Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory

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fuzziﬁed theories of uncertainty examined in Chapter 8, which are, by and

large, based on standard fuzzy sets.

7.2. BASIC CONCEPTS OF STANDARD FUZZY SETS

Since this section (and the subsequent Sections 7.3–7.7) deal only with stan-

dard fuzzy sets, the adjective “standard” is omitted for the sake of linguistic

simplicity.

As in classical set theory, the concept of subsethood is one of the most fun-

damental concepts in fuzzy set theory. Contrary to classical set theory,

however, this concept has two distinct meanings in fuzzy set theory. One of

them is a crisp subsethood, the other one is a fuzzy subsethood. These two con-

cepts of subsethood are introduced as follows.

Given two fuzzy sets A, B deﬁned on the same universal set X, A is said to

be a subset of B if and only if

for all x Œ X. The usual notation, A 債 B, is used to signify the subsethood rela-

tion. According to this deﬁnition, clearly, set A is either a subset of set B or it

is not a subset of B (and, similarly, the other way around). Therefore, this def-

inition of subsethood is called a crisp subsethood. The set of all fuzzy subsets

of X (deﬁned in this way) is called the fuzzy power set of X and is denoted by

F(X). Observe that this set is crisp, even though its members are fuzzy sets.

Moreover, this set is always inﬁnite, even if X is ﬁnite.

When X is a ﬁnite universal set, a fuzzy subsethood relation, Sub, is deﬁned

by specifying for each pair of fuzzy sets on X, A and B, a degree of subset-

hood, Sub(A, B), by the formula

Ax Bx

()

262 7. FUZZY SET THEORY

Non-boolean

algebra of

some type

Fuzziﬁed

uncertainty

theory

Monotone

measure of

some type

Figure 7.1. Components of a fuzziﬁed uncertainty theory.

(7.1)

The negative term in the numerator describes the sum of the degrees to which

the subset inequality A(x) £ B(x) is violated, the positive term describes the

largest possible violation of the inequality, the difference in the numerator

describes the sum of the degrees to which the inequality is not violated, and

the term in the denominator is a normalizing factor to obtain the range

When sets A and B are deﬁned on a bounded subset of real numbers (i.e., X

is a closed interval of real numbers), the three S terms in Eq. (7.1) are replaced

with integrals over X.

For any fuzzy set A deﬁned on a ﬁnite universal set X, its scalar cardinal-

ity, |A|, is deﬁned by the formula

(7.2)

Scalar cardinality is sometimes referred to in the literature as a sigma count.

Among the most important concepts of standard fuzzy sets are the concepts

of an a-cut and a strong a-cut. Given a fuzzy set A deﬁned on X and a par-

ticular number a in the unit interval [0, 1], the a-cut of A, denoted by

A, is a

crisp set that consists of all elements of X whose membership degrees in A

are greater than or equal to a. This can formally be written as

The strong a-cut,

A, has a similar meaning, but the condition “greater than

or equal to” is replaced with the stronger condition “greater than.” Formally,

The set

A is called the support of A and the set

A is called the core of A.

When the core A is not empty, A is called normal; otherwise, it is called sub-

normal. The largest value of A is called the height of A and it is denoted by

. The set of distinct values A(x) for all x ŒX is called the level set of A and

it is denoted by L

All the introduced concepts are illustrated in Figure 7.2. We can see that

when a

≥ a

. This implies that the set of all distinct a-cuts (as well as strong

a-cuts) is always a nested family of crisp sets. When a is increased, the new a-

aa a

AA A A

ÕÕ

and

()

{}

AxAx .

aAxAx=

()

≥

{}

AAx

()

01£

()

£Sub A B,.

Sub A B

Ax Ax Bx

xX xX

max ,

()

(){}

()

ŒŒ

ÂÂ

7.2. BASIC CONCEPTS OF STANDARD FUZZY SETS 263

cut (strong a-cut) is always a subset of the previous one. Clearly,

A = X and

A = ⭋.

It is well established that each fuzzy set is uniquely represented by the asso-

ciated family of its a-cuts via the formula

(7.3)

or by the associated family of its strong a-cuts via the formula

(7.4)

where sup denotes the supremum of the respective set and

A (or

a +

A) denotes

for each a Œ [0, 1] the special membership function (characteristic function)

of the a-cut (or strong a-cut, respectively).

