2 FUZZY SET THEORY

Fuzzy Set Theory

Sonam Devgan KaulM.Sc. (Hons.), NET-JRF

Assistant Professor of MathematicsAmity University

Noida, U.P. - 201301, India.

MUMBAI NEW DELHI NAGPUR BENGALURU HYDERABAD CHENNAI PUNE LUCKNOW AHMEDABAD ERNAKULAM BHUBANESWAR INDORE KOLKATA GUWAHATI

CRISP SETS 3

© Author

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First Edition : 2014

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4 FUZZY SET THEORY

Dedicated To

My Beloved Husband“Mr. ROHIT KAUL”

CRISP SETS 5

Preface

I am very happy to bring out the first addition of the book Fuzzy Set Theory. The book isdesigned as a textbook for the undergraduate and postgraduate students of Science and Engineering ofvarious Indian universities on the basis of the University Grants Commission curriculum. Particularly,covers the entire ambit of the course Fuzzy Mathematics for the Semester VI B.Tech. program ofAmity University. In this book, fuzzy concepts have been introduced as generalization and extensionof crisp sets. Also emphasis is given not only to the presentation of the fundamental and theoreticalconcepts of fuzzy set theory in an intelligible and easy-to-understand manner but also details thetheoretical advances and applications of fuzzy sets in the real world phenomena. This book providesstudents with a self-contained introduction that requires no preliminary knowledge of fuzzymathematics. The book covers the following contents in 9 chapters:

Chapter 1 provides fundamentals of basic crisp set theory and the mapping of crisp set to afunction. Chapter 2 deals with the introduction, basic definitions, standard operations and propertiesof fuzzy sets. It then proceeds to link the crisp set with fuzzy set theory and shows how and wherethese two theories concur as well as differ. Chapter 3 deals with the alpha cut set and the extent towhich an element is a member of a fuzzy set, followed by the features and types of membershipfunctions. Chapter 4 shows how fuzzy sets can be represented by family of crisp sets and alsoprovides the extension principle on multiple fuzzy sets. Fuzzy complement function along with theYager and Sugeno class of complement function is given in Chapter 5. Chapter 6 covers the axioms,types and characterization theorems of t-norms and t-conorms. Chapter 7 deals with the variousaggregation operations on fuzzy sets. Chapter 8 provides basic aspects of fuzzy numbers andlinguistic variables and also covers the arithmetic operations between fuzzy numbers with intervalanalysis method and with extension principle method. Finally, fuzzy equations are introduced andexamined in Chapter 9.

Each topic has been thoroughly covered in scope, content and also from the examination point ofview. For each topic, several worked out examples, simple as well as typical, carefully selected tocover all aspects of the topic, so that the reader may gain confidence in the techniques of solvingproblems of each chapter.

First and foremost, I thank Lord RAM for giving me an inspiration, encouragement andconfidence to write the book. I wish to acknowledge my profound and deep regards to Dr. Ashok K.Chauhan, Founder President, Amity University, Dr. (Mrs.) Balvinder Shukla, Acting Vice Chancellor,AUUP, Prof. Sunita Rattan, Additional Director, AIAS and Dr. Prakriti Rai, HOD Mathematics, fortheir constant encouragement and inspiration. I am grateful to all the faculty members of Departmentof Mathematics, Amity University, particularly Dr. Lakhveer Kaur, Dr. Pratima Rai, Ms. Roosel Jainand Ms. Anjali Naithani for providing me valuable suggestions and support. With great pleasure, Iwish to express my deep sense of gratitude to my research supervisor Dr. Amit Kumar Awasthi for hisconstant encouragement, valuable advice and inspiring guidance.

My family members have been my most consistent source of support in my work. I would like tothank my parents Mr. Sanjeev Devgan and Mrs. Sushma Devgan and my parents-in-law Mr. M.L. Kaul

6 FUZZY SET THEORY

and Mrs. Pushp Lata Kaul for their sound counseling, cheerful support, love, blessings and bestwishes kept my spirit up. Without their blessings, the task of writing a book could never beencompleted. A heartful thanks to my brother Mr. Sahil Devgan for his love and affection towards me.

