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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 (๐‘‡1,๐‘‡2,๐‘‡3) and T-conorms(๐‘†1,๐‘†2,๐‘†3) 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.




Cover page i
Fly leaf ii
Title page iii
Declaration iv
Certification v
Dedication vi
Acknowledgements vii
Abstract viii
Table of Contents ix
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
2.0 literature Review 28
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
4.1 Inclusion Indices 45
4.2 Partial Matching Indices 51
4.3 Similarity Indices 55
5.0 Summary 64
5.1 Conclusion 64
5.2Recommendations 64
References 65




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
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
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(๐‘‡1,๐‘‡2,๐‘‡3) and T-conorms(๐‘†1,๐‘†2,๐‘†3). We propose a new approach for comparing fuzzy objects which
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
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 ๐ดcan be described as in different ways;
i. By stating the condition for membership (๐ด={๐‘ฅโˆˆ๐‘โˆง๐‘ฅโ‰ค5})
ii. Define the elements by listing the characteristic function, in which 1 indicates
membership and 0 non membership. That is ๐œ‡๐ด ๐‘ฅ =1 if and only if ๐‘ฅโˆˆ๐ด and
๐œ‡๐ด ๐‘ฅ =0 if and only if ๐‘ฅโˆ‰๐ด.
Fuzzy set
Let ๐‘‹ be a collection of objects, the fuzzy set ๐ด in ๐‘‹ is a set of ordered pairs. ๐ด={ ๐‘ฅ,๐œ‡๐ด ๐‘ฅ โ”‚๐‘ฅโˆˆ๐‘‹}
๐œ‡๐ดis called the membership function which maps each element in the universal set ๐‘‹ to the
membership space [0,1].
That is, ๐œ‡๐ด:๐‘‹โ†’[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 ๐‘‹={1, 2, 3, โ€ฆ, 10} be the set of available type of houses described by๐‘ฅโˆˆ๐‘‹ where ๐‘ฅ= number of bedrooms in the house. The fuzzy set โ€œcomfortable type of housesโ€ for a four persons family may be described as
๐ด= {(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 ๐‘ฅโˆˆ๐‘‹ such that ๐œ‡๐ด(๐‘ฅ)>0.
In the above example, the support set of a fuzzy set ๐ด=(1,2,3,4,5,6)
ฮฑ- level set
The crisp set of elements that belong to the fuzzy set A at least to the degree ฮฑ is
๐ด๐›ผ={๐‘ฅโˆˆ๐‘‹โ”‚(๐œ‡๐ด(๐‘ฅ) )โ‰ฅ๐›ผ}.
For example,
i. ๐ด0.2={1,2,3,4,5,6}
ii. ๐ด0.5={2,3,4,5}
iii. ๐ด0.8= 3,4 .
Strong ๐œถ-level set
The strong ๐›ผ-level set is known as strong ๐›ผ-cut and is defined by
ร๐›ผ={๐‘ฅโˆˆ๐‘‹โ”‚(๐œ‡๐ด(๐‘ฅ) )>๐›ผ}.
For example, the strong ฮฑ- level set for ๐›ผ=0.8 ๐‘–๐‘  ๐ด0.8={4}
Convexity of a fuzzy set
A fuzzy set๐ด is convex if ๐œ‡๐ด(๐‘ก)โ‰ฅminโก{๐œ‡๐ด(๐‘Ÿ),๐œ‡๐ด(๐‘ )} where ๐‘ก=(๐œ†๐‘Ÿ+ 1โˆ’๐œ† ๐‘ ),๐‘Ÿ,๐‘ ,๐‘กโˆˆ๐‘‹ and ๐œ†โˆˆ 0,1 .
Alternatively, a fuzzy set is convex if all ฮฑ- level sets are convex.
For example
Convex fuzzy set ๐œ‡๐ด(๐‘ก)โ‰ฅ๐œ‡๐ด(๐‘Ÿ)
For a finite fuzzy set A, the cardinality denoted by |A| is defined as
๐ด = ๐œ‡๐ด(๐‘ฅ)๐‘ฅโˆˆ๐‘‹
๐ด =|๐ด||๐‘‹|is called the relative cardinality of A.
For example, for a fuzzy set โ€œcomfortable type of housesโ€ for a four person family, the cardinality is
๐ด =0.2+0.5+0.8+1+0.7+0.3=3.5.
The relative cardinality is ๐ด =3.510=0.35
Standard operations of fuzzy set
Complement set ๐ด , union ๐ดโˆช๐ต, and intersection ๐ดโˆฉ๐ตrepresent the standard operations of
fuzzy sets as follows ๐œ‡๐ด ๐‘ฅ =1โˆ’๐œ‡๐ด(๐‘ฅ) ๐œ‡๐ดโˆช๐ต ๐‘ฅ =maxโก[๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ ] ๐œ‡๐ดโˆฉ๐ต ๐‘ฅ =minโก[๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ ]
We describe these concepts in details as follows:
Fuzzy complement
The fuzzy complement of a fuzzy set ๐ด, is denoted as ๐ด defined by ๐œ‡๐ด ๐‘ฅ =1โˆ’๐œ‡๐ด(๐‘ฅ)
Complement function C is designed to map the membership function ๐œ‡๐ด(๐‘ฅ) of a fuzzy set A
to [0, 1] and the mapped value is written as ๐ถ(๐œ‡๐ด ๐‘ฅ )
Properties of fuzzy complement function;
i. ๐ถ 0 =1,๐ถ 1 =0 (Boundary conditions).