EXAMPLE 7.1. To illustrate the meaning of Eq. (7.3) consider a fuzzy set A

deﬁned on ⺢ by the simple stepwise membership function shown in Figure

7.3a. Since L(A) = {0, 0.3, 0, 0.6, 1} and 0·

A(x) = 0 for all x Œ ⺢, A is fully

represented by the three special fuzzy sets a·

A(a = 0.3, 0.6, 1), whose mem-

bership functions are shown in Figure 7.3b. The membership function A is

uniquely reconstructed from these three membership functions by taking for

each x Œ ⺢ their supremum. The same can clearly be accomplished by the

strong a-cuts and Eq. (7.4).

Ax Ax

()

=◊

()

[]

{}

sup , ,aa

Ax Ax

()

=◊

()

[]

{}

sup , ,aa

264 7. FUZZY SET THEORY

Support

Core

(

)

(

is normal)

Figure 7.2. Illustration of some basic characteristics of fuzzy sets.

EXAMPLE 7.2. An alternative way to illustrate the meaning of Eq. (7.3) is

to use a fuzzy set deﬁned on a ﬁnite universal set. Let

This fuzzy set is fully represented by the following four special fuzzy sets:

Ax x x x x x=+++++03 05 10 10 08 05

123456

.... . ..

7.2. BASIC CONCEPTS OF STANDARD FUZZY SETS 265

A(x) 0.3

0.6

0.3

0.6

0.3 ◊

0.3

A(x)

0.6 ◊

0.6

A(x)

1 ◊

A(x)

(a)

(b)

Figure 7.3. Illustration of the a-cut representation of Eq. (7.3): (a) given fuzzy set A; (b) decom-

position of A into special fuzzy sets.

Now taking for each x

the maximum value in these sets, we readily obtain the

original fuzzy set A.

The signiﬁcance of the a-cut (or strong a-cut) representation of fuzzy sets

is that it connects fuzzy sets with crisp sets. While each crisp set is a collection

of objects that are conceived as a whole, each fuzzy set is a collection of nested

crisp sets that are also conceived as a whole. Fuzzy sets are thus wholes of a

higher category.

The a-cut representation of fuzzy sets allows us to extend the various prop-

erties of crisp sets, established in classical set theory, into their fuzzy counter-

parts. This is accomplished by requiring that the classical property be satisﬁed

by all a-cuts of the fuzzy set concerned. Any property that is extended in this

way from classical set theory into the domain of fuzzy set theory is called a

cutworthy property. For example, when convexity of fuzzy sets is deﬁned by

the requirement that all a-cuts of a fuzzy convex set be convex sets in the clas-

sical sense, this conception of fuzzy convexity is cutworthy. Other important

examples are the concepts of a fuzzy partition, fuzzy equivalence, fuzzy com-

patibility, and various kinds of fuzzy orderings that are cutworthy.

It is important to realize that many (perhaps most) properties of fuzzy sets,

perfectly meaningful and useful, are not cutworthy. These properties cannot

be derived from classical set theory via the a-cut representation.

7.3. OPERATIONS ON STANDARD FUZZY SETS

As is well known, each of the three basic operations on sets—complementa-

tion, intersection, and union—is unique in classical set theory. However, their

counterparts in fuzzy set theory are not unique. Each of them consists of a

class of functions that satisfy certain requirements. In this section, only some

main features of these classes of functions are described.

7.3.1. Complementation Operations

Complementation operations on fuzzy sets are functions of the form

that are order reversing and such that c(0) = 1 and c(1) = 0. Moreover, they

are usually required to possess the property

c :, ,01 01

[]

03 03 03 03 03 03 03

05 00 05 05 05 05 05

08 00 00 08 0

123456

123

. ......

.....

◊=+++++

◊= + + +

Axxxxxx

Ax x x

880800

1 000010100000

456

12345 6

xxx

Ax x x x x x

◊=+++++

.......

266 7. FUZZY SET THEORY

(7.5)

for all a Œ [0, 1]; this property is called an involution. Given a fuzzy set A and

a particular complementation function c, the complement of A with respect to

A, is deﬁned for all x ŒX by the formula

(7.6)

An example of a practical class of involutive complementation functions,

, is deﬁned for each a Œ [0, 1] by the formula

(7.7)

where l is a parameter whose values specify individual complements in

this class. When l = 1, the resulting complement is usually called a standard

complement.

Which of the possible complements to choose is basically an experimental

question. The choice is determined in the context of each particular applica-

tion by eliciting the meaning of negating a given concept. This can be done

by employing a suitable parametrized class of complementation functions,

such as the one deﬁned by Eq. (7.7), and determining a proper value of the

parameter.