Last but not the least, I am deeply indebted to my husband, Mr. Rohit Kaul for helping me torealize my strengths and weaknesses. Just because of his patience and continuous support during theperiod, this book has become a reality.

I am hopeful that this exhaustive work will be helpful for students as well as teachers. If you haveany queries, please feel free to write at: skaul@amity.edu. In spite of careful efforts, some errors mighthave crept into the book. Report of any such errors and suggestions for improving the future editionsof the book are welcome and will be gratefully acknowledged.

December 11, 2013 Sonam Devgan Kaul

CRISP SETS 7

Contents

1. Crisp Sets 1 – 131.1 Introduction of Crisp Sets1.2 Mapping of Crisp Sets to a Function1.3 Representation of a Crisp Set1.4 Types of Crisp Set1.5 Operations on Crisp Sets1.6 Properties of Crisp Sets1.7 Examples1.8 Exercises

2. Fuzzy Sets 14 – 372.1 Introduction2.2 Definition2.3 Representation Methods of Fuzzy Set2.4 Cardinality of Fuzzy Set

2.4.1 Scalar Cardinality2.4.2 Relative Cardinality

2.5 Types of Fuzzy Set2.6 Standard Operations on Fuzzy Set

2.6.1 Standard Complement of Fuzzy Set2.6.2 Standard Union of Fuzzy Sets2.6.3 Standard Intersection of Fuzzy Sets

2.7 Properties of Fuzzy Sets2.8 Crisp Sets versus Fuzzy Sets2.9 Exercises

3. α-Cut Set and Membership Function 38 – 673.1 α-Cut Set

3.1.1 Strong α-Cut Set3.1.2 Level Set

3.2 Closed Interval α-Cut3.3 Features of Membership Functions3.4 Properties3.5 Convex Fuzzy Set3.6 Types of Membership Functions

8 FUZZY SET THEORY

3.7 Exercises

4. Decomposition of a Fuzzy Set and Extension Principle 68 – 994.1 Introduction4.2 Properties of α-Cut Set4.3 Decomposition of Fuzzy Set4.4 Extension Principle for Fuzzy Sets4.5 Extension Principle for Multiple Fuzzy Sets4.6 Exercises

5. Fuzzy Complement 100 – 1305.1 Fuzzy Set and its Complement5.2 Axioms of the Complement Function5.3 Sugeno and Yager Fuzzy Complement Function

5.3.1 Sugeno Function5.3.2 Yager Function

5.4 Equilibrium Value of the Complement Function5.5 Increasing and Decreasing Generator and their Pseudo-inverses

5.5.1 Increasing Generator5.5.2 Decreasing Generator5.5.3 Pseudo-inverse of an Increasing Generator5.5.4 Pseudo-inverse of a Decreasing Generator

5.6 Characterization Theorems of Fuzzy Complement5.7 Exercises

6. T-Norms and T-Conorms 131 – 1686.1 Fuzzy Intersection and Fuzzy Union

6.1.1 T-Norms or Fuzzy Intersection6.1.2 T-Conorms or S-Norms or Fuzzy Union

6.2 Requirements of the T-Norms and T-Conorms6.2.1 Axioms of the Intersection Function6.2.2 Axioms of the Union Function

6.3 Types of T-Norms and T-Conorms6.3.1 Standard Intersection and Union (Minimum T-norm and

Maximum T-conorm)6.3.2 Algebraic Product and Sum (Product T-norm and Sum

T-conorm)6.3.3 Drastic Product and Sum (Drastic T-norm and Drastic T-conorm)6.3.4 Bounded Difference and Bounded Sum (Lukasiewicz T-norm

and Lukasiewicz T-conorm)6.3.5 Einstein Product and Sum6.3.6 Hamacher Product and Sum

6.4 Dual Pair of Functions

CRISP SETS 9

6.5 Characterization Theorem of T-Norm and T-Conorm6.6 Exercises

7. Aggregation Operations 169 – 1837.1 Aggregation Operation on Multiple Fuzzy Sets7.2 Definition and Axioms of Aggregation Operation7.3 Types of Aggregation Operations

