PART 1From classical sets to fuzzy sets

1. Introduction2. Crisp sets: an overview3. Fuzzy sets: basic types4. Fuzzy sets: basic concepts

FUZZY SETS AND

FUZZY LOGICTheory and Applications

Introduction

• Crisp set:

to dichotomize the individuals in some given universe of discourse into two groups: members and nonmembers. A sharp, unambiguous distinction exists between the members and nonmembers of the set.

Introduction

• Fuzzy set:

to assign each individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar to or compatible with the concept represented by the fuzzy set. We perceive fuzzy sets as having imprecise boundaries that facilitate gradual transition from membership to nonmembership and vice versa.

Crisp sets: an overview

• Universal set: The universe of discourse, containing all the possible elements of concern in each particular context from which sets can be formed, X

• Empty set: The set containing no members,

• Member (element):

Ax

Ax

,

Crisp sets: an overview

• Family of sets:

• Three methods of defining sets:

list: A = {x,y,z}

rule: A = {x|P(x)}

characteristic function:

IiAi

Axfor

AxforxA

XA

0

1)(

}1,0{:

Crisp sets: an overview

• Subset:

• Proper subset:

• Equal sets:

• Power set: P• Cardinality: The number of members of a

finite set, |A|

• Relative complement:

ABandBABA AxxA )(

AxandBxxAB

BA

BA

Crisp sets: an overview

• Complement:

X is the universal set

• Union:

• Intersection:

• Fundamental properties

BA

IiAxxAIi

ii

},{

BA

IiAxxAIi

ii

},{

AXA

Crisp sets: an overview

Crisp sets: an overview

Crisp sets: an overview

• Partition:

• Nested family:

• Cartesian product:

• Relations: Subsets of Cartesian products

Ii

i

ji

ii

AA

jiforAA

IiAAAA

.

,

},,{)(

1

21 },,,,{

ii

n

AA

AAAA

},,,,{

},,{

211

niinini

NiAaaaaA

BbAabaBA

Crisp sets: an overview

• Convexity:

• Upper bound, lower bound:

Asr

Asr

)1(

,,

Ayly

Axux

,

,

Crisp sets: an overview

• Supremum, infimum:

sup A: upper bound of A

no smaller is an upper bound.

inf A: lower bound of A

no greater is an lower bound.

Crisp sets: an overview

Fuzzy sets: basic types

• Membership function:

• Fuzzy set

]1,0[: XA

Fuzzy sets: basic types

Fuzzy sets: basic types

Fuzzy sets: basic types

Fuzzy sets: basic types

Fuzzy sets: basic types

• Fuzzy variables: Several fuzzy sets representing linguistic concepts, such as low, medium, high, are often employed to define states of a fuzzy variable.

• States of fuzzy variable: Temperature within a range [a,b] is characterized as a fuzzy variable, with states being fuzzy sets very low, low, medium, etc.

Fuzzy sets: basic types

Fuzzy sets: basic types

• Interval-valued fuzzy sets

Fuzzy sets: basic types

• Type 2 fuzzy sets

Fuzzy sets: basic types

• Fuzzy power set: F(A), the set of all fuzzy sets that can be defined within A

• Type 2: A:X→F([0, 1])• L fuzzy sets: A:X →L

where L is a set of symbols that are partially ordered.

• Level 2 fuzzy sets: A: F(X) →[0, 1]

Fuzzy sets: basic concepts

• Concepts of young, middle-aged, and old

Fuzzy sets: basic concepts

• Discrete approximation

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

• α-cut : })({ xAxA

Fuzzy sets: basic concepts

• Strong α-cut: })({ xAxA

Fuzzy sets: basic concepts

• Level set:

• Support:

• Core:

},)({)( XxxAA

AA 0)sup(

AAcore 1)(

Fuzzy sets: basic concepts

• Height:

• Normal, subnormal:

• Convex fuzzy sets: is convex for all

α [0, 1].

)(sup)( xAAhXx

1)( Ah1)( Ah

A

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

• Standard fuzzy set operations

• Standard complement:

• Equilibrium points:

• Standard intersection:

• Standard union:

)(1 xAA

)}()(|{ xAxAx

)](),(min[))((

}|)(inf{))((

xBxAxBA

IixAxA iIi

i

)](),(max[))((

}|)(sup{))((

xBxAxBA

IixAxAIi

ii

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

Fuzzy sets: basic concepts

• Fundamental properties

• Laws of contradiction and excluded middle

• Fuzzy set inclusion:

• Scalar cardinality:

• Degree of subsethood:

XxxBxABA ),()(

Xx

xAA )(||

||

||),(

A

BABAS

))]()(,0max[|(|||

1

Xx

xBxAAA

Fuzzy sets: basic concepts

• Notation of fuzzy sets:

• Distance:

xx

xA

n

ix

a

xa

xa

xa

A

A

A

i

i

n

n

)(

1

2

2

1

1

Xx

xBxABAd |)()(|),(

Exercise 1

• 1.7

• 1.8

• 1.9

• 1.10

• 1.11