7.3.2. Intersection and Union Operations

Intersection and union operations on fuzzy sets are deﬁned in terms of func-

tions i and u, respectively, which assign to each pair of numbers in the unit

interval [0, 1] a single number in [0, 1]. These functions, i and u, are commu-

tative, associative, monotone nondecreasing, and consistent with the charac-

teristic functions of classical intersections and unions, respectively. They are

often referred to in the literature as triangular norms (or t-norms) and trian-

gular conorms (or t-conorms). It is well know that the inequalities

(7.8)

(7.9)

are satisﬁed for all a, b Œ [0, 1], where

(7.10)

(7.11)

uab

ab ab

max

max , min ,

()

() ()

when

otherwise.

iab

ab ab

min

min , max ,

()

() ()

when

otherwise,

max , , ,

max

ab uab u ab

()

i ab iab ab

min

, , min , ,

()

ca a

()

>10

Ax c Ax

()

()()

cca a

()()

7.3. OPERATIONS ON STANDARD FUZZY SETS 267

Operations expressed by min and max are usually called standard opera-

tions; those expressed by i

min

and u

max

are called drastic operations.

Given fuzzy sets A and B, deﬁned on the same universal set X, their inter-

sections and unions with respect to i and u, A «

B and A »

B, are deﬁned

for each x Œ X by the formulas

(7.12)

(7.13)

It is useful to capture the full range of intersections and unions, as expressed

by the inequalities (7.8) and (7.9), by suitable classes of functions. For example,

all functions in the classes

(7.14)

(7.15)

where l > 0, qualify as intersection and unions, respectively, and cover the

whole range by varying the parameter l.The standard operations are obtained

in the limit for l Æ•, while the drastic operations are obtained in the limit

for l Æ 0. In each particular application, a ﬁtting operation is determined by

selecting appropriate values of the parameter l by knowledge acquisition

techniques.

7.3.3. Combinations of Basic Operations

In classical set theory, the operations of complementation, intersection, and

union possess the properties listed in Table 1.1. For each universal set X, these

are properties of the Boolean algebra deﬁned on the power set of X.The oper-

ations of intersection and union are dual with respect to the operation of com-

plementation in the sense that they satisfy for any pair of subsets of X, A, and

B, the De Morgan laws

It is desirable that this duality be satisﬁed for fuzzy sets as well. It is obvious

that only some combinations of t-norms, t-conorms, and complementation

operations on fuzzy sets can satisfy this duality. We say that a t-norm i and a

t-conorm u are dual with respect to a complementation operation c if and only

if for all a, b Œ [0, 1]

(7.16)

cia b uca cb,,

()

[]

() ()

[]

ABAB ABAB«=» »=«and .

uab a b

,min, ,

()

{}

iab a b

,min, ,

()

=- -

()

[]

{}

111 1

ABxuAxBx

()()

() ()

[]

ABxiAxBx

()()

() ()

[]

268 7. FUZZY SET THEORY

and

(7.17)

These equations describe the De Morgan laws for fuzzy sets. Triples ·i, u, cÒ

that satisfy these equations are called De Morgan triples.

None of the De Morgan triples satisﬁes all properties of the Boolean

algebra on P(X). Depending on the operations used, various weaker algebraic

structures are obtained. In some of them, for example, the laws of contradic-

tion and excluded middle are violated. In others, these laws are preserved, but

the laws of distributivity are violated.

7.3.4. Other Operations

Two additional types of operations are applicable to fuzzy sets, but they have

no counterparts in classical set theory. They are called modiﬁers and averag-

ing operations.

Modiﬁers are unary operations that are order preserving. Their purpose is

to modify fuzzy sets that represent linguistic terms to account for linguistic

hedges such as very, fairly, extremely, more or less, and the like. The most

common modiﬁers either increase or decrease all values of a given member-

ship function. A convenient class of functions, m

, that qualify as increasing or

decreasing modiﬁers is deﬁned for each a Œ [0, 1] by the simple formula

(7.18)

where l > 0 is a parameter whose value determines which way and how

strongly m

modiﬁes a given membership function. Clearly, m

(a) > a when

l Œ (0, 1), m

(a) < a when l Œ (1, •), and m

(a) = a when l = 1. The farther

the value of l from 1, the stronger the modiﬁer m

Averaging operations are monotone nondecreasing and idempotent, but are

not associative. Due to the lack of associativity, they must be deﬁned as func-

tions of n arguments for any n ≥ 2. It is well known that any averaging oper-

ation, h, satisﬁes the inequalities

(7.19)

for any n-tuple (a

, a

,...,a

) Œ [0, 1]

. This means that the averaging opera-

tions ﬁll the gap between intersections (t-norms) and unions (t-conorms).