7.3.1 Arithmetic Mean Aggregation Operation7.3.2 Geometric Mean Aggregation Operation7.3.3 Harmonic Mean Aggregation Operation7.3.4 Generalized Mean Averaging Operation7.3.5 Ordered Weighted Averaging Operation

7.4 Additional Operations on Fuzzy Sets7.4.1 Disjunctive Sum7.4.2 Disjoint Sum7.4.3 Simple Difference7.4.4 Bounded Difference

7.5 Exercises8. Arithmetic Operations on Fuzzy Numbers 184 – 218

8.1 Fuzzy Numbers8.1.1 Fuzzy Interval

8.2 Types of Fuzzy Numbers8.2.1 Triangular Fuzzy Number8.2.2 Trapezoidal Fuzzy Number8.2.3 Quasi Fuzzy Number

8.3 Linguistic Variables8.3.1 Linguistic Hedges

8.4 Interval Fuzzy Arithmetic Operations8.5 Fuzzy Arithmetic with Extension Principle8.6 Exercises

9. Fuzzy Equations 219 – 2379.1 Lattice of Fuzzy Numbers9.2 Fuzzy Equations9.3 Solution of Fuzzy Equation of type A + X = B9.4 Solution of Fuzzy Equation of type A.X = B9.5 ExercisesTutorial Sheet 1 238 – 243Tutorial Sheet 2 244 – 246Tutorial Sheet 3 247 – 248Tutorial Sheet 4 249 – 253

10 FUZZY SET THEORY

1.1 Introduction of Crisp SetsCrisp set is a well defined collection of objects selected from the universe of discourse or

universal set. Let the universal set X be the set of all objects that are needed in a particular context ofstudy or application or situation. The individual objects of a crisp set are known as elements ormembers of the set. Each single element can either be an element of a crisp set A, x Î A (x belongs to A)or not an element of crisp set A, x Ï A (x does not belongs to A). The total number of elements in afinite set is the cardinality of that set and it is denoted by |A|. Some examples of crisp sets are asfollows:

(i) Crisp set of Integers, = {..., −3, −2, −1, 0, 1, 2, 3, ...}.(ii) Crisp set A of Natural numbers which are divisible by 4, A = {4, 8, 12, 16, ...}.

(iii) Crisp set B of Vowels in English Alphabet, B = {a, e, i, o, u}

(iv) Crisp Set C = {1

:?

?n

nxx , n Î and 1 ≤ n ≤ 7} = {87,

76,

65,

54,

43,

32,

21 }.

(v) Crisp Set D = {x : x is a positive integer and x2 < 30} = {1, 2, 3, 4, 5}.(vi) Crisp Set E = {x : x Î and x2 = 9} = {3}.

1.2 Mapping of Crisp Sets to a FunctionMapping of set theoretic forms to function theoretic forms is an important concept. In general it

can be used to map elements or subsets of one universe of discourse to elements of sets in anotheruniverse.

Let X be the universal set and A be a crisp set on X, each element of the universal set eitherbelongs to or not belongs to the crisp set A. If an element x Î X is a member of crisp set A then themembership function or characteristic function of crisp set A assigns 1 to x otherwise if an elementx Î X is not a member of crisp set A then the membership function or characteristic function of crisp setA assigns 0 to x, where the characteristic function represents the membership in set A for an element xin the universal set. The membership function associated with crisp set A is denoted by μA(x) or A(x).The characteristic function μA is defined by:

μA : X → {0, 1}Such that

CRISP SETS

Chapter 1

CRISP SETS 11

???

??

?AxAx

xA if0 if1

)(?

The membership mapping for the crisp set A is shown in figure 1.1.

Figure 1.1: Membership Mapping of Crisp Set A

Note. The membership function of a crisp set can take value either 0 or 1, while the membershipfunction of fuzzy set can take any value in the closed interval [0, 1].