ii. ๐‘Ž,๐‘โˆˆ 0,1 ๐‘–๐‘“ ๐‘Ž<๐‘,๐‘ก๐‘•๐‘’๐‘› ๐ถ(๐‘Ž)โ‰ฅ๐ถ ๐‘ (Monotonic non- increasing).
iii. ๐ถ is a continuous function.
iv. ๐ถ is involutive i.e. ๐ถ ๐‘Ž =๐‘Ž ๐‘“๐‘œ๐‘Ÿ ๐‘Ž๐‘™๐‘™ ๐‘Žโˆˆ[0,1]
Fuzzy union
โˆช ฮผA ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ =maxโก[๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ ]or ๐œ‡๐ดโˆช๐ต ๐‘ฅ =maxโก[๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ ]
The fuzzy union of two sets ๐ดand ๐ต can be expressed by a function of the form โˆช: 0,1 ร— 0,1 โ†’[0,1]
The membership degree of union ๐ดโˆช๐ต arises from the union function
Properties of fuzzy union function;
i. โˆช 0,0 =0,โˆช 0,1 =1,โˆช 1,0 =1,โˆช 1,1 =1 (boundary conditions).
ii. โˆช ๐‘Ž,๐‘ =โˆช ๐‘,๐‘Ž commutativity.
iii. ๐‘–๐‘“ ๐‘Žโ‰ค๐‘Ž ๐‘Ž๐‘›๐‘‘ ๐‘โ‰ค๐‘ ๐‘ก๐‘•๐‘’๐‘›โˆช ๐‘Ž,๐‘ โ‰คโˆช ๐‘Ž ,๐‘ .
iv. โˆช โˆช ๐‘Ž,๐‘ ,๐‘ =โˆช(๐‘Ž,โˆช ๐‘,๐‘ ).
v. โˆช๐‘–๐‘  ๐‘Ž ๐‘๐‘œ๐‘›๐‘ก๐‘–๐‘›๐‘ข๐‘œ๐‘ข๐‘  ๐‘“๐‘ข๐‘›๐‘๐‘ก๐‘–๐‘œ๐‘›.
vi. โˆช ๐‘Ž,๐‘Ž =๐‘Ž (Idempotency).
Fuzzy intersection
I ๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ =min ๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ ๐‘œ๐‘Ÿ ฮผAโˆฉB ๐‘ฅ =minโก[๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ ]
The intersection of two fuzzy sets A and B is defined by the function; ๐ผ: 0,1 ร— 0,1 โ†’[0,1]
Properties of fuzzy set intersection function;
i. ๐ผ 1,1 =1,๐ผ 1,0 =0,๐ผ 0,1 =0 ๐‘Ž๐‘›๐‘‘ ๐ผ 0,0 =0 (Boundary conditions).
ii. ๐ผ ๐‘Ž,๐‘ =๐ผ ๐‘,๐‘Ž . Commutativity.
iii. ๐‘–๐‘“ ๐‘Žโ‰ค๐‘Ž ๐‘Ž๐‘›๐‘‘ ๐‘โ‰ค๐‘ ๐‘ก๐‘•๐‘’๐‘› ๐ผ(๐‘Ž,๐‘)โ‰ค๐ผ(๐‘Ž ,๐‘ ). ๐ผis a monotonic non decreasing function.
iv. ๐ผ ๐ผ ๐‘Ž,๐‘ ,๐‘ =๐ผ ๐‘Ž,๐ผ ๐‘,๐‘ . Associativity.
v. ๐ผ is a continuous function
vi. ๐ผ ๐‘Ž,๐‘Ž =๐‘Ž. (Idempotency).
Properties of complement, union and intersection
Let ๐ด and ๐ต be two fuzzy sets with membership function ๐ด ๐‘ฅ ๐‘Ž๐‘›๐‘‘ ๐ต ๐‘ฆ respectively, then
the following properties hold:
(1) Commutativity
i. ๐ดโˆช๐ต=๐ตโˆช๐ด
ii. ๐ดโˆฉ๐ต=๐ตโˆฉ๐ด
i. maxโก{๐ด ๐‘ฅ ,๐ต ๐‘ฆ }=maxโก{ ๐ต ๐‘ฆ ,๐ด ๐‘ฅ =๐ด ๐‘ฅ โˆจ๐ต ๐‘ฆ =๐ต ๐‘ฆ โˆจ๐ด ๐‘ฅ .
This can be verified by considering the two possibilities as follows;
๐ด ๐‘ฅ <๐ต ๐‘ฆ ๐‘œ๐‘Ÿ ๐ด ๐‘ฅ >๐ต ๐‘ฆ .
That is, if ๐ด ๐‘ฅ <๐ต ๐‘ฆ , we have ๐ด ๐‘ฅ โˆจ๐ต ๐‘ฆ =๐ต ๐‘ฆ โˆจ๐ด ๐‘ฅ =๐ต ๐‘ฆ .
