Fuzzy Sets and Fuzzy InformationGranulation Theory Key selected papers by Lotfi A. Zadeh
Edited by Da Ruan Chongfu Huang
Beijing Normal University Press
Beijing
FfTB~$H** CIP R @ % ? < 2 0 0 0 ) % 02100 $-
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The International Committee of the Series Etienne E. Kerre (Belgium)
Da Ruan (Belgium) Madan M . Gup ta ( Canada)
Liu Yingming (China) Wang Peizhuang (China) Huang Chongfu (China) Didier Dubois ( France) Hans J . Zimmermann (German y ) Yee Leung (HK, China) Hiroshi Inoue (Japan) Masahara Mizumoto (Japan) Zeungnam Bien ( Korea) A . J . van der Wal ( Netherlands) Janusz Kacprzyk ( Poland) Bart Kosko (USA) George Klir ( USA )
Paul P. Wang (USA) (Note: the order is according to the countries' names)
Preface
Advances in Fuzzy Mathematics and Engineering is a new
international series dedicated to the support and development of the theory of fuzzy mathematics and related areas and their industrial applications in general and in engineering in particular.
The series i s supported and published by Beijing Normal University Press, Beijing ,China.
This book
.
Fuzzy
Sets and Fuzzy In fr3rmation-€~runtc~utr'~m
Theory, is the third volume of Collected Papers by Lotfi A.
Zadeh. The first volume, entitled Fuzzy Sets and Applications, was published in 1987 by John Wiley. I t s editors, Ronald R. Yager, Sergei Ovchinnikav, Richard M. Tong, and Hung T. Ngugen undertook the project on the occasion of the 20th anniversary of t h e publication of the f i r s t paper on fuzzy set by Lotfi A + Zadeh. The second volume , entitled Fuzzy Sets, Fuzzy L o g i c , and Fuzzy Systems + was published in 1996 by World Scientific, Its editors, George J. Klir and h Yuan selected, from among all papers b y Lotfi Zadeh not included in the first volume. those papers on fuzzy sets,fuzzy logic,and fuzzy systems whose easy accessibility would likely be of benefit to those working in these areas. The previous two volumes in English have proved t o be great utility to anyone interested in fuzzy set theory and its applications*
Considering the largest number of the readers related to
Fuzzy Mathematics and Engineering in China, we felt that
a
Chinese version of the key selected papers by Lotfi Zadeh Fuzzy Sets and Fuzzy I I t J i ~ r n l u t i c ~ ~ ~ - U r u ? r r r l rTheory t t i ~ ~ ~ r would fit well
Fuzzy Mathematics and Engineering by Beijing Normal University Press. The book collects Zadeh's original perception which may be viewed as an evolution of ideas rooted in his 1965 paper on fuzzy sets; 1971 into the book series on Advances in
paper on fuzzy
systems;
1973
- 1976
z ~ u r i c r h € ~fs ,LmzY i j - t h ~r ~t ~k and /Lery
rvts
gerteru€ized
U
N
~i?lfLrt?1~tiu?t
r.{mstruirz.~;
The
ftlzz_),grdphs r
purpose of this
book
is
1979 paper on
gril~t~kurity;1986
1 9 96 pa per on
1997 papers on thuory r,J' fuzzy
papers on dirtgcristir paper
on
c . , ~ ~ ? ~ b r r a i rw~r'rA r# u w ~ d su rrd
irrfirmmutir~n gruwalutii~r.
twofold. Firstly-it is intended as
a quick reference for those working in F a z q M u r h ~ ~ ~ u run~- rsd Esginwring
in China
as
well a?; in
the w{,rld.
Secondly + it
is
expected to play a major role in-Research and Development of F u z z y M~rcthrmotks a~rd Engi~wuring,a s n useful source of supplementary readings in this new book series. volume
will benefit many readers arr~undt h e world.
n a Ruan ,Chongfu Huang
Editors
We
hope t h i s
Note to the Reader
?'his book is the third volume of collected papers by 1,otfi
A. Zadeh, The first volume ,entitled Ftr:ry Sets srnd i 4 ~ p l i c . ~ t i r n ~ s was published in 1987 by John W iley (Editors :Ronald R,Yslger , Sergei Ovchinnikov, Richard M , Tong, and Hung T, Ngugen 1. The second volume ,entitled Fuzzy Sets. Fuzzy 1,ogi~.,u?td FWZZY Sysrsnts, was published in 1996 by World Scientific ( Editors : George J. Klir and &, Yuan>, +
The current edited-book is a set of key selected papers by I-otfi Zadeh. Both English and Chinese versions of these papers are available,
Acknowledgements
This book consists of the following reprinted (both retyped and translated ) papers. The relevant copyright owners whose permissions to reproduce the papers in this book are gratefully acknowledged. A c d t u ? c P-:
L.A. Zadeh.HFuazy
sets. "It2fomatim and Corojrd .8f
3) ,pp+ 338- 353 t
1965.
L,A+Zadeh ,"A fumy -algorithmic a pprmch to the definition of complex or imprecise concepts + "Internat. J . Man-Machine Stud. 8 t pp. 24 9 -- 291 +
1976+
Ektrkr Seitlwt r
L.A* Zedeh +"Theconcept of
a
linguistic variable and its application to
approximate reasoning. l , I I , 111, w 3 d ~ a ~ t ' Scimc~s, oa 8 (3
t
pp. 199 -- 25 1 +
(4)301--35719,pp, 43-80,1975. L, A. Zadeh, *Fuzzy sets and intormation granularity, ')In r Gupta M.
M. ,R. K*Ragadc *and R. R. Yager *eds. + Advahc~.ci n
Fuzzy S P ~ Theory and
Applicatians. Nort h-Hr>llsd New York *pp. 3% 89 1979.
L, A. 2adeh "Toward a theory of fuzzy information granulation and i t s centrality in human reasoning and fuzzy logic*w Fuzzy SPJ.Tand Sy.~t~rns 90 I23,pp. 111-127m1997* Itrditute a€ EIectrZTaI and Eltdmnics Emginten:
L, A * Zadeh,"Outline of
a new approach to the analysis of complex
5
system ilnd decision
processes. " l E E E Y'rans. On Sysrpmr. Mlm. und
t,y?wrn~cfics.1 ( 1 1 ,yp. 28- - 46.1973.
I,, A . Zadth, "Fuzzy logic =computing
.
with
words. "IELE 'I'rdn.~.[In
F'uzzy .7-y.~~~rns 4 ,pp. 103- 11 1 1 996.
Oxford University P m s Inc.
:
I,. A . Zadeh ,*'Towardsa t henry of fuzzy systems. "1n:Kalmn ,R.E+and
N. ~)ecl&ri?;+cds. *A,~prcls of , V u i w r k s utrd
Systr,~r.+ Theory.
Hoit ,Rinehart 8.