1.3 Representation of a Crisp SetCrisp set A defined on universal set X can be represented by any of the following three ways:(i) Tabular or Roaster Form (Listing Method):

In Roaster form, a crisp set can be represented by listing all its elements being separated bycommas and enclosed within braces.

(a) A = {1, 2, 3, 4, ...}(b) B = {a, e, i, o, u}

(ii) Set Builder or Rule Method Form:In set builder form, crisp set is defined by a property satisfying by its members, i.e.

A = {x : P(x)}Thus the crisp set A is a set of all elements x that satisfy the property P(x).

(a) A = {x : x Î and x > 0}(b) B = {x : x is a vowel in English Alphabet}

(iii) Membership or Characteristic Function Method:A crisp set A is defined by a function, usually called the characteristic or membershipfunction, that declares which elements of the universal set X are the members of crisp set A.Crisp set A is defined by its membership function as follows:

μA : X → {0, 1}such that

???

??

?AxAx

xA if0 if1

)(?

Crisp Set A

x

y

o

1

12 FUZZY SET THEORY

(a)??? ??

?otherwise0

0 if1)(

xandxxA?

(b)???

?otherwise0

AlphabetEnglish in vowela is if1)(

xxB?

1.4 Types of Crisp Set

Universal SetThe universal set or the universe of discourse X be the set of all objects that are needed in a

particular context of study or application or situation. If there are some crisp sets under consideration,then the set of each of these is a subset of universal set or universe of discourse. For instance, the crispsets A and B defined on universal set X are as follows:

X = {x : x is a real number}A = {x : x is a rational number}

B = {x : x is an irrational number}

Singleton SetA crisp set having only one element is called a singleton set. For instance,(i) A = {a}.

(ii) B = {x : x ≥ 0 and x ≤ 0} = {0}.(iii) C = {x : x Î and x2 = 9} = {3}

Null SetA crisp set having no element is called the null set. It is also called the empty set or void set and

it is denoted by f or {}. For instance,(i) A = {x : x Î and x2 = 2}.

(ii) B = {x : x > 0 and x < 0}.(iii) C = {x : x Î and 2x − 3 = 0}

SubsetsLet A and B be two crisp sets. The set A is said to be the subset of set B if every element of A is

also an element of B. Symbolically,A Í B Û if x Î A then x Î B

For instance, the crisp set of positive odd integers is a subset of crisp set of positive integers,while the crisp set of negative integers is not a subset of crisp set of positive integers.

Remark. (i) Every crisp set is a subset of itself, A Í A.(ii) Every crisp set is a subset of universal set, A Í X.

CRISP SETS 13

(iii) The empty set is a subset of every crisp set, f Í A.(iv) For crisp sets A, B and C if A Í B and B Í C then A Í C.(v) A crisp set A is a Superset of B if and only if B is a subset of A.

(vi) A crisp set A is a proper subset of crisp set B if there is at least one element of Bwhich is not in A, i.e., A Í B but A ¹ B.

(vii) A crisp set A is said to be the subset of crisp set B if the membership function of Ais less than or equal to the membership function of B, i.e.

A Í B Û μA(x) ≤ μB(x)

Equal SetsTwo crisp sets A and B are said to be equal if they have the exactly same elements, Symbolically,

A = B iff A Í B and B Í AFor instance,(i) If A = {1, 2, 3, 4} and B = {x : x Î and x < 5}, then A = B.

(ii) If A = {x : x − 5 = 0} and B = {x : x is a positive integral root of x2 − 2x − 15 = 0}, thenA = B = {5}.

Power SetThe collection of all possible subsets of a finite crisp set A, is called the power set of A and it is

denoted by P(A). For instance, if A = {a, b, c}, then power set of A isP(A) = {f, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}}

Note. If A has n elements, then power set P(A) has 2n elements, i.e., cardinality of power set of A is|P(A)| = 2|A|

Countable and Uncountable SetsA crisp set A is called countable if either it is finite (finite number of elements) or it is countably

infinite, i.e., there exist one to one mapping from the set of natural numbers onto the crisp set A,f : → A

otherwise crisp set A is said to be uncountable.For instance, the set of natural numbers, the set of integers, the set of all ordered pair of integers,

the set of rational numbers are countable sets while the set of irrational numbers, the set of realnumbers, the closed interval [0, 1], the set of all points lying on a circle are uncountable set.