Also if ๐ด ๐‘ฅ >๐ต ๐‘ฆ , we have ๐ด ๐‘ฅ โˆจ๐ต ๐‘ฆ =๐ต ๐‘ฆ โˆจ๐ด ๐‘ฅ =๐ด ๐‘ฅ .
ii. min ๐ด ๐‘ฅ ,๐ต ๐‘ฆ =min ๐ต ๐‘ฆ ,๐ด ๐‘ฅ =๐ด ๐‘ฅ โˆง๐ต ๐‘ฆ =๐ต ๐‘ฆ โˆง๐ด ๐‘ฅ
๐ด ๐‘ฅ <๐ต ๐‘ฆ ๐‘œ๐‘Ÿ ๐ด ๐‘ฅ >๐ต ๐‘ฆ .
If ๐ด ๐‘ฅ <๐ต ๐‘ฆ , then we have ๐ด ๐‘ฅ โˆง๐ต ๐‘ฆ =๐ต ๐‘ฆ โˆง๐ด ๐‘ฅ =๐ด ๐‘ฅ .
Also if ๐ด ๐‘ฅ >๐ต ๐‘ฆ , then ๐ด ๐‘ฅ โˆง๐ต ๐‘ฆ =๐ต ๐‘ฆ โˆง๐ด ๐‘ฅ =๐ต ๐‘ฆ .
(2) Associativity :
๐ดโˆช ๐ตโˆช๐ถ = ๐ดโˆช๐ต โˆช๐ถ, ๐ดโˆฉ ๐ตโˆฉ๐ถ = ๐ดโˆฉ๐ต โˆฉ๐ถ.
Proof; the proof is almost the same as above, this also apply to properties (3) to (7).
(3) Idempotency:
๐ดโˆช๐ด=๐ด, ๐ดโˆฉ๐ด=๐ด.
(4) Distributivity:
๐ดโˆช ๐ตโˆฉ๐ถ = ๐ดโˆช๐ต โˆฉ ๐ดโˆช๐ถ , ๐ดโˆฉ ๐ตโˆช๐ถ = ๐ดโˆฉ๐ต โˆช ๐ดโˆฉ๐ถ .
(5) ๐ดโˆฉโˆ…=โˆ…, ๐ดโˆช๐‘‹=๐‘‹.
(6) Identity: (๐ดโˆชโˆ…=๐ด, ๐ดโˆฉ๐‘‹=๐ด.
(7) Absorption:
๐ดโˆฉ ๐ดโˆช๐ต =๐ด, ๐ดโˆช ๐ดโˆฉ๐ต =๐ด.
(8) De Morganโ€™s laws:
i. ๐ดโˆช๐ต =๐ด โˆฉ๐ต
ii. ๐ดโˆฉ๐ต =๐ด โˆช๐ต.
Proof; since we know that ๐ด =1โˆ’๐ด,๐‘Ž๐‘›๐‘‘ ๐ต =1โˆ’๐ต. Therefore, we have also two cases ๐ด ๐‘ฅ <๐ต ๐‘ฆ ๐‘œ๐‘Ÿ ๐ด ๐‘ฅ >๐ต ๐‘ฆ
For ๐ด(๐‘ฅ)<๐ต(๐‘ฆ) we have 1โˆ’max ๐ด ๐‘ฅ ,๐ต ๐‘ฆ =1โˆ’ ๐ด ๐‘ฅ โˆจ๐ต ๐‘ฆ =min 1โˆ’๐ด ๐‘ฅ ,1โˆ’๐ต ๐‘ฆ =1โˆ’๐ต ๐‘ฆ
Also for ๐ด ๐‘ฅ >๐ต ๐‘ฆ ,๐‘ค๐‘’ ๐‘•๐‘Ž๐‘ฃ๐‘’
1โˆ’max ๐ด ๐‘ฅ ,๐ต ๐‘ฆ =1โˆ’ ๐ด ๐‘ฅ โˆจ๐ต ๐‘ฆ =1โˆ’๐ด ๐‘ฅ . Proved
(9) Involution:
๐ด =๐ด.
Proof; 1โˆ’ 1โˆ’๐ด ๐‘ฅ = 1โˆ’1+๐ด ๐‘ฅ =๐ด ๐‘ฅ . โ‡’๐ด =๐ด
(10) Equivalence formula:
๐ด โˆฉ๐ต โˆฉ ๐ดโˆช๐ต = ๐ด โˆฉ๐ต โˆช ๐ดโˆฉ๐ต .
