝐴∩𝐶 .

(5) 𝐴∩∅=∅, 𝐴∪𝑋=𝑋.

(6) Identity: (𝐴∪∅=𝐴, 𝐴∩𝑋=𝐴.

(7) Absorption:

𝐴∩ 𝐴∪𝐵 =𝐴, 𝐴∪ 𝐴∩𝐵 =𝐴.

(8) De Morgan’s laws:

i. 𝐴∪𝐵 =𝐴 ∩𝐵

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ii. 𝐴∩𝐵 =𝐴 ∪𝐵.

Proof; since we know that 𝐴 =1−𝐴,𝑎𝑛𝑑 𝐵 =1−𝐵. Therefore, we have also two cases 𝐴 𝑥 <𝐵 𝑦 𝑜𝑟 𝐴 𝑥 >𝐵 𝑦

For 𝐴(𝑥)<𝐵(𝑦) we have 1−max 𝐴 𝑥 ,𝐵 𝑦 =1− 𝐴 𝑥 ∨𝐵 𝑦 =min 1−𝐴 𝑥 ,1−𝐵 𝑦 =1−𝐵 𝑦

Also for 𝐴 𝑥 >𝐵 𝑦 ,𝑤𝑒 𝑎𝑣𝑒

1−max 𝐴 𝑥 ,𝐵 𝑦 =1− 𝐴 𝑥 ∨𝐵 𝑦 =1−𝐴 𝑥 . Proved

(9) Involution:

𝐴 =𝐴.

Proof; 1− 1−𝐴 𝑥 = 1−1+𝐴 𝑥 =𝐴 𝑥 . ⇒𝐴 =𝐴

(10) Equivalence formula:

𝐴 ∩𝐵 ∩ 𝐴∪𝐵 = 𝐴 ∩𝐵 ∪ 𝐴∩𝐵 .

(11) Symmetrical formula:

𝐴 ∩𝐵 ∪ 𝐴∩𝐵 =(𝐴 ∪𝐵 )∩(𝐴∪𝐵)

Nonstandard operations of a fuzzy set

The following are the nonstandard operators of fuzzy set;

Union operations

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i. Probabilistic sum 𝐴+ 𝐵 (Algebraic sum)

∀𝑥∈𝑋,𝜇𝐴+𝐵 𝑥 =𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −𝜇𝐴(𝑥)𝜇𝐵(𝑥) ∀𝑥∈𝑋,𝜇𝐴+𝐵 𝑥 =𝜇𝐵+𝐴 𝑥 𝑐𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑖𝑡𝑦

Since 𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −𝜇𝐴 𝑥 𝜇𝐵 𝑥 =𝜇𝐵 𝑥 +𝜇𝐴 𝑥 −𝜇𝐵(𝑥)𝜇𝐴(𝑥) ∀𝑥∈𝑋, 𝜇𝐴+𝐵 𝑥 =1− 𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −𝜇𝐴 𝑥 𝜇𝐵 𝑥 𝐷𝑒𝑀𝑜𝑟𝑔𝑎𝑛′𝑠𝑙𝑎𝑤

ii. Bounded sum A⨁B (bold union)

∀𝑥∈𝑋,𝜇𝐴⨁𝐵 𝑥 =min[1,𝜇𝐴 𝑥 +𝜇𝐵 𝑥 ] ∀𝒙∈𝑿,𝜇𝐴⨁𝐵 𝑥 =𝜇𝐵⨁𝐴 𝑥 𝑐𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑖𝑡𝑦

Sincemin 1,𝜇𝐴 𝑥 +𝜇𝐵 𝑥 =min[1,𝜇𝐵 𝑥 +𝜇𝐴 𝑥 ] ∀𝑥∈𝑋, 𝜇𝐴⨁𝐵 𝑥 =1− min 1,𝜇𝐴 𝑥 +𝜇𝐵 𝑥 𝐷𝑒 𝑀𝑜𝑟𝑔𝑎𝑛′𝑠 𝑙𝑎𝑤

iii. Drastic sum (𝐴⨃𝐵)