Convex SetA crisp set A defined on universal set X is said to be a convex set if x Î A and y Î A then lx

+ (1 − l)y Î A for any l Î [0, 1]. In other words, a set A on X is said to be convex, iff, for any pair x, y ÎA, all points located on the straight line segment connecting x and y are also in A. In figure 1.2 and 1.3,crisp set A1 and A2 are convex sets while crisp sets B1 and B2 are not convex set in 2.

14 FUZZY SET THEORY

Figure 1.2: Convex Set A1 and A2

Figure 1.3: Non Convex Set of B1 and B2

1.5 Operations on Crisp SetsThe result of the operation performed on the classical or crisp sets will be definite. The

operations that can be performed on the classical sets are described as follows:

UnionThe union of two crisp sets A and B is the set of all the elements in the universe that reside in (or

belongs to) either the set A or the set B or both sets A and B. It is denoted by A È B. In set theoreticform it is represented as

A È B = {x : x Î A or x Î B}or the membership function of A È B is defined as follows:

μAÈB(x) = max[μA(x), μB(x)], " x Î XIn general, the union of n fuzzy sets A1, A2, A3, ..., An where n ≥ 1 is

},3,2,1:{1

nisomeforAxxA ii

n

i ???

?

and the membership function of the union of n fuzzy sets A1, A2, A3, ..., An where n ≥ 1 isXxxxxmaxx

nn AAAAAA ?????? )],(,),(),([)(2121

????

For instance, if A = {1, 3, 5, 7, 9} and B = {1, 2, 4, 8, 9}, then A È B = {1, 2, 3, 4, 5, 7, 8, 9}.The Venn diagram of A È B is shown in figure 1.4

A1

x

y

o

A2

x

y

o

y

x yx

yx

B1

x

y

o

B2

x

y

o

y

x

CRISP SETS 15

Figure 1.4: Union of Two Crisp Sets

IntersectionThe intersection of two crisp sets A and B is the set of all those elements in the universe that

simultaneously reside in (or belongs to) both sets A and B. It is denoted by A Ç B. In set theoretic formit is represented as:

A Ç B = {x : x Î A and x Î B}or the membership function of A Ç B is defined as follows:

μAÇB(x) = min[μA(x), μB(x)], " x Î XIn general, the intersection of n fuzzy sets A1, A2, A3, ..., An where n ≥ 1 is

},3,2,1:{1

niAxxA ii

n

i ????

?

and the membership function of the intersection of n fuzzy sets A1, A2, A3, ..., An where n ≥ 1 isXxxxx

nn AAAAAA ?????? )],(,),(),(min[2121

????

For instance , if A = {1, 3, 5, 7, 9} and B = {1, 2, 4, 8, 9}, then A Ç B = {1, 9}.The Venn diagram of A Ç B is shown in figure 1.5

Figure 1.5: Intersection of Two Crisp Sets

Note. Two crisp sets A and B are said to be disjoint if they have no elements in common, i.e., A Ç B = f.

Complement

x

y

oA È B

A

B

y

o

A Ç BA

B

x

16 FUZZY SET THEORY

The complement of crisp set A is the set of all those elements in the universe X that do not residein the crisp set A. It is denoted by A . In set theoretic form it is represented as

A = {x : x Î X and x Ï A}

or the membership function of A is defined as

A? (x) = 1 − μA(x), " x Î X.

For instance , if A = {1, 3, 5, 7, 9} defined on universal set X = {1, 2, 3, ..., 10}, then

A = {2, 4, 6, 8, 10}.