(11) Symmetrical formula:
๐ด โˆฉ๐ต โˆช ๐ดโˆฉ๐ต =(๐ด โˆช๐ต )โˆฉ(๐ดโˆช๐ต)
Nonstandard operations of a fuzzy set
The following are the nonstandard operators of fuzzy set;
Union operations
i. Probabilistic sum ๐ด+ ๐ต (Algebraic sum)
โˆ€๐‘ฅโˆˆ๐‘‹,๐œ‡๐ด+๐ต ๐‘ฅ =๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’๐œ‡๐ด(๐‘ฅ)๐œ‡๐ต(๐‘ฅ) โˆ€๐‘ฅโˆˆ๐‘‹,๐œ‡๐ด+๐ต ๐‘ฅ =๐œ‡๐ต+๐ด ๐‘ฅ ๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ
Since ๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’๐œ‡๐ด ๐‘ฅ ๐œ‡๐ต ๐‘ฅ =๐œ‡๐ต ๐‘ฅ +๐œ‡๐ด ๐‘ฅ โˆ’๐œ‡๐ต(๐‘ฅ)๐œ‡๐ด(๐‘ฅ) โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ด+๐ต ๐‘ฅ =1โˆ’ ๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’๐œ‡๐ด ๐‘ฅ ๐œ‡๐ต ๐‘ฅ ๐ท๐‘’๐‘€๐‘œ๐‘Ÿ๐‘”๐‘Ž๐‘›โ€ฒ๐‘ ๐‘™๐‘Ž๐‘ค
ii. Bounded sum AโจB (bold union)
โˆ€๐‘ฅโˆˆ๐‘‹,๐œ‡๐ดโจ๐ต ๐‘ฅ =minโก[1,๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ ] โˆ€๐’™โˆˆ๐‘ฟ,๐œ‡๐ดโจ๐ต ๐‘ฅ =๐œ‡๐ตโจ๐ด ๐‘ฅ ๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ
Sincemin 1,๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ =minโก[1,๐œ‡๐ต ๐‘ฅ +๐œ‡๐ด ๐‘ฅ ] โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโจ๐ต ๐‘ฅ =1โˆ’ min 1,๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ ๐ท๐‘’ ๐‘€๐‘œ๐‘Ÿ๐‘”๐‘Ž๐‘›โ€ฒ๐‘  ๐‘™๐‘Ž๐‘ค
iii. Drastic sum (๐ดโจƒ๐ต)
โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโจƒ๐ต ๐‘ฅ = ๐œ‡๐ด ๐‘ฅ ,๐‘ค๐‘•๐‘’๐‘› ๐œ‡๐ต ๐‘ฅ =0๐œ‡๐ต ๐‘ฅ ,๐‘ค๐‘•๐‘’๐‘› ๐œ‡๐ด ๐‘ฅ =01,๐‘œ๐‘ก๐‘•๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’
iv. Hamacherโ€™s sum (๐ดโˆช๐ต)
โˆ€๐‘ฅโˆˆ๐‘‹,๐œ‡๐ดโˆช๐ต ๐‘ฅ =๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’(2โˆ’๐›พ)๐œ‡๐ด(๐‘ฅ)๐œ‡๐ต(๐‘ฅ)1โˆ’(1โˆ’๐›พ)๐œ‡๐ด(๐‘ฅ)๐œ‡๐ต(๐‘ฅ),๐›พโ‰ฅ0
Intersection operations
i. Algebraic product ๐ดโˆ™๐ต (probabilistic product).
โˆ€๐‘ฅโˆˆ๐‘‹,๐œ‡๐ดโˆ™๐ต ๐‘ฅ =๐œ‡๐ด(๐‘ฅ)โˆ™๐œ‡๐ต(๐‘ฅ) โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโˆ™๐ต ๐‘ฅ =๐œ‡๐ตโˆ™๐ด ๐‘ฅ ๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ.
Since ๐œ‡๐ด(๐‘ฅ)โˆ™๐œ‡๐ต(๐‘ฅ)=๐œ‡๐ต(๐‘ฅ)โˆ™๐œ‡๐ด(๐‘ฅ). โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโˆ™๐ต ๐‘ฅ =1โˆ’ ๐œ‡๐ด ๐‘ฅ โˆ™๐œ‡๐ต ๐‘ฅ ๐ท๐‘’ ๐‘€๐‘œ๐‘Ÿ๐‘”๐‘Ž๐‘›โ€ฒ๐‘  ๐‘™๐‘Ž๐‘ค.
ii. Bounded product ๐ดโจ€๐ต (bold intersection).
โˆ€๐‘ฅโˆˆ๐‘‹,๐œ‡๐ดโจ€๐ต ๐‘ฅ =maxโก[0,๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’1] โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโจ€๐ต ๐‘ฅ =๐œ‡๐ตโจ€๐ด ๐‘ฅ ๐‘๐‘œ๐‘š๐‘š๐‘ข๐‘ก๐‘Ž๐‘ก๐‘–๐‘ฃ๐‘–๐‘ก๐‘ฆ
Since max 0,๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’1 =max 0,๐œ‡๐ต ๐‘ฅ +๐œ‡๐ด ๐‘ฅ โˆ’1 . โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโจ€๐ต ๐‘ฅ =1โˆ’ max 0,๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’1 ๐ท๐‘’ ๐‘€๐‘œ๐‘Ÿ๐‘”๐‘Ž๐‘›โ€ฒ๐‘  ๐‘™๐‘Ž๐‘ค
. iii. Drastic product ๐ดโฉ€๐ต โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโฉ€๐ต ๐‘ฅ = ๐œ‡๐ด ๐‘ฅ ,๐‘ค๐‘•๐‘’๐‘› ๐œ‡๐ต ๐‘ฅ =1๐œ‡๐ต ๐‘ฅ ,๐‘ค๐‘•๐‘’๐‘› ๐œ‡๐ด ๐‘ฅ =10,๐‘ค๐‘•๐‘’๐‘› ๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต(๐‘ฅ)<1
iii. Hamacherโ€™s intersection (๐ดโˆฉ๐ต)
โˆ€๐‘ฅโˆˆ๐‘‹, ๐œ‡๐ดโˆฉ๐ต ๐‘ฅ =๐œ‡๐ด ๐‘ฅ ๐œ‡๐ต ๐‘ฅ ๐›พ+ 1+๐›พ ๐œ‡๐ด ๐‘ฅ +๐œ‡๐ต ๐‘ฅ โˆ’๐œ‡๐ด ๐‘ฅ ๐œ‡๐ต ๐‘ฅ , ๐›พโ‰ฅ0.