∀𝑥∈𝑋, 𝜇𝐴⨃𝐵 𝑥 = 𝜇𝐴 𝑥 ,𝑤𝑒𝑛 𝜇𝐵 𝑥 =0𝜇𝐵 𝑥 ,𝑤𝑒𝑛 𝜇𝐴 𝑥 =01,𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒

iv. Hamacher’s sum (𝐴∪𝐵)

∀𝑥∈𝑋,𝜇𝐴∪𝐵 𝑥 =𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −(2−𝛾)𝜇𝐴(𝑥)𝜇𝐵(𝑥)1−(1−𝛾)𝜇𝐴(𝑥)𝜇𝐵(𝑥),𝛾≥0

Intersection operations

i. Algebraic product 𝐴∙𝐵 (probabilistic product).

∀𝑥∈𝑋,𝜇𝐴∙𝐵 𝑥 =𝜇𝐴(𝑥)∙𝜇𝐵(𝑥) ∀𝑥∈𝑋, 𝜇𝐴∙𝐵 𝑥 =𝜇𝐵∙𝐴 𝑥 𝑐𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑖𝑡𝑦.

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Since 𝜇𝐴(𝑥)∙𝜇𝐵(𝑥)=𝜇𝐵(𝑥)∙𝜇𝐴(𝑥). ∀𝑥∈𝑋, 𝜇𝐴∙𝐵 𝑥 =1− 𝜇𝐴 𝑥 ∙𝜇𝐵 𝑥 𝐷𝑒 𝑀𝑜𝑟𝑔𝑎𝑛′𝑠 𝑙𝑎𝑤.

ii. Bounded product 𝐴⨀𝐵 (bold intersection).

∀𝑥∈𝑋,𝜇𝐴⨀𝐵 𝑥 =max[0,𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −1] ∀𝑥∈𝑋, 𝜇𝐴⨀𝐵 𝑥 =𝜇𝐵⨀𝐴 𝑥 𝑐𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑖𝑡𝑦

Since max 0,𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −1 =max 0,𝜇𝐵 𝑥 +𝜇𝐴 𝑥 −1 . ∀𝑥∈𝑋, 𝜇𝐴⨀𝐵 𝑥 =1− max 0,𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −1 𝐷𝑒 𝑀𝑜𝑟𝑔𝑎𝑛′𝑠 𝑙𝑎𝑤

. iii. Drastic product 𝐴⩀𝐵 ∀𝑥∈𝑋, 𝜇𝐴⩀𝐵 𝑥 = 𝜇𝐴 𝑥 ,𝑤𝑒𝑛 𝜇𝐵 𝑥 =1𝜇𝐵 𝑥 ,𝑤𝑒𝑛 𝜇𝐴 𝑥 =10,𝑤𝑒𝑛 𝜇𝐴 𝑥 ,𝜇𝐵(𝑥)<1

iii. Hamacher’s intersection (𝐴∩𝐵)

∀𝑥∈𝑋, 𝜇𝐴∩𝐵 𝑥 =𝜇𝐴 𝑥 𝜇𝐵 𝑥 𝛾+ 1+𝛾 𝜇𝐴 𝑥 +𝜇𝐵 𝑥 −𝜇𝐴 𝑥 𝜇𝐵 𝑥 , 𝛾≥0.

Disjunctive sum; 𝐴⨁𝐵=(𝐴∩𝐵 )∪(𝐴 ∩𝐵)

.

.