Winston.New pp. 469- 490,1971. Springer Vorlag : I,. A+ Zadeh, "Outline of a computational approach t o meaning and
k n o w l c d ~ rrepresentation based on the concept oi a generalized assignment statement. "In : T h o m a ,
. .
M. .4nd W + Wyner
eds.
Prr~.+ Of t k ~I n t ~ r n ~ t i o n u l
Scmt'nar on A I and Mun-Mdchin~Sysma~,Springer-Verlag, Heidelberg, y y .
198-211 *1986.
Lotf i A. Zadeh's biography
A. Zadth joined the Department r>f Elecrrirai Enginerring a t the University of California + f i r k e l e y,in 1959 ,and served as its chairman (rum I,trtf~
1963 to 1968. Earlier, he was a member of the clertricnl rngineering faculty at
Columbia University. In 1956.he
was a visiting
member of the
for Advanced Study in Princeton, New Jersey. In nddltton ,lie held o l other
visit in^
appointments. among them a
Electrical Engineering ar MI']; in 1962 and
.
visit in^ 196Rr
;I
Institute ii
number
professorship in visiting s r w n t i s t
appointment at IBM Research 1,a boratory %in Jose +CA, in 1968 * 1 973 ,;rnd
1977 rand visiting scholar appointments at the A1 Center .SRI International. in 1981, and at the Center for the Study of 1,anguage and Inf{rrm~ltion+ Stanford University, in 1987 Graduate
School .and
1988. Currently he is a Professt>r in the
i s serving as
the CJirectur of Bl%:(hrkelcy ln~tiativr.in
Soft Computing). Until 1965 ,Dr. Zndth's work had been centered o n system theory rind
decision analysis. Since thcn,his research interests have shifted oI
to the thrrjry
Iuzzy sets and i t s apylicarinns to artificial intelligencc, iinguistics+ltrgii..
decision analysis* control theory, expert systems and neural nerwr,rks+
Currently. his research is focused on fuzzy l(.,Ric. s o f t computing and computing with words.
An alumnus of the University of 'Teheran. MI'T. and Columbin
7
Universit y . D r . Zadeh is
R
fellbw OI the
TEEE,AAAS,ACM and AAAI+arrd a
member of thc Maticma! Academy of Engineering. H e was the recipient of the
IEEE Education Mcdal in 19'73 and a in 1984. In 1989. Dr. Zadeh was
IEEE Centennial Medal awarded the Honda Prize by the Hrmda recipient of the
Foundation, amd in 1991 received the Berkeley Citation, University of
Ca l i f o r n i ~ ~ . In 1992. u
Dr. Zadeh was uwarded t h e ]EKE Richard W.Hamming Mcbal
For seminal contributions to information science and systems.including the
conceptuaLization of fuzzy sets. Russian Academy
"
Ht became a Foreign Member of the
of Natural Sciences (Computer Sciences and Cybernetics
Section )in 1992 and received the Certificate of Commendation for A1 Special Contributions Award from the International Foundation for Artificial Intelligence+Also in 1992,he was awarded the Karnye de Ferict Medal and
became an Honorary Member of the Austrian S c i e t y of Cybernetic Studies, In 1993, Dr. Zadeh received the Rufus Oldenburger Mcdal from the American Society of Mechanical Engineers "For seminal contributiuns in
system theory ,decision ana!ysis,and theory of fuzzy sets and its lipplications to
Al , linguistics, logic +expert systems and neural networks. He was also H
awarded the Crigore Moisil Prize for Fundamental Researches. and the
Premier B e s t Paper Award by the Second International Conference on Fuzzy
and Technology. In 1995 ,Dr. Zadth was awarded the IEEE Medal of Honor "For pioneering development of fuzzy logic and its many diverse applications, ln 1996, Dr. Zadeh was awarded the Okawa Prize "For Outstanding contribution to informtion science through the development of fuzzy logic and its applications. v* Theory
*)
In 1997, Dr* Zadeh was awarded the B. b l z a n o Medal by the Academy of Sciences of the Czech Repub1ic"For outstanding achievements in fuzzy
mathematics. * H e also received the J, P. Wohl Outstanding Career
8
Achievpment Award
oi the IEEE
Systems. Man and I'ybernt?tics Snuiet y. Ide
served fis a I,ec Kuan Yew Distinguisl~edVisitor, l ~ r l u r i ~ar~ gthe Naticmal U n i v e r s ~ t of ~
Singapore and the Nanyang Technr,lr~gical Univrrsity in
Singapurc ,and as the Gulben kian Foundatinn Visiting Professor at the New
Univers,ty of Lisbon in
Portugal.
Dr. Zadeh holds honorary doctotatcs from Paul-Sabatier University. Toulouse, France; State University of New York, Uinghamton, NY
;
University of h r t m u n d ,r)ortmund.Cernlany; University af Oviedn + O v i e d u . Spain; University of G r ~ n a d a+Granada+Spainr Lakehead University ,Canada
;
University oi I~ouisville, K Y r. Baku State University. Azerbaijan ; and the
Silersian Technical University .Gliwice, Poland. Dr. Zadth has authored close to t w o hundred papers and serves on the
editorial boards of over f i i l y journals. He is a member of the Technology Advisory b a r d , U. S. Postal Service; Advisory Committee, Ikpartmcm
nf
Electrical and Computer Engineering * UC Santa b r b a r a r Advisory h a r d ,
FUZZYInitiative, North Rhine-Westialia , Germany r Fuzzy L,ogic Research Center, T e x a s A 8.. M University. College Station, Texas; +4dviscrry Committee + Center for Education and Research in Fuzzy Systems and
Artificial Intelligenr.r. Iavi , Romania r Senior hdvisr>ry b a r d , International Institute for
General Systems Studies; the b a r d of Governors + Interna tional
Neural Networks Society.
Contents
P a r t I :Fuzzy Sets Fuzzy
sets
3
Part 2 :Fuzzy Systems Towards a theory of fuzzy systems
29
Outline of a new approach to the analysis of complex systems and decision processes 62 Part 3 :Linguistic variable and approximate reasoning
The concept of a linguistic variable and its application approximate reasoning. I
.