Disjunctive sum; ๐ดโจ๐ต=(๐ดโˆฉ๐ต )โˆช(๐ด โˆฉ๐ต)
Definition (simple disjunctive sum) ๐œ‡๐ด ๐‘ฅ =1โˆ’๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ =1โˆ’๐œ‡๐ต(๐‘ฅ) ๐œ‡๐ดโˆฉ๐ต ๐‘ฅ =minโก[๐œ‡๐ด ๐‘ฅ ,1โˆ’๐œ‡๐ต ๐‘ฅ ] ๐œ‡๐ด โˆฉ๐ต ๐‘ฅ =minโก[1โˆ’๐œ‡๐ด ๐‘ฅ ,๐œ‡๐ต ๐‘ฅ ] ๐ดโจ๐ต= ๐ดโˆฉ๐ต โˆช ๐ด โˆฉ๐ต ,๐‘ก๐‘•๐‘’๐‘› ๐œ‡๐ดโจ๐ต ๐‘ฅ =maxโก{min ๐œ‡๐ด ๐‘ฅ ,1โˆ’๐œ‡๐ต ๐‘ฅ ,min 1โˆ’๐œ‡๐ด ๐‘ฅ }
For example, let
๐ด= ๐‘ฅ1,0.2 , ๐‘ฅ2,0.7 , ๐‘ฅ3,1 , ๐‘ฅ4,0 ,and
๐ต={ ๐‘ฅ1,0.5 , ๐‘ฅ2,0.3 , ๐‘ฅ3,1 , ๐‘ฅ4,0.1 }.
Then, ๐ด ={ ๐‘ฅ1,0.8 , ๐‘ฅ2,0.3 , ๐‘ฅ3,0 , ๐‘ฅ4,1 } ๐ต ={ ๐‘ฅ1,0.5 , ๐‘ฅ2,0.7 , ๐‘ฅ3,0 , ๐‘ฅ4,0.1 } ๐ด โˆฉ๐ต={ ๐‘ฅ1,0.5 , ๐‘ฅ2,0.3 , ๐‘ฅ3,0 , ๐‘ฅ4,0.1 } ๐ดโˆฉ๐ต ={ ๐‘ฅ1,0.2 , ๐‘ฅ2,0.7 , ๐‘ฅ3,0 , ๐‘ฅ4,0 }
Therefore ๐ดโจ๐ต= ๐ดโˆฉ๐ต โˆช ๐ด โˆฉ๐ต ={ ๐‘ฅ1,0.5 , ๐‘ฅ2,0.7 , ๐‘ฅ3,0 , ๐‘ฅ4,0.1 }
Disjoint sum ๐œ‡๐ดฮ”๐ต ๐‘ฅ = ๐œ‡๐ด ๐‘ฅ โˆ’๐œ‡๐ต ๐‘ฅ .
For example, let
๐ด= ๐‘ฅ1,0.2 , ๐‘ฅ2,0.7 , ๐‘ฅ3,1 , ๐‘ฅ4,0 ,and
๐ต={ ๐‘ฅ1,0.5 , ๐‘ฅ2,0.3 , ๐‘ฅ3,1 , ๐‘ฅ4,0.1 }.
Then, ๐ดฮ”๐ต={ ๐‘ฅ1,0.3 , ๐‘ฅ2,0.4 , ๐‘ฅ3,0 , ๐‘ฅ4,0.1 }
Difference fuzzy set;
๐ดโˆ’๐ต=๐ดโˆฉ๐ต .
In fuzzy set, there are two ways of obtaining the difference:
(a) Simple difference
For example, let
๐ด= ๐‘ฅ1,0.2 , ๐‘ฅ2,0.7 , ๐‘ฅ3,1 , ๐‘ฅ4,0 ,and
๐ต={ ๐‘ฅ1,0.5 , ๐‘ฅ2,0.3 , ๐‘ฅ3,1 , ๐‘ฅ4,0.1 }.
Then,๐ต ={ ๐‘ฅ1,0.5 , ๐‘ฅ2,0.7 , ๐‘ฅ3,0 , ๐‘ฅ4,0.9 }, and
๐ดโˆ’๐ต=๐ดโˆฉ๐ต ={ ๐‘ฅ1,0.2 , ๐‘ฅ2,0.7 , ๐‘ฅ3,0 , ๐‘ฅ4,0 }.
(b) Bounded difference;
๐œ‡๐ด๐œƒ๐ต ๐‘ฅ =maxโก[0,๐œ‡๐ด ๐‘ฅ โˆ’๐œ‡๐ต ๐‘ฅ ], and ๐ด๐œƒ๐ต= ๐‘ฅ1,0 , ๐‘ฅ2,0.4 , ๐‘ฅ3,0 , ๐‘ฅ4,0 .