The Venn diagram of A is shown in figure 1.6

Figure 1.6: Complement of Crisp Set A

DifferenceThe difference of crisp set A with respect to crisp set B is the set of all those elements in the

universe that reside in crisp set A but simultaneously do not reside in crisp set B. It is denoted by A − B.In set theoretic form it is represented as

A − B = {x : x Î A and x Ï B}or the membership function of A − B is defined as follows:

μA − B(x) = min[μA(x), B? (x)], " x Î X

For instance, If A = {1, 3, 5, 7, 9} and B = {3, 7} then A − B = {1, 5, 9}.The Venn diagram of A − B is shown in figure 1.7

Figure 1.7: Difference of Two Crisp Sets

Note. The difference of crisp set B with respect to crisp set A is

x

BA

A – B

o

y

o x

A

Xy

A

CRISP SETS 17

B − A = {x : x Î B and x Ï A}or the membership function of B − A is

μB −A(x) = min[μB(x), A? (x)], " x Î X

Symmetric DifferenceThe symmetric difference of crisp sets A and B is the set of all those elements in the universe that

reside in crisp set A È B but simultaneously do not reside in crisp set A Ç B. It is denoted by A D B.Symbolically, it is represented as

A D B = |A − B| = (A − B) È (B − A) = {x : (x Î A and x Ï B) or (x Î B and x Ï A)}or the membership function of A D B is defined as follows:

μADB(x) = max(min[μA(x), B? (x)], min[μB(x), A? (x)]), " x Î X

For instance, If A = {1, 3, 5, 7, 9} and B = {2, 3, 7, 8} then A − B = {1, 5, 9} and B − A = {2, 8},therefore,

A D B = (A − B) È (B − A) = {1, 5, 9} È {2, 8} = {1, 2, 5, 8, 9}The Venn diagram of A D B is shown in figure 1.8

Figure 1.8: Symmetric Difference of Two Crisp Set

Note. The symmetric difference of crisp set A with respect to A is A D A = f.

1.6 Properties of Crisp SetsThe following properties hold for crisp sets A, B and C defined on universal set X and having null

set f:(i) Commutativity

A È B = B È AA Ç B = B Ç A

(ii) AssociativityA È (B È C) = (A È B) È CA Ç (B Ç C) = (AÇ B) Ç C

(iii) DistributivityA È (B Ç C) = (A È B) Ç (A È C)

x

BA

A D

o

y

18 FUZZY SET THEORY

A Ç (B È C) = (A Ç B) È (A Ç C)(iv) Idempotency

A È A = AA Ç A = A

(v) Involution

AA ??

(vi) IdentityA È f = AA Ç X = A

(vii) Absorption by XA È X = X

(viii) Absorption by fA Ç f = f

(ix) AbsorptionA È (A Ç B) = AA Ç (A È B) = A

(x) De Morgan’s Law

BABA ???BABA ???

(xi) TranstivityIf A Ì B Ì C then A Ì C

(xii) Law of Contradiction

A Ç A = f(xiii) Law of Excluded Middle

A È A = X

1.7 ExamplesExample 1.7.1. Let a crisp set be A = {a, b, c, d}. Find the cardinality of crisp set A and its power setP(A).Solution. Let A = {a, b, c, d}. Cardinality of crisp set A is

|A| = 4and

P(A) = {f, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d},{a, b, c}, {b, c, d}, {a, c, d}, {a, b, d}, {a, b, c, d}}

Cardinality of power set of A isP(A) = 2|A| = 24 = 16

Example 1.7.2. Find the corresponding characteristic function for the following crisp sets:(i) A, the set of an odd integers.

CRISP SETS 19

(ii) B, the set of natural numbers which are multiple of 3.Solution. Crisp set A is defined by its membership function as follows:

μA : X → {0, 1}such that

???

??

?AxAx

xA if0 if1

)(?

(i) If A is the set of odd integers then the membership function μA assigns 1 to all odd integersand 0 to all even integers. Thus,

????

integereven an is if0integer oddan is if1

)(xx

xA?

that is,μA(x) ≡ x mod 2

(ii) If B is the the set of natural numbers which are multiple of 3, i.e.,B = {3, 6, 9, 12, ...}

then the membership function μB assigns 1 to all those elements which belongs to the set {3,6, 9, 12, ...} and 0 to all those elements which belong to the set {1, 2, 4, 5, 7, 8, 10, 11, ...}.Thus,

???