1+ 1
to
121
Part 4 ; Fuzzy-algorithmic approach and information granularity
A fuzzy-algorit hmic approach
to
the definition of
complex or imprecise concepts 307 Fuzzy sets and information granularity 384 Outline of a computational approach to meaning and
knowled g e representation based on the concept of assignment statement
41 2
a
Part 5 :Soft computing with words Fuzzy logic=Computing with words
435
Toward a theory of fuzzy information granulation and its
centrality in human reasoning and fuzzy logic What is soft computing? 504
461
Part 1 :Fuzzy Sets
Fuzzy Sets
A fuzzy
set i s a class
of objects with
a cunrinuum ui grade!:
of membership, Such a set is charac~erized by a membership
Icharacteristic)functton which assigns to each object
a gade
OI
membership ranging between zero and one. T h e nutions r ~ f inclusion * union, intersectiun,
complement, relation. convexity,
etcr,. are extended to such sets+ and various properties of
these
notions in the context of fuzzy sets are established, In particuinr ,
tor convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
a separation theurem
l*Introduction More often than not + the classes o f objects encountered in
the real physical world do not have precisely defined criteria of
membership. For example. the class of animals clearly incl udes dogs, horses, birds,etc. as its members, a n d clearly excludes
such objects as rocks, fluids ,plants. etc. However, such objects as starfish, bacteria, etc. have an ambiguous status with respect
to the class of animals. The same kind of ambiguity arises in the case of a number such as 10 in relation to the "class" of all real
numbers which are much greater than 1. Clearly, the
d
class of all real numbers
which are much
greater than I + " or "the class of beautiful women, 7 t or "the class of tall men, -9 do not constitute classes or sets in the usual
mathematical sense uf these terms.
such imprecisely defined
u
Yet, the fact remains that
classes" play an
important role in
in the domains of pattern recognition, communication of information and abstraction. The purpose of this note is explore in a preliminary way some of the basic properties and implications of a concept which may be of use in dealing with "classes " of the type cited above. The concept in question is that of a h z z y set ,@ that is, a uclass" with a continuum of grades of membership. As will be seen in the sequel, the notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which paralIels in m a n y respects the framework used in t h e case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables. W e begin the discussion of fuzzy sets with several basic definitions. human thinking,
particularly
I
Let X be a space of points (objects ),with
a generic element
An application of this conceptto the formulation of B class ofproblems in pattern classification is described in RAND Memorandum RM-4307-PR , "Abstrac tion and PatternClassification+ "by R. kliman IR. Kalaba and L. A. Zadeh, Oerolxr. 1964.
of X denoted by
Thus,X=(xE, A fuzzy set (t(eiass) A in X is characterized by a mernbrship ( X 1 whit h a s s ~ ia c tes with each poi nr '!? ( CAamc~~risti6 )function 3,
-'\
in X a
real number in the interval [0,1]&.
(r ) a t x representing
with the value of f,,
the "grade ot membership" of x in A.
Thus, the nearer the value of S, ( a >to unity, the higher
the
grade of membership of x in A. When A is a set in the ordinary sense
of the term, i t s membership function can take on only t w o
values O and
I , with
fA ( x )= 1 or
not bebrig to A. Thus, in this case
0 according as x does or does
f i( 2 )reduces to the familiar
characteristic function of a set A. I When there is a need to
differentiate between such sets and fuzzy
sets, the sets
with two-
valued characteristic functions will be referred to as nrdinary sets
o r simply sets, )
Examp&e,Let X be the real line R' and let A be a fuzzy set of numbers which are much greater than 1. Then, one can give a
precise, a l beit subjective, characterization of A by specifying J I
(xt-) as a function on R 1 . Representative values of such a function might b t : h I O ) = & $ A r l ) = O i S p ( ( W = ~ * O 1 ; f A ( l O ) = O .
2rj-,4
(100)=0* 95;fA(500)=1.
It should be noted that, although the membership function of a ~ U W Yset has some resemblance t o a probability function More
the domain of definition of
f.4 (1) may
be restricted
10 a
s u k t of X. @ I n a more general setting, the range of the membership function can be taken to be a suitable partially ordered set P. For out purwses, it i s convenient and sufficitnt to restrict the m r w of / to the unit interval. tf the values of fA [ I ) a r t interpreted as tmth values, the latter rase corresponds to a tnultivnlued Logic with a con~inuumd truth values in tht interval 10-I].
when X is a countable set (or a probability density function when
X i s a continuum 1, there are essential differences between these concepts which will become clearer in the sequel once the rules of combination of membership functions and their basic properties have been established, In fact, the notion of a fuzzy set i s completely nonstatistical in nature, W e begin with several definitions involving f u u y sets which are obvious extensions of the corresponding definitions for ordinary sets,
A fuzzy set is m+ty if and only if i t s membership function is identically zero on X. T w o fuzzy sets A and B are equal, written as A = & i f and only if f A (x)= E n ( x ) for all x iri X, (ln the sequel ,instead of writing f A ( x ) = f s ( . x )for all x in X , we shall write more simply
f~'f&+ The complement of
a fuzzy set A is
denoted by A' and is
defined by fA,=lfA*
(1)
As in the case of ordinary sets, the notion of containment
plays a central role in the case of fuzzy sets. This notion and the
related notions of union and intersection are defined as follows. Containment, A is contained in B ( o r , equivalently, A is a
subset of
Bt or
A is smaller thun w e q w l to B)if and only if
fm45
In symbols
fD.
ACBW-ASfBh
(2)
Unicm. The union of two fuzzy sets A and B with respective
membership functions f,(x) and fB<x) is a fuuy set C , written a s C=A
6
UB .whose membership function i s related to those of A
and
B by fc.(s) =Max[.f,.,(s) , f n T x > ]
XEX
(3)
o r , in abbreviated form
(4) j;.= f ,v fl, Note that (J has the associative property. that is. A U (But.-)=
( A U U ) UCI Uf~ntnrent. A more intuitively appealing way of defining the
mion is the fol10wing:The union af A and Bis the smallest fuzzy
A and U,Mare precisely, if D is any fuzzy set which contains both A and S , then i t aisu contains the union of
set cantaining both
A and B,
T o show that this definition is equivalent to 3 > ,we
note.
f i r s t , that C as defined by I 3 I contains both A and U . sinre
M a x C J i +fE112J,
and Max[f:+ tf;iIZfr; Furthermore, if D is any fuzzy set containing both A and U. then . f f J > f .-l
fnZf ,l and
hence fnZMaxCf,4,f
ill
=f(.
which implies that CCD. Q. E. D,
The notion of an intersection o f fuzzy
sets can
be defined in
a n analogous manner. Specifically:
Inter.~rc&ion. The intcr,wctir,r~o l two iuzzy sets A and l$ with
respective membership functions f , . , ( x )and f r ; ( ~ ? is fuzzy written as C = A
set (',
n B, whose membership function is related 7
lu
those of A and B by
(51
f c ( x ) = ~ i n [ f ~ ( x ) r f ~ ( x ) ] tx E X I
or, in abbreviated form h = f A
Af ~ -
(6)
As in the case of the union* it is casy to show that the intersection of A and 3 is the largest fuzzy set which is contained in b t h e A and 3. As in the case of ordinary sets, A and 3 are dispint if A B is empty. Note that like U, has the asatxiat ive property, The intersection and union of two fuzzy sets in R' are illustrated in Fig. 1. The membership function of the union is comprised of curve segments 1 m d 2 t that of the intersection is mmprised of segments 3 and 4 (heavy lines).
n
n,
u Fig. I. fllustration of the union and intersection of fumy sets in
Cmmorc. Notc that the notion of 'belonging
RL
," which
plays
fundamental role in the case of ordinary sets, does not have the same role in the case of fuzzy sets. Thus, it is not memingful to
A except in the trivial sense of fa ( x 1 being positive. Less trivially, one can introduce two icvels a and /?(O
p) and agree
speak
d
a point x ubelonging" to a fuzzy set
to say that ( 1 ) " ~ belongs to A"if , f A ( x ) 2 a +(
2 ) " does ~ not
belong to A" if fA(x)5/38and ( 3 I Y xhas an indeterminate stat us relative to A' it /3
Sp) + -
three truth values: T I f A { r ) ) a ) , F{fA(x)
and UI#3.
