A call for crispness in fuzzy set theory

Fuzzy Sets and Systems 29 (1989) 57-65 North-Holland

57

A C A L L F O R C R I S P N E S S IN F U Z Z Y S E T T H E O R Y Etienne E. KERRE Seminarfor Mathematical Analysis, State University of Gent, 9000Gent, Belgium Received January 1987 Revised June 1987

.4bstract: This paper is a reflection of some rather negative experiences I have had during the preparation of lecture notes for a general course on fuzzy set theory: the lack of standard definitions for fundamental concepts, many fundamental concepts are not expanded fully enough and several proofs of important properties are incorrect. Keywords: Fuzzy set theory; fuzzy real number; fuzzy topology; fuzzy convex relations.

1o |ntrodu~on

Since 1985 I have been giving a new course on fuzzy set theory and its applications to computer sciences at the graduate level in mathematics at The State University of Gent. In order to write some lecture notes for this 30 hour course, I carefully read and studied several papers and books on fuzzy sets. In this paper I will list some rather negative experiences I have had during this job. This paper should not be seen as personal criticism but only as a plea for cleaning up and completing the basic material on fuzzy sets and furthermore as a plea for a more careful reading of the papers by the referees. In the first part we will show that there exists a lack of standard definitions for many fundamental concepts. Examples from three different domains will be given. The second point that will be made is the following: many fundamental or basic concepts are not fully expanded. Since we all are familiar with ordinary set theory it is of great importance to know very well the deviations of fuzzy set theory with respect to basic notions used to develop higher ones. As examples we give: a complete list of the elementary properties of direct and inverse image of a fuzzy set under an (ordinary) mapping and secondly the properties of the Cartesian product of two fuzzy sets. In the last part we illustrate by several examples that proofs of fundamental properties, for example with respect to convex relations, are too often given in an incorrect way. 2. The lack of standard de~ifions for fuzzy concepts One of the main difficulties I have met during the preparation of lecture notes on some basic material concerning fuzzy set theory, consisted of a lack of standard definitions for basic elementary notions. 0165-0114/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

58

E. Kerre

As a first example I mention the notion of a fuzzy real number. One of the first papers I have found on this subject was written by Mizumoto and Tanaka [1]. They simply defined a fuzzy real number as a fuzzy set A on R satisfying the convexity condition A(Axt + (1 - ~.)x2) ~>min(A(xt), A(x2)),

V(xt, x2) ¢ R 2, Vg ¢ [0, 1].

Several definitions have been given by Dubois and Prade [2, 3] putting monotony conditions and different forms of continuity ranging from piecewise to global continuity. Thirdly, a lot of papers use the more abstract definition of Rodabaugh and Hutton, [4,5, 6]: they define a fuzzy real number as an equivalence class of decreasing fuzzy sets on R satisfying some boundary conditions, obtaining in this way a concept that resembles the one=dimensional distribution functions from probability theory. I often met the following situation: authors who used some definition but in fact applied the properties of another one, especially the frequently used property that the oMevel of a sum of two numbers equals the sum of their (r-levels. Examples can be found in [1] and [7]. The importance of the notion of a fuzzy real number is almost self-evident, since it appears in all kind of applications. Hence it is of crucial importance to have a solid notion from a theoretical as well as from a practical point of view. Moreover, the properties needed to do some practical calculus should be described in detail. As far as I know these things are missing in fuzzy literature. In order to make fuzzy set theory more popular it is essential to agree about the definition of elementary basic concepts needed for instruction. A second example concerns the concept of a fuzzy topology. Two approaches have been extensively studied: Chang and Goguen who define a fuzzy topology on a universe X as a class of fuzzy sets on X that is closed under finite intersections and arbitrary unions and secondly the approach of Lowen who also requires that every constant map on X be an open fuzzy set. This again is a very dangerous situation because papers on fuzzy topology frequently start with the sentence "Let (X, r) be a fuzzy topological space" without mentioning which definition has to be considered. A third example concerns the fuzzification of the notions of a point or a singleton from ordinary set theory. Two concepts and corresponding relations of membership (inclusion ~_, fuzzy membership ~, quasicoincidence relation q) have been proposed: fuzzy point and fuzzy singleton. I refer to [8] for an extensive list of the properties of these concepts with respect to equality, inclusion and operations between fuzzy sets. It is evident that this list is far from being complete: the many variations of separation axioms and compactness notions provide worse examples of proliferation. I am quite aware of the fact that choosing the most suitable definition for a fundamental notion remains a very difficult job to perform. It has to be made by a consensus based both on frequency of usage and on applicability. Perhaps there has not been sufficient time for this process. Nevertheless, I would recommend a stronger approach to discourage proUfera-

Crbpness in fuzzy set theory

59

tion of definitions. As suggested by one of the referees of this paper, one could adopt the following policy: a paper which studies a new definition of a previously studied property should not be accepted for publication unless either it contains a comparative section arguing for the change or it contains a significant application of the new variation. Analogs of crisp theorems should not, by themselves, be regarded as significant applications.