Distances in fuzzy set theory
(a) Hamming distance
๐‘‘๐‘• ๐ด,๐ต = ๐œ‡๐ด ๐‘ฅ๐‘– โˆ’๐œ‡๐ต ๐‘ฅ๐‘– ๐‘›๐‘–=1,๐‘ฅโˆˆ๐‘‹
For example, let
๐ด= ๐‘ฅ1,0.4 , ๐‘ฅ2,0.8 , ๐‘ฅ3,1 , ๐‘ฅ4,0 ,and
๐ต={ ๐‘ฅ1,0.4 , ๐‘ฅ2,0.3 , ๐‘ฅ3,0 , ๐‘ฅ4,0 }.
Then,๐‘‘ ๐ด,๐ต = 0 + 0.5 + 1 + 0 =1.5
This definition satisfies the usual mathematical notion of distance;
i. ๐‘‘(๐ด,๐ต)โ‰ฅ0
ii. ๐‘‘ ๐ด,๐ต =๐‘‘ ๐ต,๐ด ๐‘ ๐‘ฆ๐‘š๐‘š๐‘’๐‘ก๐‘Ÿ๐‘–๐‘.
iii. ๐‘‘ ๐ด,๐ถ โ‰ค๐‘‘ ๐ด,๐ต +๐‘‘ ๐ต,๐ถ ๐‘ก๐‘Ÿ๐‘–๐‘Ž๐‘›๐‘”๐‘™๐‘’ ๐‘–๐‘›๐‘’๐‘ž๐‘ข๐‘Ž๐‘™๐‘–๐‘ก๐‘ฆ.
iv. ๐‘‘ ๐ด,๐ด =0
Relative hamming distance ๐‘‘๐‘Ÿ๐‘• ๐ด,๐ต =1๐‘›๐‘‘๐‘•(๐ด,๐ต)
Hamming distance can be called symmetrical distance by using the operator โˆ‡;
โˆ€๐‘ฅโˆˆ๐‘‹,๐œ‡๐ดโˆ‡๐ต ๐‘ฅ =|๐œ‡๐ด ๐‘ฅ โˆ’๐œ‡๐ต ๐‘ฅ |
(b) Euclidean distance
๐‘‘๐‘’ ๐ด,๐ต = (๐œ‡๐ด ๐‘ฅ โˆ’๐œ‡๐ต(๐‘ฅ))2๐‘›๐‘–=1
From the example taken above ๐‘‘๐‘’ ๐ด,๐ต =(02+0.52+12+02)12
Relative Euclidean distance; ๐‘‘๐‘Ÿ๐‘’ ๐ด,๐ต =๐‘‘๐‘’(๐ด,๐ต) ๐‘›
(c) Minkowski distance;
๐‘‘๐‘š ๐ด,๐ต =( |๐‘ฅโˆˆ๐‘‹๐œ‡๐ด ๐‘ฅ โˆ’๐œ‡๐ต(๐‘ฅ)|๐‘š)1๐‘š ๐‘šโˆˆ[1,โˆž]
Hamming distance and Euclidean distance can be obtain fromMinkwoski distance
When w=1 it becomes Hamming distance.
When w=2 it becomes Euclidean distance.
Fuzzy relation
Let X, YโŠ† โ„be universal sets then;
๐‘…={( ๐‘ฅ,๐‘ฆ ,๐œ‡๐‘… ๐‘ฅ,๐‘ฆ )|(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘Œ}is called a fuzzy relation in ๐‘‹ร—๐‘Œ โŠ†๐‘….
Or ๐‘‹ and ๐‘Œ are two universal sets, the fuzzy relation ๐‘… ๐‘ฅ,๐‘ฆ is given as ๐‘… ๐‘ฅ,๐‘ฆ ={๐œ‡๐‘… ๐‘ฅ,๐‘ฆ ๐‘ฅ,๐‘ฆ |(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘Œ}
Fuzzy relations are often presented in the form of two dimensional tables. A fuzzy relation ๐‘… can be represented by a๐‘šร—๐‘› matrix:
๐‘ฆ1โ€ฆ ๐‘ฆ๐‘› ๐‘…=๐‘ฅ1โ‹ฎ๐‘ฅ๐‘š ๐œ‡๐‘…(๐‘ฅ1,๐‘ฆ1)โ‹ฏ๐œ‡๐‘…(๐‘ฅ1,๐‘ฆ๐‘›)โ‹ฎโ‹ฑโ‹ฎ๐œ‡๐‘…(๐‘ฅ๐‘š,๐‘ฆ1)โ‹ฏ๐œ‡๐‘…(๐‘ฅ๐‘š,๐‘ฆ๐‘›)
For example,๐‘™๐‘’๐‘ก ๐‘‹= 1,2,3 ๐‘Ž๐‘›๐‘‘ ๐‘Œ={1,2}
the membership function is defined by
๐œ‡๐‘… ๐‘ฅ,๐‘ฆ =๐‘’โˆ’(๐‘ฅโˆ’๐‘ฆ)2.