3. Some properties of
the
n, U + and complementation
union, intersection, and complementation defined as in i31, ( 5 'I,and ( 1 1, i t is easy to extend many oi the basic identities which hold for ordinary sets to fuzzy sets. A s examples, we have (AUB)'=AfnLY ~e Morgan's laws r With
operations
of
( A n B ) ' =A uB'
/
C ~ C A U B ~ = ( C ~ A ) U ( CDistributivelaws. ~B) (9) CU(AnB)=ICUAIn(cuB, (10) These and similar equalities can readily be established by showing that the corresponding relations for the membership functions of A ,B, and C are identities, For example, in the case of (71, we have 1 - M a x v A t f ~ f= M i n [ l - f ~ + l -fB] (11) which can be easily verified to be an identity by testing for the two possible casts; f A C x ) > f ~ ( x ; r )and fA(x)
,.,
(12, which can be verified to be an identity by considering the six cases $
9
f A ( ~ ) > f f i I x ) > f c ( x~ ) ~ , ~ ( x ) > J I ' ( x 9) > ~ B I x ~ fH(x)>fA(x)>fc.(.~)
9
ffi(~)>f~(~)>ff~(~)
rf['(~.)>fA(~)>fB(x)
*
f('(~>>ffj'i~j>f.4(~)#
Essentially , fuzzy sets in X constitu'te a distributive lattice with a 0 and 1 (Birkhoff ,1948L
An Interpretation for Unions and Intersections
In the case of ordinary sets,a set C which is expressed in terms of a family of sets Al, * * - , A , , A, through the connectives U and 0 , can be represented as a network of switches a,, * * * , uH,with A, A, and A. l.j Aj ~orresponding. respective1 y , t o series and parallel combinations of a, and a,. In the case of fuzzy sets ,one can give an analogous interpretation in terms of sieves. Specifically, let f i ( x 1, i = 1 + .** , n , denote the value of the membership function of A; at x * Associate with f , ( x ) a sieve S,(xz-)whose meshes are of size fi(r) + Then ,fi fi(r) V f, (x) and fi( x ) A f, (XI correspond, respectively, to parallel and series combinations of S i ( t 1 and S , ( x ) .as shown in Fig 2. Dm.,
31Id
O9;. 1"'
SICS)
n
Fig. 2- Parallel and series connection of sieves simultating IJ and
More
An, U,and
a well-formed expression involving A , ,
n corresponds to a network of sieves S , {r),
- - +
,
,Sm
++*
(XI which can be found by the conventional synthesis techniques for switching circuits. A s a very simple example, 10
c=[(A,UA:)~A,IUA,
(1:)
corresponds to the network shown in Fig. 3 ,
Note that the mesh sizes of the sieves in the network depend on .tand that the network as rz whole is
whose
equivalent to a single sieve
meshes are of size f;.(x).
4 Fig*3. A network of
sieves simdtating { [j',
V .f,(.r)]h , f ? I ~ l -Vl , f , { . r . )
4+Algebraic operations on fuzzy sets I n addition to
the operations of union and intersection
r>nP
can define a number of other ways of forming combinations of
fuzzy
sets and relaring them to one another.
Among the more
important o i these arc the following.
Algehrak prducr. The u I ~ P O ~ U ~ Cprrdltci . of A and 1j is
denoted by AB and i s defined in terms of the membership functir~nsof A and B by the relation =J.4.f-1i
(14)
nB.
(15)
Clearly, ABCA
Adgehaie sum. @The a l g c h i c sun, of A and B is denoted by
A - t B and is defined by
, (16)
~ A + B = ~ A + ~ B
provided the sum f A
+$B is less than ox equal to unity. Thus
9
unlike the algebraic prduct ,the algebraic sum is meaningful only when the condition fA Ix)+fs(x)
The
absdutc diffsmce of A and
B
is
denoted bylA-Bland is defined by flA-81=
If4-fSl.
Note that in the cast of ordinary
sets
I A - B I reduces
to
the
relative complement of A f') B in A UB.
Convex cmbinatim. By a convex combination of two vectors f and g is usually meant a linear combination of f and g of the form Af
+ ( 1- A)g,in which O
1. This mode of combining f
and g can be generalited to fuuy sets in the following manner.
Let A , B, end A be arbitrary f u u y sets. The convex c o d i ~ ~ t i oaf t A,B, and A is denoted by (A,BfA)and is defined
by the relation IA,B;A>=AA+A'B
where A' is the complement of A. Written out in terms of
membership functions, Ii7 )reads A basic property af the convex combination of A , 3,and A is expressed by OJ Tht dual d tk algebraic p d w t is the 3um A@B = (A'H 1' = A 4- B - AB. (Tbi was pointed out by T.Cover. >Note that for ordinary sets n ~ t l dthe al~tbraic p r d u e t are quimlent operetions,ss art U 3rd @.
12
This
A n b C ( A + B : A ) C A U B , f o r a l l A* (19) property is an immediate consequence of the
Inequajities.
Min[fA(x) ,ftdx)]
+
1-A}fdx)
<Ma~[f.~(x),f,dr)] , r E X (20)
which hold for all A in [a, 11. It is of interest to observe that, given any fuzzy set C satisfyin g A B C C C A U B , one can always find a fuzzy set A such that C ( A t B ; A 1 , The membership function of this set i s given by
n
Furq
relution, Thc
concept
generalization of that of a
of
a
-
relutirm
I which
functionlhas a natural
is
a
extension to
fuzzy sets and plays an important role in the theory of such sets and their applications---just as i t does in the case of ordinary
In the seque1,wc shall merely define the nation of a fuzzy relation and touch upon a few related concepts. Ordinarily, a relation i s defined as a set of ordered pairs (Halmos, 1960);e. g . the set of all ordered pairs of real numbers x and y such that x>y. In the canrext ol fuzzy sets,a .fuzzy relation in X is a fuzzy set in the product space X X X . For
sets.
example +therelation denoted by x>>y,x + y f R1,may
be regarded
as a fuzzy set A in RZlwiththe membership function of A ,f;l(x,
y 1 , having the following ( subjective representative values ! j,+
(10~5)=Orf~~l00~lO3=0.7sf~~l00,1~=1;etr:.
More generally, one can define an n-ary f u z z y r~iutionin X as a fuzzy set A in the product spacts X X X X X X. For such relations,the membership function is of the form fA(s,, - * * , x , ) , --•
'13
where x i E X , i = 1 , * * * , P I .