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Definition (simple disjunctive sum) 𝜇𝐴 𝑥 =1−𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 =1−𝜇𝐵(𝑥) 𝜇𝐴∩𝐵 𝑥 =min[𝜇𝐴 𝑥 ,1−𝜇𝐵 𝑥 ] 𝜇𝐴 ∩𝐵 𝑥 =min[1−𝜇𝐴 𝑥 ,𝜇𝐵 𝑥 ] 𝐴⨁𝐵= 𝐴∩𝐵 ∪ 𝐴 ∩𝐵 ,𝑡𝑒𝑛 𝜇𝐴⨁𝐵 𝑥 =max{min 𝜇𝐴 𝑥 ,1−𝜇𝐵 𝑥 ,min 1−𝜇𝐴 𝑥 }

For example, let

𝐴= 𝑥1,0.2 , 𝑥2,0.7 , 𝑥3,1 , 𝑥4,0 ,and

𝐵={ 𝑥1,0.5 , 𝑥2,0.3 , 𝑥3,1 , 𝑥4,0.1 }.

Then, 𝐴 ={ 𝑥1,0.8 , 𝑥2,0.3 , 𝑥3,0 , 𝑥4,1 } 𝐵 ={ 𝑥1,0.5 , 𝑥2,0.7 , 𝑥3,0 , 𝑥4,0.1 } 𝐴 ∩𝐵={ 𝑥1,0.5 , 𝑥2,0.3 , 𝑥3,0 , 𝑥4,0.1 } 𝐴∩𝐵 ={ 𝑥1,0.2 , 𝑥2,0.7 , 𝑥3,0 , 𝑥4,0 }

Therefore 𝐴⨁𝐵= 𝐴∩𝐵 ∪ 𝐴 ∩𝐵 ={ 𝑥1,0.5 , 𝑥2,0.7 , 𝑥3,0 , 𝑥4,0.1 }

Disjoint sum 𝜇𝐴Δ𝐵 𝑥 = 𝜇𝐴 𝑥 −𝜇𝐵 𝑥 .

For example, let

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𝐴= 𝑥1,0.2 , 𝑥2,0.7 , 𝑥3,1 , 𝑥4,0 ,and

𝐵={ 𝑥1,0.5 , 𝑥2,0.3 , 𝑥3,1 , 𝑥4,0.1 }.

Then, 𝐴Δ𝐵={ 𝑥1,0.3 , 𝑥2,0.4 , 𝑥3,0 , 𝑥4,0.1 }

Difference fuzzy set;

𝐴−𝐵=𝐴∩𝐵 .

In fuzzy set, there are two ways of obtaining the difference:

(a) Simple difference

(𝐴−𝐵)

For example, let

𝐴= 𝑥1,0.2 , 𝑥2,0.7 , 𝑥3,1 , 𝑥4,0 ,and

𝐵={ 𝑥1,0.5 , 𝑥2,0.3 , 𝑥3,1 , 𝑥4,0.1 }.

Then,𝐵 ={ 𝑥1,0.5 , 𝑥2,0.7 , 𝑥3,0 , 𝑥4,0.9 }, and

𝐴−𝐵=𝐴∩𝐵 ={ 𝑥1,0.2 , 𝑥2,0.7 , 𝑥3,0 , 𝑥4,0 }.

(b) Bounded difference;

𝜇𝐴𝜃𝐵 𝑥 =max[0,𝜇𝐴 𝑥 −𝜇𝐵 𝑥 ], and 𝐴𝜃𝐵= 𝑥1,0 , 𝑥2,0.4 , 𝑥3,0 , 𝑥4,0 .

Distances in fuzzy set theory

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(a) Hamming distance

𝑑 𝐴,𝐵 = 𝜇𝐴 𝑥𝑖 −𝜇𝐵 𝑥𝑖 𝑛𝑖=1,𝑥∈𝑋

For example, let

𝐴= 𝑥1,0.4 , 𝑥2,0.8 , 𝑥3,1 , 𝑥4,0 ,and

𝐵={ 𝑥1,0.4 , 𝑥2,0.3 , 𝑥3,0 , 𝑥4,0 }.