A fuzzy relation can be defined as follows;
R = ๐‘’โˆ’(1โˆ’1)2(1,1),๐‘’โˆ’(1โˆ’2)2(1,2),๐‘’โˆ’(2โˆ’1)2(2,1),๐‘’โˆ’(2โˆ’2)2(2,2),๐‘’โˆ’(3โˆ’1)2(3,1),๐‘’โˆ’(3โˆ’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
Let R and Z be two fuzzy relations in the same product space. The union of
R with Z is defined by:
๐œ‡๐‘…โˆช๐‘ ๐‘ฅ,๐‘ฆ =max ๐œ‡๐‘… ๐‘ฅ,๐‘ฆ ,๐œ‡๐‘ ๐‘ฅ,๐‘ฆ ,(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘Œ.
Let R and Z be two fuzzy relations in the same product space. The intersection of
R with Z is defined by:
๐œ‡๐‘…โˆฉ๐‘ ๐‘ฅ,๐‘ฆ =min ๐œ‡๐‘… ๐‘ฅ,๐‘ฆ ,๐œ‡๐‘ ๐‘ฅ,๐‘ฆ ,(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘Œ.
The complement relation ๐‘… for fuzzy relation ๐‘… is defined by the following function: โˆ€ ๐‘ฅ,๐‘ฆ โˆˆ๐ดร—๐ต, ๐œ‡๐‘… ๐‘ฅ,๐‘ฆ =1โˆ’๐œ‡๐‘…(๐‘ฅ,๐‘ฆ)
Projection of fuzzy relations
Let๐‘…={[ ๐‘ฅ,๐‘ฆ ,๐œ‡๐‘… ๐‘ฅ,๐‘ฆ ]โ”‚(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘Œ} be a fuzzy relation. The projection of ๐‘…(๐‘ฅ,๐‘ฆ) on
๐‘‹denoted by ๐‘…1 is given by ๐‘…1= ๐‘ฅ,๐‘š๐‘Ž๐‘ฅ๐œ‡๐‘… ๐‘ฅ,๐‘ฆ ,(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘Œ
and the projection of ๐‘… ๐‘ฅ,๐‘ฆ on ๐‘Œ denoted by ๐‘…2 is given by ๐‘…2= ๐‘ฆ ,๐‘š๐‘Ž๐‘ฅ๐œ‡๐‘… ๐‘ฅ,๐‘ฆ , ๐‘ฅ,๐‘ฆ โˆˆ๐‘‹ร—๐‘Œ.
Similarly, we calculate the grade of membership for all pairs, so the projection on ๐‘‹ is given by๐‘…1={ ๐‘ฅ1,1 , ๐‘ฅ2,1 , ๐‘ฅ3,1 }
Cylindrical extension of fuzzy relation
The cylindrical extension of ๐‘‹ร—๐‘Œ of a fuzzy set ๐ด of ๐‘‹ is a fuzzy relation cylA whose
membership function is equal to;
๐‘๐‘ฆ๐‘™๐ด ๐‘ฅ,๐‘ฆ =๐ด ๐‘ฅ , โˆ€๐‘ฅโˆˆ๐‘‹,โˆ€๐‘ฆโˆˆ๐‘Œ.
Cylindrical extension from ๐‘‹- projections means filling all the columns of the related matrix
by the ๐‘‹-projections. Similarly cylindrical extension from ๐‘Œ-projections means filling all the
rows of the relational matrix by the ๐‘Œ-projections.
Fuzzy maximum-minimum composition of relations
Let ๐‘‹,๐‘Œ and ๐‘ be universal sets and let ๐‘… and ๐‘„ be relations given by
๐‘…= ๐‘ฅ,๐‘ฆ ,๐œ‡๐‘… ๐‘ฅ,๐‘ฆ ๐‘ฅโˆˆ๐‘‹,๐‘ฆโˆˆ๐‘Œ,๐‘…โŠ‚๐‘‹ร—๐‘Œand
๐‘„= ๐‘ฆ,๐‘ง ,๐œ‡๐‘… ๐‘ฆ,๐‘ง ๐‘ฆโˆˆ๐‘Œ,๐‘งโˆˆ๐‘,๐‘„โŠ‚๐‘Œร—๐‘.
Then ๐‘† will be a relation that relates elements in ๐‘‹ that ๐‘… contains to the elements in ๐‘ that ๐‘„
contains, i.e.,๐‘†=๐‘…โˆ˜๐‘„.
Here โ€œโˆ˜โ€ means the composition of membership degrees of ๐‘… and ๐‘„ in the max-min sense.
๐‘†= ๐‘ฅ,๐‘ง ,๐œ‡๐‘… ๐‘ฅ,๐‘ง ๐‘ฅโˆˆ๐‘‹,๐‘งโˆˆ๐‘,๐‘†โŠ‚๐‘‹ร—๐‘.