In the case of binary fuzzy relations, the compo..ririon of t w o fuzzy relations A and B is denoted by B A and is defined as a fuzzy relation in X whose membership function is related to those of A and B by fH.A(x,y)=Sup. Min[fA(r,w)
,f~(v,~)].
Note that the operation of composition has the associative property A
IB-C)-(AoB)aC,
Fuzzy s e ~ sinduced by ntapfiing.~.l.et'I' be a map pin^ from
Y.Let B be a
X
in Y with membership function f&). T'ha inverse mapping l'-' induces a fuzzy set A in X whose membership function is defined by to a
space
fuzzy set
f A ( ~ ) = f f i < ~r )y E Y
I221
for all x in X which are mapped by 7' into y.
Consider now a converse problem in which ,4 i s a given fuzzy set in X , and 'Im, as before, i s a mapping from X . to Y+ The question is :What is the membership function for the fuzzy s e t LJ in Y which is induced by this mapping? If 'l'i s not one-one , then an ambiguity a rises when two or more distinct points in X,say x, and x,,with different grades of membership in A .are mapped into the same point y in Y . In this case, the question i s :What grade of membership in B s h ~ u l dbe assigned t o y?
To
resalve this ambiguity, we agree to assign the larger of
the two grades of membership to y, More generally. the membership function for
B will
be defined by
ffi(~)=Max,~-I.-l,y,fA(xt ) yf
14
Y
where 7 ' - ' ( y )is the set of points in X which are mapped intu
J~
by '7'.
5, Convexity
As will be seen in the sequel, the nvtiun of convexity can readily be extended to fuzzy sets in such a w a y a s to preserve many t ~ fthe properties which it has in ihe cuntext of ordinary sets. This notion peam to be particularly useful in applications involving pattern classificc?tiun, optimizal iun an[? related problems,
I
ymvtr
fuzzy set
Fig+4, Convrx and nunconvex fuzzy sets in fil
In What.follows ,we assume for concreteness that X is
rl
real
Euclidean space EM.
Ci/nvrxi~j~. A fuzzy set A i s co?rzv.r*if and only
if
the sets
F..
defined b y
> To= { J - Ij:q(.r)=tt~ are convex for all a in the interval ( 0 , I].
(24 1
A n alternative and more direct definition of convexity is the -. f o l l o w i n g a : ~is convex if and only if (25) fA[Lr, (1-.\)x2]z~in[fAcxl).fA(x2)]
+
for all rland x2in X and all A in [0.1]. Note that this definition does not imply that
fA
.
( x must be a a convex function of s.This
is illustrated in Fig. 4 for n= 1.
To show the
equivalence between the above definitions note
that if A is convex in the sense of the first definition and a=&
( x I ) S f A ( x 2 ) , t h e nx t E reand Axl+ ( l - A ) c z f convexity of
.,'I
by the
Hence
f A [ k ,( + ~ - A ) x ~ J ~ ~ = ~ ~ ( x ~ ) = M ~ ~ ~ A ( x ~ } + ~ A ( x Conversely, if A is convex in the sense of the second definition and a = h (xl1, then
may be regarded as the set of
all points x, for which f A ( x , ) 2 f A ( x i ) .I n virtue of ( 2 5 ) ~ e v e r y point af the form Ax1$- (1-A)x2, 0 5 1 5 1 +isalso in r, and hence is a convex set* Q. E. D.
A basic property of convex fuzzy sets is expressed by the Theorem. If A a n d B are convex,sr, i s rhrir intersectim. Prmf :Let C = A n B ,
Then
fcC~l-t~~-a)d2J =Min[f~[LI
+ (1-Ux,]
+
,fD[kL., (1- a x , ] ] .
Now ,since A and B are convex
Q This way of exxpwwing convexity was suggested to the writer by his mlkague* I?. krlekamp.
and hence
+
L[h-[ ( 1 -41x2] 2 ~ i n [ ~ i n [ j ; d , . r ,frr(~~):,Min[f;r(.r,) *~ .,,t'&i: ) ] ( 2 8 )
and thus
+ < 1 -A).r2
f>[Lxi
JkMin[.f; (1,+ j i < z ?Q* E*D, (30)
U<~zdn&dncss.A fuzzy set +4is l ? ~ u ~ idf and ~ d only if the sers {xlf
tr)Lai are bounded lor all
there exists a finite R(a)such that
is,for every a > O
o>O;that
11 .r 11 S K ( a j
for all
If A is a bounded set, then for each c > O hyperplane H such that f14(.x)
r,,=
in
I,
then exists a
for all J- on the side of H which does not contain the origin. For,consider the set L = (2:/ <E
f A t x ) z ~ jBy . h y p ~ t h e ~ i s ~ t hset i s is contained in
a sphere
radius R,,,. Let H be any hyper --plane suppvrtiog
S of
S. Then,all
points on the side of H which does not contain the origin lie outside a r on S .and hence fat all such points
(.I).
f., ( X
) ~ E +
Idemma. I k r A be u ri~)trrzd~d St(=? .rct und M ==SUP.~J:, ( M wilL be r e f ~ r r ~to d u?; thd> n2u.z-intul jim~lrdi f i A. )7'b~r1
there is u t h ~ s nrtu t point J-,, C
t h SPI ~
- ~ I ~ U ~ pi~irrt.~ P ~ S 7 ~ 1
Q(E)
(XIZM-E}, Prr~)f+ *consider a nested sequence of bounded * * ~ , w b e rI e' , = ( + r I S A ( x ) 2 M - - , W / I r r + :I) This P T ~ Cwas
sugge,.;~erlby A .
=
sets
I .f:.,
(-2%
I?, .F2
1 ) ) . n - - 1 * 2 * * * . ~ o t that e
J. Thomasisin.
T. is nonempty Ior all finite n as a consequence of the definition of M as M=Sup,-fAl~). (We assume that M>O. ) Let J, be a n arbitrarily chosen point in r, ,?t= 1 , 2 , * * * . Then,
.
x i , x 2 ,*** is a sequence of points in a closed bounded set rl.B y h e Bolzano-Weierstrass theorem, this sequence must have a t least one limit point, say x,, i n r , , Consequently, every spherical neighborhood of x, will contain infinitely many points from the sequence x l , x 2 .+--,and *more particularly, from the subsequence XHCI X N + ~ 9 "*
, where N 2 M / E . Since the
points of this
subsequencefall w~thinthe set Q ~ E ) = {r I f 4 ( _ r ) 2 M - ~ } , t h e
lemma is proved, Strict a n d strong c o n v ~ ~ tA y .fuzzy set A is strr'ctdy c u n w x if t h e sets
r,,O < a S
1 are strict Iy convex Ithat is ,if the midpoint of
any two distinct paints in
Falies in
the interior of
r,),Nore that
this definition reduces to that oi strict convexity for ordinary sets when A is such a set.
A fuzzy set A is slrmgLy convex i f , for any two distinct points XI and x2,and any Ain the open interval ( 0 , l ) f,q[Ax, + ( l - ~ ) r l ] > M i n [ f A ( x , ) , f A ( r , ) ] .