One class of averaging operations, h

, which covers the entire interval

between min and max operations, is deﬁned for each n-tuple (a

, a

,...,a

) in

[0, 1]

by the formula

(7.20)

haa a

aa a

ll l

, ,...,

...

()

+++

min , ,..., , ,..., max , ,...,aa a haa a aa a

nn n12 12 12

()

ma a

()

= ,

cuab ica cb,,.

()

[]

() ()

[]

7.3. OPERATIONS ON STANDARD FUZZY SETS 269

where l is a parameter whose range is the set of all real numbers except 0.

For l = 0, function h

is deﬁned by the limit

(7.21)

which is the well-known geometric mean. Moreover

(7.22)

(7.23)

This indicates that the standard operations of intersection and union also may

be viewed as extreme opposites in the range of averaging operations.

Other classes of averaging operations are now available, some of which use

weighting factors to express the relative importance of the individual fuzzy

sets involved. For example, the function

where the weighting factors w

usually take values in the unit interval [0, 1]

and

expresses for each choice of values w

the corresponding weighted average of

values a

(i = 1,2,...,n). Again, the choice is an experimental issue.

7.4. FUZZY NUMBERS AND INTERVALS

Fuzzy sets that are deﬁned on the set of real numbers, ⺢, have a special sig-

niﬁcance in fuzzy set theory. Among these sets, the most important are cut-

worthy fuzzy intervals. They are deﬁned by requiring that each a-cut be a

single closed and bounded interval of real numbers for all a Œ (0, 1]. A fuzzy

interval, A, may conveniently be represented for each x Œ ⺢ by the canonical

form

(7.24)

fx x ab

xbc

gx x cd

()

[

)

[

)

()

(

]

when

0 otherwise,

ha w i n wa

ii ii

, , ,..., ,=

()

lim , ,..., max , ,..., .

Æ-•

()

haa a aa a

nn12 12

lim , ,..., min , ,..., ,

Æ-•

()

haa a aa a

nn12 12

lim , ,..., , ,..., ,

()

12 12

haa a aa a

270 7. FUZZY SET THEORY

where a, b, c, d are speciﬁc real numbers such that a £ b £ c £ d, f

is a real-

valued function that is increasing, and g

is a real-valued function that is

decreasing. In most applications, functions f

and g

are continuous, but, in

general, they may be only semicontinuous from the right and left, respectively.

When A(x) = 1 for exactly one x Œ ⺢ (i.e., b = c in the canonical representa-

tion), A is called a fuzzy number.

Some common shapes of membership functions of fuzzy numbers or inter-

vals are shown in Figure 7.4. Each of them represents, in a particular way, a

fuzzy set of numbers described in natural language as “close to 1” (or “around

1”). Whether a particular membership function is appropriate for represent-

ing this linguistic description can be determined only in the context of each

given application of the linguistic expression. Usually, however, the member-

ship function that is supposed to represent a given linguistic expression in the

context of a given application is constructed from the way in which the lin-

guistic expression is interpreted in this application. The issue of constructing

membership functions is addressed in Section 7.9.

In practical applications, the most common shapes of membership functions

of fuzzy intervals are the trapezoidal ones, illustrated by the membership func-

tion in Figure 7.2 and also function B in Figure 7.4.They are easy to represent

and manipulate. Each trapezoidal-shaped membership function is uniquely

deﬁned by the four real numbers a, b, c, d, in Eq. (7.24). Deﬁning a trapezoidal

fuzzy interval A by the quadruple

means that

in Eq. (7.24). Clearly, triangular-shaped fuzzy numbers are special cases of the

trapezoidal-shaped fuzzy intervals in which b = c.

For any fuzzy interval A expressed in the canonical form, the a-cuts of A

are expressed for all a Œ(0, 1] by the formula

(7.25)

where f

-1

and g

-1

are the inverse functions of f

and g

, respectively. When a

membership function of fuzzy intervals has a trapezoidal shape, such as the

function A in Figure 7.2, the a-cuts can readily be expressed in terms of the

four real numbers a, b, c, d by the formula

(7.26)

aaAabaddc=+-

()

[]

aa a

() ()

[]

()

[]

--11

,,,

when

()

and

Aabcd= ,,,

7.4. FUZZY NUMBERS AND INTERVALS 271