?????

?xmodxmodx

xmodxxB and)32or 31( if0

and30 if1)(?

Example 1.7.3. If A = {1, 3, 5, 7, 9} defined on universal set X = {1, 2, 3, ..., 10}, then find themembership value of all the elements of the universal set corresponding to crisp set A .Solution. If A = {1, 3, 5, 7, 9} defined on universal set X = {1, 2, 3, ..., 10}, then the membershipfunction of A is defined as

A? (x) = 1 − μA(x), " x Î X.

Thus,

A? (1) = 1 − μA(1) = 1 − 1 = 0

A? (2) = 1 − μA(2) = 1 − 0 = 1

A? (3) = 1 − μA(3) = 1 − 1 = 0

A? (4) = 1 − μA(4) = 1 − 0 = 1

A? (5) = 1 − μA(5) = 1 − 1 = 0

A? (6) = 1 − μA(6) = 1 − 0 = 1

A? (7) = 1 − μA(7) = 1 − 1 = 0

A? (8) = 1 − μA(8) = 1 − 0 = 1

20 FUZZY SET THEORY

A? (9) = 1 − μA(9) = 1 − 1 = 0

A? (10) = 1 − μA(10) = 1 − 0 = 1

Thus, A has the set of those elements whose membership value is 1,

A = {2, 4, 6, 8, 10}Example 1.7.4. If A = {3, 4, 5, 6, 7}, B = {2, 5, 7} and C = {2, 4, 5} are the crisp sets defined on X ={1, 2, 3, ..., 10}, then find the membership value of 3, 4, 5 and 7 corresponding to crisp set A Ç B Ç C.Solution. The membership function of A Ç B Ç C is defined as follows:

μAÇBÇC(x) = min[μA(x), μB(x), μC(x)]Therefore,

μA∩B∩C(3) = min[μA(3), μB(3), μC(3)] = min[1, 0, 0] = 0μA∩B∩C(4) = min[μA(4), μB(4), μC(4)] = min[1, 0, 1] = 0μA∩B∩C(5) = min[μA(5), μB(5), μC(5)] = min[1, 1, 1] = 1μA∩B∩C(7) = min[μA(7), μB(7), μC(7)] = min[1, 1, 0] = 0

Example 1.7.5. If A = {3, 4, 5, 6, 7}, B = {2, 5, 7} and C = {2, 4, 5} are the crisp sets defined onX = {1, 2, 3, ..., 10}, then find the membership value of 3,4,5,7 and 9 corresponding to crisp setA È B È C.Solution. The membership function of A È B È C is defined as follows:

μAÈBÈC(x) = max[μA(x), μB(x), μC(x)]Therefore,

μAÈBÈC(3) = max[μA(3), μB(3), μC(3)] = max[1, 0, 0] = 1μAÈBÈC(4) = max[μA(4), μB(4), μC(4)] = max[1, 0, 1] = 1μAÈBÈC(5) = max[μA(5), μB(5), μC(5)] = max[1, 1, 1] = 1μAÈBÈC(7) = max[μA(7), μB(7), μC(7)] = max[1, 1, 0] = 1μAÈBÈC(9) = max[μA(9), μB(9), μC(9)] = max[0, 0, 0] = 0

Example 1.7.6. Check whether the crisp set A = [0, 2] È [3, 5] is convex or not.Solution. If a crisp set is convex, then for any pair x, y Î A implies

λx + (1 − λ)y Î Afor any λ Î [0, 1].

Let x = 1 and y = 4 and λ = 0.4, thenλx + (1 − λ)y = 0.4(1) + (1 − 0.4)4 = 2.8 Ï A

Thus crisp set A is not a convex set.Example 1.7.7. Show that law of contradiction and law of excluded middle holds for the crisp setA = {3, 4, 5, 6, 7} defined on universal set X = {1, 2, 3, ..., 8}.