Note that strong convexity does not imply strict convexity or vice -versa. Note also that if A and B are bounded, so is their union
and intersection. Similarly, if A and B are strictly (strongly) convex, their intersection is strictly(strongly)convex. Let A be a convex fuzzy set and let M = Sup, fA (x). If A i s bounded ,then ,as shown above ,either M is attained for some X , say x,,or there i s at least one point x, at which M is essentially attained in the sense that t for each E > 0, every spherical neighborhood of isonrains points in the s e t Q(E) = { x I M-fA 18
(r)st). In particular . i f
A
is
strongly convex and x,, is attained,
then .x0is unique. For.if M = f A ( . ~ O ) a nM d= then
fA
Max,,f,
f A ( ~ - , )
,with x , f s,,,
1M ior r -- 0.5 x o $- O. Ssl,which contradicts M
(1:)
=
(x),
More generally, let C ( A ) be the set of all points in X at which M is essentially attained. T h i s set will be referred to as t h e core of A . In the case of convex fuzzy sets. w e can assert the fo120wing property of C(.4 ), Theorem. lf -4 i s a
fuzzy set, then i t s cur-6 i s o
ct>?rvPs
Sel*
Prwf ;It will suffice
to show that
ii M is essentially attained
at x , and x,,lrl+x, ,then i t is also essentially attained at all n- nf the form r - - A x , + ( l - A ) x , , O ~ A ~ l .
To
the end, let p be a cylinder of r a d i u ~E with the line
passing through x, and x , as its axis. Ider x,,' be a point in
sphere of radius
E
:I
centering on x,, and a l f bc a point in a sphere of
radius E centering on x,such [hat f A 4 ( x , , ' l Z M -and ~ ~ A I x , ~ ) ~ M - s. Then, by the convexity of A t for a n y point tr on the segment x, r x, I , we have f A 4 ( rl j 2 M - E. Furt her-more by the convexity of P.all points a n x,,'xlr will lie i n P,
Now let x be any point in the segment
fn . X I + The
distance of
this point from the segment r , , ' x I ' must be less than or equal to 6,since sl;r,' must be less than or equal to &,sincexfi> X I ? lies in
P.Consequently,a sphere of radius E centering on x at [cast one point of the
segment
least one point ,say w , a t which that
M is essentially
J:~'X~ and ' fA
will contain
hence will contain at
(-ru)ZM--e.This establishes
attained at x and thus proves [he theorem.
Corollary. If X =E L ~ l A~ f d . strongly ~ c o * l v r r +thcln ikp
/wr'~zi
19
ur wlltich
M
is essenizrllly aituined i s urtzqw.
S k u d m of a furzy membership function f
5et. L e t
,. ( x 1 -
fA
A be a fuzzy set in E" with
I XI,
**-
, x, 1, For
simplicity, the notion of the shadow ( projection
notational
> of
A on a
hyperplane H will be defined below for the special case where H is a coordinate hyptrplane,e. g . , H = {x(K,-0 j .
SpecificalJy,the ~ A U ~ I Y L Uof A on H = { x l x l = O } i s defined to
be
a fuzzy set SFr(A)in En-'with fsH(A,(x)givenby
fs,(~~(x)=~
S , ~ ~ ~ ~ ( X ~ , * * * * X ~ ) = J U P* ~* -~ * ~r ~~ l () (. X ~
Note that this definition is consistent with ( 2 3 ).
When A is a convex fuzzy set, the fullowing property of SH (A)is a n immediate consequence of the above d e f i n i t i ~ n ~ lAf is a convex fuzzy set, then its shadow on any hyperplane is alsu a
convex fuzzy set . An interesting property of the shadows of two convex fuzzy sets
is expressed by the fr>llowi.ng implicarian
S H ( A ) = S H ( B ) + f o r all H * A = B . ,'7. To prove this assertion ,!%t is sufficient to show that if there exists a point ,say xo,such that f A I x ~ ) # ~ B ( x ~ ,then ) their exists a hyperplane
f-l such that
f3,(A,
x,, " IfJsf,(o, (xo'
.where I,
1
'
1s
H. Suppose that f d x , ) =i-.r>f,(x,) =jX Since B is a convex fuzzy set * the set & = 1 fB(x)>/3}is convex, end hence there exists a hyper-plane F supporting Fpand passing through xz. 1,et H be a hyperplane orthogonal to F ,and let r, be the projection the projection of xa on
{J+
Q This proof is ksed on an idea suggested by G. h n t z i g for the and B are ordinary tonvex sets.
20
c a w where A
of r, on H. Then .since f n ( r1 ) 3 for all r on ( xr * )
5 /3.
On the orher hand.
I+ f
/z,,,r<j(~fi'
;,,f.r,
F .we have
(:rod) >a. Consequently,
f:q,,,Al
(.rb' ) ,and similarly for the case where a<& form of the above assertion is the
A somewhat more
follcswing:I,et A , but not necessarily B, be a convex fuzzy
set ,and
let S H ( A l=SH(B)fmall H. Then A=conv B , where cnnv B is
.
the convex hull of B ,that is
B. More
the smallest ,convex set containing
general1y ,Ssr( A ) = Sri ( B ) for all
conv U. Stparutitm of
ftl2z-y
CO?~~CS
H implies conv A =
The classical separation
.SVCS.
theorem for ordinary convex sets states ,in essence,that iF A and B are disjoint con
- vex
S ETS .
then there
exists
a separating
hyperphnc 11 such that -4 i s on one side nf H and B is o n the
other side. It is natural to inquire if this theorem can be extended t o cunvex fuzzy sets, withour requiring that A and
B be disjuint,
since the condition of disjaintness is much too restrictive in thr case of fuzzy sets. It turn?i o u t ,as will be seen in the sequel.that
rhe answer to this question is in the af-firmntive. As a preliminary,we shall have to make a few definitions. Specifically,let A and B be two bounded fuzzy sets and let M be a hypersurface in E" defined by an equation h (J-)= 0, with all
points for which h ( x 1 2 0 being an one
jar which h ( s ) 5 O being on
s i c k of
H and all points
..
the
other side. ;!!Ilet K!! be a number
dependent on H such rhat f ~ , ( x ) S K o, n~ one side of
( x 3 S K H on t h e other side. I,et .
Mi!be
H and
-
f'h:
Inf KfI.The n u m b ~ rD,,
Vote that the sets in : r ~ ~ e s t i uhave n k! in common,
21
'
1-MH will be called the degrpe of seporurion rcf A and k? by HH. In general, one is concerned not with a given hypersurface H,but with a family of hypersurfaces:hll), with A ranging o v e r , say. P".The problem, then, is to find a member of rhis family which realizs the highest possible degree of separation. A special case of this problem is one where the HAare hyperplanes in E" .with A ranging over E? 1n this case. we define the degree of separability of A and B by the relation ~=l--m where
(31)
-
M = h f HMH
with the subscript A omitted for simplicity.