The max-min composition is defined as ๐œ‡๐‘† ๐‘ฅ,๐‘ง =maxyโˆˆY(minโก(ฮผR ๐‘ฅ,๐‘ฆ ,๐œ‡๐‘„ ๐‘ฆ,๐‘ง )โก
and max product composition is then defined ๐œ‡๐‘† ๐‘ฅ,๐‘ง =maxyโˆˆY(minโก(๐œ‡๐‘… ๐‘ฅ,๐‘ฆ โˆ™๐œ‡๐‘„ ๐‘ฆ,๐‘ง )โก
Fuzzy max-min composition operation
Let R1(๐‘ฅ,๐‘ฆ),(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘Œ and R2 ๐‘ฆ,๐‘ง , ๐‘ฆ,๐‘ง โˆˆ๐‘Œร—๐‘ be two fuzzy relations. The max-
min composition of R1 and R2 is then the set:
R1โˆ˜R2 ๐‘ฅ,๐‘ง ={[ ๐‘ฅ,๐‘ง ,maxโก{minโก{๐œ‡๐‘…1 ๐‘ฅ,๐‘ฆ ,๐œ‡๐‘…2(๐‘ฆ,๐‘ง)}}]โ”‚๐‘ฅโˆˆ๐‘‹,๐‘ฆโˆˆ๐‘Œ,๐‘งโˆˆ๐‘}
Fuzzy max-product operation[Rosenfeld, 1975]
The max- product composition R1โˆ˜R2 is defined as ๐‘…1ยฐ๐‘…2 (๐‘ฅ,๐‘ง)={[(๐‘ฅ,๐‘ง),maxโก{๐œ‡๐‘…1(๐‘ฅ,๐‘ฆ)โˆ™๐œ‡๐‘…2(๐‘ฆ,๐‘ง)} ]โ”‚๐‘ฅโˆˆ๐‘‹,๐‘ฆโˆˆ๐‘Œ,๐‘งโˆˆ๐‘}
Fuzzy max- average composition operation[Rosenfeld, 1975]
The max- ave composition R1โˆ˜R2 is defined by
R1โˆ˜R2 ๐‘ฅ,๐‘ง ={[ ๐‘ฅ,๐‘ง ,12โˆ™(maxโก(๐œ‡๐‘…1(๐‘ฅ,๐‘ฆ)+๐œ‡๐‘…2(๐‘ฆ,๐‘ง))]โ”‚๐‘ฅโˆˆ๐‘‹,๐‘ฆโˆˆ๐‘Œ,๐‘งโˆˆ๐‘}
Properties of Fuzzy Relations
Reflexive relation
Let R be a fuzzy relation in ๐‘‹ร—๐‘‹. Then R is called reflexive, if
๐œ‡R ๐‘ฅ,๐‘ฅ =1 โˆ€๐‘ฅโˆˆ๐‘‹
Antireflexive relation
Fuzzy relation ๐‘…โŠ‚๐‘‹ร—๐‘‹ is antireflexive if
๐œ‡R ๐‘ฅ,๐‘ฅ =0,๐‘ฅโˆˆ๐‘‹.
Symmetric Relation
A fuzzy relation R sis called symmetric if,
๐œ‡R ๐‘ฅ,๐‘ฆ =๐œ‡R ๐‘ฆ,๐‘ฅ โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹.
Antisymmetric Relation
Fuzzy relation ๐‘…โŠ‚๐‘‹ร—๐‘‹ is antisymmetric iff
If ๐œ‡R ๐‘ฅ,๐‘ฆ >0 ๐‘ก๐‘•๐‘’๐‘› ๐œ‡R ๐‘ฆ,๐‘ฅ =0 ๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘ฅโ‰ ๐‘ฆ.
Transitive Relation
Fuzzy relation ๐‘…โŠ‚๐‘‹ร—๐‘‹ is transitive in the sense of max-min iff
๐œ‡R(๐‘ฅ,๐‘ง)โ‰ฅmaxโก(minโก(๐œ‡R ๐‘ฅ,๐‘ฆ ,๐œ‡R(๐‘ฆ,๐‘ง))) ๐‘ฅ,๐‘งโˆˆ๐‘‹
Since ๐‘…2=๐‘…โˆ˜๐‘…, if
๐œ‡R2 ๐‘ฅ,๐‘ง =maxโก(๐œ‡R ๐‘ฅ,๐‘ฆ ,๐œ‡R(๐‘ฆ,๐‘ง))
then ๐‘… is transitive if ๐‘…โˆ˜๐‘…=๐‘…(๐‘…โˆ˜๐‘…โŠ†๐‘…)
and๐‘…2โŠ‚๐‘… means that ๐œ‡R2(๐‘ฅ,๐‘ฆ)โ‰ค๐œ‡R ๐‘ฆ,๐‘ฅ .
Fuzzy Compatibility Relation ๐‘…:๐‘‹ร—๐‘‹โ†’{0,1} ๐‘… ๐‘–๐‘  ๐‘Ž ๐‘๐‘œ๐‘š๐‘๐‘Ž๐‘ก๐‘–๐‘๐‘–๐‘™๐‘–๐‘ก๐‘ฆ ๐‘Ÿ๐‘’๐‘™๐‘Ž๐‘ก๐‘–๐‘œ๐‘› ๐‘–๐‘“ ๐‘–๐‘ก ๐‘–๐‘ 
i. Reflexive
โˆ€๐‘ฅโˆˆ๐‘‹โ‡’๐œ‡R ๐‘ฅ,๐‘ฅ =1
ii. Symmetric
โˆ€(๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘‹โ‡’๐œ‡R ๐‘ฅ,๐‘ฆ = ๐œ‡R(๐‘ฅ,๐‘ฆ)
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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