Then,𝑑 𝐴,𝐵 = 0 + 0.5 + 1 + 0 =1.5

This definition satisfies the usual mathematical notion of distance;

i. 𝑑(𝐴,𝐵)≥0

ii. 𝑑 𝐴,𝐵 =𝑑 𝐵,𝐴 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐.

iii. 𝑑 𝐴,𝐶 ≤𝑑 𝐴,𝐵 +𝑑 𝐵,𝐶 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦.

iv. 𝑑 𝐴,𝐴 =0

Relative hamming distance 𝑑𝑟 𝐴,𝐵 =1𝑛𝑑(𝐴,𝐵)

Hamming distance can be called symmetrical distance by using the operator ∇;

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∀𝑥∈𝑋,𝜇𝐴∇𝐵 𝑥 =|𝜇𝐴 𝑥 −𝜇𝐵 𝑥 |

(b) Euclidean distance

𝑑𝑒 𝐴,𝐵 = (𝜇𝐴 𝑥 −𝜇𝐵(𝑥))2𝑛𝑖=1

From the example taken above 𝑑𝑒 𝐴,𝐵 =(02+0.52+12+02)12

Relative Euclidean distance; 𝑑𝑟𝑒 𝐴,𝐵 =𝑑𝑒(𝐴,𝐵) 𝑛

(c) Minkowski distance;

𝑑𝑚 𝐴,𝐵 =( |𝑥∈𝑋𝜇𝐴 𝑥 −𝜇𝐵(𝑥)|𝑚)1𝑚 𝑚∈[1,∞]

Hamming distance and Euclidean distance can be obtain fromMinkwoski distance

When w=1 it becomes Hamming distance.

When w=2 it becomes Euclidean distance.

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Fuzzy relation

Let X, Y⊆ ℝbe universal sets then;

𝑅={( 𝑥,𝑦 ,𝜇𝑅 𝑥,𝑦 )|(𝑥,𝑦)∈𝑋×𝑌}is called a fuzzy relation in 𝑋×𝑌 ⊆𝑅.

Or 𝑋 and 𝑌 are two universal sets, the fuzzy relation 𝑅 𝑥,𝑦 is given as 𝑅 𝑥,𝑦 ={𝜇𝑅 𝑥,𝑦 𝑥,𝑦 |(𝑥,𝑦)∈𝑋×𝑌}

Fuzzy relations are often presented in the form of two dimensional tables. A fuzzy relation 𝑅 can be represented by a𝑚×𝑛 matrix:

𝑦1… 𝑦𝑛 𝑅=𝑥1⋮𝑥𝑚 𝜇𝑅(𝑥1,𝑦1)⋯𝜇𝑅(𝑥1,𝑦𝑛)⋮⋱⋮𝜇𝑅(𝑥𝑚,𝑦1)⋯𝜇𝑅(𝑥𝑚,𝑦𝑛)

For example,𝑙𝑒𝑡 𝑋= 1,2,3 𝑎𝑛𝑑 𝑌={1,2}

the membership function is defined by

𝜇𝑅 𝑥,𝑦 =𝑒−(𝑥−𝑦)2.

Solution

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A fuzzy relation can be defined as follows;

R = 𝑒−(1−1)2(1,1),𝑒−(1−2)2(1,2),𝑒−(2−1)2(2,1),𝑒−(2−2)2(2,2),𝑒−(3−1)2(3,1),𝑒−(3−2)2(3,2)

from the example considered above

R = 1.0(1,1),0.37(1,2),0.37(2,1),1.0(2,2),0.02(3,1),0.37(3,2) .

Operations of fuzzy relation

Union

Let R and Z be two fuzzy relations in the same product space. The union of

R with Z is defined by:

𝜇𝑅∪𝑍 𝑥,𝑦 =max 𝜇𝑅 𝑥,𝑦 ,𝜇𝑍 𝑥,𝑦 ,(𝑥,𝑦)∈𝑋×𝑌.