Fig, 5. Illustration of the separation theorem for iuzzy s e t s in E'
Among the various assertions that can be made concerning D , the following statement@ is, in effect, a n extension of the separation theorem to convex fuzzy
sets.
Theorem. Let A ar~dB be bun& This siattment
is based on a auggesiion of
convex fuzzy sets in
E. Berlekarnp.
E",
with nrnzimul grodrs M A und MI,,ruspe~.tivul~ [M~=s up,,f~( x 1
Mfi=sup,frr(x)]-Lei M hu i h v nrarimul grub for chr A n,U(M=sup,
~ i n V ' , , (, rf l)r ( ~ ? ] ) .
inturvuctitnr
7'hrlt D = 1 - M .
In plain words ,the theorem
the highest degree of separation of t w o convex fuzzy sets A and B that can be Grnmunt.
states that
achieved with a hyperplane in E"is one minus the maximal grade
in the intersection A fl B.This is illus~rntedin Fig. 5 for
Prwf:
rr = 1.
consider separately t h e folluwing two cases: ( 1 ) M = M i n ( M d q!MK)and(2jM<Min(rsA,I.MR). Note It is convenient tu
that the latter case rules out A C U or B C A .
Cc;l.re 1. For concreteness, assume that M,,<Mir, so that M = M A .
Then, by the property of bounded sets already stated there
exists a hyperplane H s u c h that f ' H (M~for ,5 all x on one side of H. On the other side of H . ~ ; , ( T ) Sbecause M f , 4 ( x > 5 M , 4 =M for all x, It remains to be shown that there do not exist a n ,W'< M
and a hyperplane H' such that f.,,ix)SM' on one side oi H'ancl
f H I x ) S h f ton the other side.
This follows a t once from the following observation. Suppose that such H' and M' exist, and assume for concreteness that the core of A(that i a , t h e set o f points a t which MA4==1bf is ~ s s c n t i a i l yattained)is on the plus side of H ' . This rules o u t t h e
f , , ( r ) S M t for all x on t h e plus side of H' ,and necessitates that f . 4 ( x ) S M ' for alt LT on he minus side of
possibility that hence
H ' , and
Cs> 5
ffi
M'
for a l l
I.
on t h e plus side of H ' ,
Consequently+overa l l x on t h e plus side of H' Sup, M i n [ f 4 W q f l c h ) j S M '
and likewise for all r on t h e minus side of H'. This implies t h a t , 23
over all r in X ,sup,Min[ f A ( x ) .fs { x ) ] 5 M ' +whichcontradicts the assumption that sup,Min[fA(x).
If
(I1
>MI.These
sets
]=kf>bf.
Mland rH are nonempty and disjuint , for if
Case 2. Consider the convex sets = {x
fB<x)
r A= { x IfA ( X I >
they were not there would be a paint ,say u,such that fA(tr)>M
and fr+(u) > M . and hence
fAns
(n)
>M, which contradicts
the
assumption that M=sup, fAns(x),
Since FAand
rg are disjoint, by the separation
theorem for
ordinary convex sets there exists a hyperplane H such that
rq4 is
rB is on the other side ( the minus side). Furthermore, by the definitions of rA and rL, for all points on the minus side of H,f, ( x 1 5 M, and for all on one side of H ( s a y , the plus side land
points on
the plus side of H7fN{-2)5M.
Thus, we have shown that there exists a hyperplane H which realizes I M as the degree of separation ol A and B. The conclusion that a higher degree of separation of A and B cannot -
be realized follows from the argument given in Case 1. This concludes the proof of the theorem. The separation theorem for convex fuzzy sets appears to be of particular relevance to the problem of pattern discrimination. Its application to this class of problems as well as ro problems of optimization will be explored in subsequent notes on fuzzy sets and their properties.
References
Brkhoff +C,(1948) ,"l,attice Theory. "Am. Math. %c. C o l I q .
Publ. +Val. 25,New Ynrk. Hairnos ,P+R. [ 1960) * "Naivs
Set 'Theory "Van Nostrand. New
York. K1eene.S.C. I1952)~"Introducrion to Me~amarhematics.~P. 334. Van Nostrand ,New York.
Part 2:Fuzzy Systems
Toward a Theory of Fuzzy Systems
Many of the advances in network theory and system t h e ~ , r y during the past three decades are traceable to the influence and conrribu~ions of
Ernst
Gutliernin, Norbert Wiener, Richard
kllman,Rudolph Kalman,and their students. In sum, we now possess
and impressive armanentarium
of techniques for
the
analysis and synthesis nf linear and nonlinear systems nf various types
- techniques
t h a t Rre particularly effective in dealing with
systems characterized by ordinary differential o r difference equations of moderately
high order such
as
those encountered in
network the~ry,controltheory .and r d a t ~ dfields.
What w e still lack ,and lack rather acuteIy+are methods for dealing with systems which are tuo complex or too ill-defined tr, admit of precise analysis. Such systems pervade life science3, social sciences, philosophy, economics, phychology and many
other "soft"fields. Furthermore, t hey are encountered in what are normally regarded
AS
"nonsoft"fields when thc complexity nf a
system rules out the possibility oi analyzing it by conventional mathematical
means, whether with or without the aid of
computers. Many examples of such systems are found among large-scale traffic control systems, pattern-recognition systems.
.
mac him translators large-sea t e informa t itrn-prtwessing
sys terns,
29
I,erge-scale power-distribution networks, neural networks, and
games such as chess and checkers. Perhaps the major reason lor the ineffectiveness of classical
mathematical techniques in dealing with systems of high order of complexity lies in their failure to come to grips with the issue of
with imprecision
from randomness bui from a l a c k of sharp transition from membership
fuzziness, that
is,
stems not
that
in a class to nonmembership in it. It is t h i s type of imprecision that arises when one speaks, ~
O Texample, of
the class of real
numbers much larger than 10,since the real numbers can not be
divided dichotomously into those that are much larger than 10 and those that are not, T h e same applies to classes such asutall men , good strategies far playing chess,""pairs af numbers that are approximately equal ta one another, 44 systems that are approximately linear, "and so forth. Actually, most of the classes encountered in the real world are of this fuzzy, imprecisely defined kind. What sets such classes apart from classes that arc well-defined in h e conventional mathematical sense is the fuzziness of their boundaries. In effect ,in the case of a class with a fuzzy boundary,an obje6t may have a grade of membership in it *Id4
99
that
lies
somewhere
between
full
membership
and
nonmembership, A class tl-tat admits of the possibility of partial membership
in it is called a f u u y set.
this sense +theclass of tall men .for example,is a fuzzy s e t , as is the class of real numbers that are much larger than 10. We make a fuzzy staberne~t or msertiun when some of the words appearing in the statement or assertion in question are names f o r fuzzy sets. This is true +forexampie ,oi 30
such statements asYJohn is roll.