Solution. As A = {3, 4, 5, 6, 7}, then A = {1, 2, 8}Consequently,

CRISP SETS 21

A Ç A = {3, 4, 5, 6, 7} Ç {1, 2, 8} = fThus Law of Contradiction is satisfied.Similarly

A È A = {3, 4, 5, 6, 7} È {1, 2, 8} = {1, 2, 3, 4, 5, 6, 7, 8} = XThus Law of Excluded Middle is satisfied.

Example 1.7.8. Show that De Morgan’s law hold for crisp sets A = {3, 4, 6, 7, 9} and B = {1, 2, 4, 7, 9}defined on universal set X = {1, 2, 3, ..., 10}Solution. De Morgan’s Law is

BABA ???

BABA ???If A = {3, 4, 6, 7, 9} and B = {1, 2, 4, 7, 9}, then

A = {1, 2, 5, 8, 10}

B = {3, 5, 6, 8, 10}

A Ç B = {5, 8, 10}

A È B = {1, 2, 3, 5, 6, 8, 10}A È B = {1, 2, 3, 4, 6, 7, 9}A Ç B = {4, 7, 9}Thus

BABA ???? }10,8,5{

BABA ???? }10,8,6,5,3,2,1{

1.8 Exercises1. Show that Associative law and Distributive law hold for crisp sets A = {5, 6, 8, 9, 10},

B = {1, 2, 3, 7, 9} and C = {1, 6} defined on the universal set of natural numbers.2. Show that Law of contradiction and Law of excluded middle holds for the crisp set A = {a, c,

d, g, h} defined on universal set X = {a, b, c, d, e, f, g, h, i}.3. If A = {10, 30, 40, 50, 70, 80}, B = {20, 50, 80, 100} and C = {50, 80, 90} are the crisp sets

defined on X = {10, 20, 30, ..., 100}, then find the membership value corresponding to crispset A È B È C and A Ç B Ç C for all elements of universal set.

4. If A = {25, 45, 85, 95} and B = {15, 35, 85} are crisp sets defined on universal set X = {15,25, 35, ..., 95}, then find A , A È B and A Ç B by characteristic function method. Also findtheir cardinality.

5. If the crisp sets A, B and C are defined on universal set X = {0, 1, 2, 3, ..., 9}, whereA = {2, 3, 5, 6, 7}

22 FUZZY SET THEORY

B = {1, 2, 4, 7, 9}and

C = {0, 3, 4, 8}Show that these laws hold for crisp sets A, B and C: Commutativity, Associativity,Distributive, Idempotency, Involution, Identity, Absorption, Absorption by Universal Set,Absorption by Empty Set, De Morgan’s Law, Transtivity, Law of Contradiction and Law ofExcluded Middle.

6. Check whether the crisp set A = [0, 1] È [2, 3] is convex or not.7. Let a crisp set be A = {a, b, c, d, e, f }. Find the cardinality of crisp set A and its power set

P(A).8. Find the corresponding characteristic function for the following crisp sets:

(i) A, the set of an even integers.(ii) B, the set of integers which are multiple of 7.

(iii) C, the set of rational numbers.(iv) D = {x : x is a positive integral root of x2 − 2x − 15 = 0}.

(v) E = {x : x ∈ and x2 = 2}.

9. Show that for any two crisp sets A and B;A − (A Ç B) = A − B

10. Show that for any two crisp sets A and B;(i) A − B = B − A iff A = B

(ii) A − B = A iff A Ç B = f(iii) (A − B) Ç (B − A) = f(iv) A = (A Ç B) È (A − B)(v) (A − B) È (B − A) = (A È B) − (A Ç B)

11. Show that for any two crisp sets A and B;(i) A È B = A iff B Í A

(ii) A Ç B = A iff A Í B12. Show that for any three crisp sets A, B and C;

(i) (A Ç B) − C = (A − C) Ç (B − C)(ii) A Ç (B − C) = (A Ç B) − (A Ç C)

(iii) (A − B) − C = A − (B È C)

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