Intersection

Let R and Z be two fuzzy relations in the same product space. The intersection of

R with Z is defined by:

𝜇𝑅∩𝑍 𝑥,𝑦 =min 𝜇𝑅 𝑥,𝑦 ,𝜇𝑍 𝑥,𝑦 ,(𝑥,𝑦)∈𝑋×𝑌.

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Complement

The complement relation 𝑅 for fuzzy relation 𝑅 is defined by the following function: ∀ 𝑥,𝑦 ∈𝐴×𝐵, 𝜇𝑅 𝑥,𝑦 =1−𝜇𝑅(𝑥,𝑦)

Projection of fuzzy relations

Let𝑅={[ 𝑥,𝑦 ,𝜇𝑅 𝑥,𝑦 ]│(𝑥,𝑦)∈𝑋×𝑌} be a fuzzy relation. The projection of 𝑅(𝑥,𝑦) on

𝑋denoted by 𝑅1 is given by 𝑅1= 𝑥,𝑚𝑎𝑥𝜇𝑅 𝑥,𝑦 ,(𝑥,𝑦)∈𝑋×𝑌

and the projection of 𝑅 𝑥,𝑦 on 𝑌 denoted by 𝑅2 is given by 𝑅2= 𝑦 ,𝑚𝑎𝑥𝜇𝑅 𝑥,𝑦 , 𝑥,𝑦 ∈𝑋×𝑌.

Similarly, we calculate the grade of membership for all pairs, so the projection on 𝑋 is given by𝑅1={ 𝑥1,1 , 𝑥2,1 , 𝑥3,1 }

Cylindrical extension of fuzzy relation

The cylindrical extension of 𝑋×𝑌 of a fuzzy set 𝐴 of 𝑋 is a fuzzy relation cylA whose

membership function is equal to;

𝑐𝑦𝑙𝐴 𝑥,𝑦 =𝐴 𝑥 , ∀𝑥∈𝑋,∀𝑦∈𝑌.

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Cylindrical extension from 𝑋- projections means filling all the columns of the related matrix

by the 𝑋-projections. Similarly cylindrical extension from 𝑌-projections means filling all the

rows of the relational matrix by the 𝑌-projections.

Fuzzy maximum-minimum composition of relations

Let 𝑋,𝑌 and 𝑍 be universal sets and let 𝑅 and 𝑄 be relations given by

𝑅= 𝑥,𝑦 ,𝜇𝑅 𝑥,𝑦 𝑥∈𝑋,𝑦∈𝑌,𝑅⊂𝑋×𝑌and

𝑄= 𝑦,𝑧 ,𝜇𝑅 𝑦,𝑧 𝑦∈𝑌,𝑧∈𝑍,𝑄⊂𝑌×𝑍.

Then 𝑆 will be a relation that relates elements in 𝑋 that 𝑅 contains to the elements in 𝑍 that 𝑄

contains, i.e.,𝑆=𝑅∘𝑄.

Here “∘” means the composition of membership degrees of 𝑅 and 𝑄 in the max-min sense.

𝑆= 𝑥,𝑧 ,𝜇𝑅 𝑥,𝑧 𝑥∈𝑋,𝑧∈𝑍,𝑆⊂𝑋×𝑍.

The max-min composition is defined as 𝜇𝑆 𝑥,𝑧 =maxy∈Y(min(μR 𝑥,𝑦 ,𝜇𝑄 𝑦,𝑧 )

and max product composition is then defined 𝜇𝑆 𝑥,𝑧 =maxy∈Y(min(𝜇𝑅 𝑥,𝑦 ∙𝜇𝑄 𝑦,𝑧 )

Fuzzy max-min composition operation

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Let R1(𝑥,𝑦),(𝑥,𝑦)∈𝑋×𝑌 and R2 𝑦,𝑧 , 𝑦,𝑧 ∈𝑌×𝑍 be two fuzzy relations. The max-

min composition of R1 and R2 is then the set:

R1∘R2 𝑥,𝑧 ={[ 𝑥,𝑧 ,max{min{𝜇𝑅1 𝑥,𝑦 ,𝜇𝑅2(𝑦,𝑧)}}]│𝑥∈𝑋,𝑦∈𝑌,𝑧∈𝑍}

Fuzzy max-product operation[Rosenfeld, 1975]

The max- product composition R1∘R2 is defined as 𝑅1°𝑅2 (𝑥,𝑧)={[(𝑥,𝑧),max{𝜇𝑅1(𝑥,𝑦)∙𝜇𝑅2(𝑦,𝑧)} ]│𝑥∈𝑋,𝑦∈𝑌,𝑧∈𝑍}

Fuzzy max- average composition operation[Rosenfeld, 1975]

The max- ave composition R1∘R2 is defined by

R1∘R2 𝑥,𝑧 ={[ 𝑥,𝑧 ,12∙(max(𝜇𝑅1(𝑥,𝑦)+𝜇𝑅2(𝑦,𝑧))]│𝑥∈𝑋,𝑦∈𝑌,𝑧∈𝑍}

Properties of Fuzzy Relations

Reflexive relation

Let R be a fuzzy relation in 𝑋×𝑋. Then R is called reflexive, if

𝜇R 𝑥,𝑥 =1 ∀𝑥∈𝑋

Antireflexive relation

Fuzzy relation 𝑅⊂𝑋×𝑋 is antireflexive if

𝜇R 𝑥,𝑥 =0,𝑥∈𝑋.

Symmetric Relation

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A fuzzy relation R sis called symmetric if,

𝜇R 𝑥,𝑦 =𝜇R 𝑦,𝑥 ∀𝑥,𝑦∈𝑋.

Antisymmetric Relation

Fuzzy relation 𝑅⊂𝑋×𝑋 is antisymmetric iff

If 𝜇R 𝑥,𝑦 >0 𝑡𝑒𝑛 𝜇R 𝑦,𝑥 =0 𝑥,𝑦∈𝑋,𝑥≠𝑦.

Transitive Relation

Fuzzy relation 𝑅⊂𝑋×𝑋 is transitive in the sense of max-min iff

𝜇R(𝑥,𝑧)≥max(min(𝜇R 𝑥,𝑦 ,𝜇R(𝑦,𝑧))) 𝑥,𝑧∈𝑋

Since 𝑅2=𝑅∘𝑅, if

𝜇R2 𝑥,𝑧 =max(𝜇R 𝑥,𝑦 ,𝜇R(𝑦,𝑧))

then 𝑅 is transitive if 𝑅∘𝑅=𝑅(𝑅∘𝑅⊆𝑅)

and𝑅2⊂𝑅 means that 𝜇R2(𝑥,𝑦)≤𝜇R 𝑦,𝑥 .

Fuzzy Compatibility Relation 𝑅:𝑋×𝑋→{0,1} 𝑅 𝑖𝑠 𝑎 𝑐𝑜𝑚𝑝𝑎𝑡𝑖𝑏𝑖𝑙𝑖𝑡𝑦 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑓 𝑖𝑡 𝑖𝑠

i. Reflexive

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∀𝑥∈𝑋⇒𝜇R 𝑥,𝑥 =1

ii. Symmetric

∀(𝑥,𝑦)∈𝑋×𝑋⇒𝜇R 𝑥,𝑦 = 𝜇R(𝑥,𝑦)

Note that a compatibility relation is not transitive in general.

1.6 Organization of Dissertation

The dissertation is organized as follows;

Besides the background of the study, objective of the research and methodology given in chapter one, chapter two provides a detailed literature review on the subject matter. Chapter threeprovides various concepts related to similarity of objects and some existing models such as Jaccard, Simple Matching Coefficient, Vector, and Tversky for comparing fuzzy objects. Chapter four gives a new approach for comparing fuzzy objects which included the degree of

inclusion, partial matching and similarity and some of their properties. A modified ratio model is also presented in chapter four. Chapter five contains summary and some future research directions.

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