5,
9t U
r is upprorinto~tlyequal to
?? 4
yismuchLorger thcrn 10."lnthesestatements~rhesources
of fuzziness are the italicized words?which, in effect, are labels for fuzzy sets. Why is fuzziness so relevant to complexity? Because no matter what the nature of a system is, when its complexity exceeds a certain threshold it becomes impractical or compu tationally infeasible to make precise assertions about
it.
For example ,in the case of chess the size 01 the decision tree is so large that it is impossible ,in general ,to find a precise algorithmic sulcltinn to the following problem :Given the pfisition of pieces on
the board ,determine a n optimal next move, Similarly +in the case
of a large-scale traf fic-control system , the romplexi t y of the system precludes the possibility of precise evaluation of its
Thus , any si g nificant assert ion a bout the performance of such a system must necessarily be IUZEY in nature. with the degree of fuzziness increasing wit11 t h e compkxity of the system. How can fuzziness be made a part of system theory? A performance.
tentative step
in this direction was taken in recent papers[''L31in
which the notions of a fuzzy sustem'iand fuzzy algorithm were
introduced. In what follows, we shall proceed somewhat further in this direction 9 focusing our attention on the definition of a fuzzy system and its state. It should be emphasized however?that the task of canstructing a complete theory of fuzzy systems is one
a!
The
maximin automata or W e e and S a n t o ~ ; ~ : ~ ~ :be r nregarded a~ EIP. insrancest,f
fuzzy systems.
31
of very considerable magnitude,and that what we shall have to say about fuzzy systems in the sequel is merely a first step
toward devising a conceptual framework for dealing with such systems in
both qualitative and quantitative
ways.
Elementary properties of fuzzy sets
The concept of a fuzzy system is intimately related to that of a fuzzy set. En order to make our discussion self-contained, it will be helpful to begin with a brief summary of some of the basic definitions pertaining to such sets. @ Definition of a Fuzzy Set Let X = { x ) denote a space of points (objects ), with x denoting a generic element of X I Then a fuzzy set A in X is a set of ordered pairs A={[X,L(A{X)]I
XEX
(1)
where , u ~ ( xis) termed the grade of rnewkrshifi of x in A, T h u s ,
if
p,, ( x 1 takes values in a space M-termed
the nr~mhership
space-then A is essentially a function from X to M. The
function
FA:
X* M+which defines A , is called the
function of A.
When M contains only two points
membership
o and
1, A is
nonfuzzy and its membership function reduces to the conventional characteristic function of a nonfuzzy set. Intuitively , a fuzzy set A in X is a class without sharply
defined boundaries-that is,a class in which a point (object) x may have a grade of membership intermediate between full O
detailed discussions of fuzzy sels and the refer~ttccslisted a t the end of this chapter.
32
Mole
their properties
may br found in
membership and nonmembership. The important point to note is that such a fuzzy set can be defined precisely by associating with
each
* its
grade of membership in
assume for simplicity that
M
A. In what follows. we shall
i s the interval
LO, 11, with
the
grades O and 1 representing, respectively, nonmembership and
full membership in a fuzzy
M
set. ( More
can be a
p r t i a l l y ordered s e t or ,more particularly .a lartire. CE1)Thus,our
will be that a fuzzy set A in X ,though lacking in sharp1 defined boundaries ,can be precisely characterized by a membership funcrion t h a t associates with each x in X a number in the interval[0,1 ]representing the grade of membership of r in basic assumption
A.
Exurnpie
Let A =
{J- ] I>>1) (that
is, A is the fuzzy set of real numbers
that are much larger than 1 ). Such a set may be defined subjectively by a
membership function such as ;
pA(x)~0
f
S< 1
=[1+(1-1)-~]-~ for x > l (2) It is important t o note that in the case of a fuzzy set it is not meaningful to speak of an object as belonging or not belonging ro
that
whose grade of membership in the set i s unity or zero. Thus , i f A is the fuzzy set of tall men +then the
.
set ,except for objects
sratementUJohni s tal1"should not be interpreted
as meanin g
that
A. Rather, such a statement should be interpreted as an association of John with the fuzzy set A- -an association which will be denoted b y John E A to distinguish i t from an assertion of belonging in the usual nonfuzzy sense- that
John belongs to
-
is .John
E A ,which is meaningful only
when
A is nonfuzzy.
@
Cmiainmrnt
Let A and B be fuezy sets in X. Then A is contut'ned in BIor A i s a subset of 3)written as A c B , i f and only if pAIx)5&B(x)
for all x in X* {In the sequel, to simplify the notation we shall omit x when an equality or inequality holds for all values of x in
x*1 Equality 2
T w o fuzzy s e t s are
equal, written as
A =B,if and only if
A fuzzy set A' is the co~nplernentOI a fuzzy set A if and only
if px= 1 --FA, Example
The fuzzy sets A = { xIx>>l) and A'
-
(aIs not>>l 'l are
complements of one another.
Unim
The unim of A and 3 is denoted by A l.J Band is defined as the smallest fuzzy set containing both 4 and 3. The membership function of A U B is given by p ~ ~ f i = M & ~ [ T~ h~u ,s i~f ~at ] a. point x ~ p ~ ( x ) - 0 ~ 9 , s a y , a n d p,(z;r)==O.4,then at that point p A U ~ ( x ) 9. @ Here and elsewhere in this chapter we shall employ the convention of underscoring a symbol with a wavy bar to represent a fuzzified version of
the meaning
of that symbol. For examplt.xy y will denote a fuzzy equality of x? y will denote Zuzzy implication ,etc.
34
A s in the case of nonfuzzy
of the union is * closely related to that of the connective or . Thus , i f A is n class af tall m e n , Z3 is a class oi fat men and " John is tall "or"John is sets, the notion ?I
t*
f a t , then John is associated with the union of A and B+ More
generally ,expressed in symbols we have
The intersection of A and B is denoted by A
n B and
is
defined as the largest fuzzy ser contained in both A and B. The
nLI is given by MinEpl, pl,J. It is easy to verify that A n Er = ( A f U L3' 1'. The relation between the connective"and"and nis expressed by
members hip function of A
xG -A
and
Algdn-uic P ~ I Y C L
The
/.tA4rfi=
-
A-€B*J€A~B n
A and B is denoted by A B and is defined by p ~ n = p - ~ pNote l ~ +that the product distributes over the union but nor vice-versa. algehruic p r d u c t of
Algehruii.- Stmi
The algebraic sum of A and B is denoted by A @ B and is defined by p A ~ l l =+pitp~ p I S . It is trivial to verify rhat A @ U = IAW 11. Redut im
A fuzzy r ~ l a l i o n,R .in the product spare X X Y = I ( ~ . . y )), x f X , y E Y ,is a fuzzy set in X X Y characterized b y a membership function p~ that associates with each ordered pair (r,y )a grade of membership p,((x,y)in R. More generally.an n-ary fuzzy relation in a product space X = X' x XZx * * * x Xu is a fuzzy set in X characterized by an n-variatc membership function p~{ r,, * - -
.