3. The problem of full expansion of elementary basic concepts Since we all are very familiar with ordinary set theory, it is of great importance to know very well the deviations of fuzzy set theory with respect to the basic notions. Moreover the properties of these basic concepts must be fully outlined: in :rty opinion this is often not the case. I give two examples. The first one regards the concepts of direct and inverse image of fuzzy sets under an ordinary mapping. These concepts are fundamental to express important notions such as continuity and measurability. Let us first recall some definitions. Let f b e a mapping from X to Y, A ¢ [0, 1]x, B ¢ [0, 1]r. The direct image f(A) of A under f is defined by:

f(A):Y-->[O, 1]

f sup A(x), y ~ ~I(~)=y LO,

Vy ¢ ran(f), Vy e Y\ran(f).

The inverse image f-~(B) of B under f is given by:

f - ' ( B ) : X ~ [ O , 1],

x ~-~B(f(x)),

Vx ¢ X.

I lacked a complete list of elementary properties of direct and inverse image such as the following one: 1. f-l(co B) -- co f-l(B). 2. ran(f) N cof(A) ~_f(coA).

3. e, 4. 5. 6. 7. 8.

B2 f-'(e,)

A1 ~_A2=~f(A~)~_f(A2). f ( f - l ( B ) ) ~_B, with equality if f is surjective. A c_f-l(f(A)), with equality if f is injective. (g *f)-'(C) • f-'(g-'(C)) with f ~ yX, g ¢ Z r, C ¢ [0, II z. f-'(Uj~jBj)ffiUj~jf-'(Bj) with (VjeJ) (Bie[O, 1]r) and J an arbitrary index set.

9. f-l(Nj~.t Bj) = NjE.I f-I(Bj). 10. f(UieJ Aj)= Ujejf(Aj) with (Vj e Y) (Aj ~ [0, l] x) and J an arbitrary index set. 11. f(N~ J Aj) c_ N j~jf(Aj), with equality if f is injective. The knowledge of these properties is indispensable in case one wants to do some calculus with direct and inverse images. The first two properties describe

E. Kerr,

the interaction between images and Zadeh complementation (co A(x)= 1 - A(x)) [13]. Properties 3 and 4 describe the monotony of images. Properties 5 and 6 tell us what happens if a direct image is followed by an inverse image and vice versa. Property 7 is t~,e key for proving chain rules as for example continuity and ditierentiability. Finally, properties 8 to 11 state the interaction between arbitrary unions and intersections in Zadeh's sense (using suprema and infima) [13] on one hand and direct and inverse images on the other hand. I want to draw the attention to property 2. Very often in fuzzy literature the intersection with the range of f is omitted obtaining in this way an incorrect result, since if y ¢ ran(f) then f(co A) ( y ) = 0, f(A)(y) ffi 0 hence cof(A)(y)ffi 1 and thus .f(coA)(y)
e [0, 1],

G [0, 11

At×A2:X×X--*[O, 1],

(x,y)~min(At(x),A,(y)),

V(x,y)eX'.

The following properties are easily proved.

1. At × A 2 = ¢ ~ A t = ¢ o r A2=f£ 2. At =A2::~At X A 2 = A 2 X A t , At × A2 = A2 × At:~At = A2. 3. At × (A2 U A3) = (At XA2)O(At ×A3), At × (A2NA3)=(At × A2) N (At ×A3), At × (A2\A3)G (At × A2)\(At ×A3), (At × A2)\(At ×A3)f~_At × (A2\A3). 4. A2GA3:~,At xA2c::At ×A3, At ×A2GAt ×A3:~A2~A3 (where At :#~). At ×A2=_Ai ×A~:~At ¢:_A1'and A2c:A~ (where ,41 × A2:#~). These elementary properties show several deviations from ordinary set theory. The second property reveals the Cartesian product of two fuzzy sets may be commutative without the two sets being equal. The third property shows the distributivity of Cartesian product with respect to Zadeh's union and intersection but no longer with respect to the difference, again a deviation. Finally, property 4 indicates that the cancellation law is no longer satisfied for fuzzy sets. 4. Proofs of important theorems are onen incorrect This is the third disappointment I have had during the preparation of my lecture notes. Once again I will give an example from three different domains: convex fuzzy relations, fuzzy topology and the fuzzy real line. As a first example we consider two properties of convex fuzzy relations as mentioned by Chang [9]. Let X, Y be two ordinary subsets of the n-dimensional

61

Crispness in fuzzy set theory

Euclidian space R". A binary fuzzy relation R from X to Y is a fuzzy set on X x Y, i.e. an element of [0, 1]xx r. The domain of R is defined as the fuzzy set in X: dora(R): X---, [0, 1],

x~sup{R(x,Y)lyGY},

YxeX.

The, range of R is defined as the fuzzy set in Y: ran(R): Y--* [0, 1],

y~sup(R(x,Y)lxeX},

VyeY.

The direct image of a fuzzy set A on X under the relation R is defined using the extension principle:

R(A):Y--,[O, 1],

y~sup{min(A(x),R(x,y))[xeY},

Vy~ Y.

Finally, we recall the notions of convexity: R is convex ¢~ (¥(x~, Yt) ~ X x V)(V(x,, y,) ¢ X x Y)('~°A~ [0, 1]) (R(A(x,, y,) + (1 - 3.)(x2, y ) ) ~>min(R(x,, y,), R(x2, Y2))), dom(R) is convex ¢~ (~xt ¢ X)(Vx: ~ X)(~. ¢ [0, 1]) (dom(R). (~.~ + (1 - ~.)x2)~> min(dom(R) • x~, dora(R), x2)). Property 1. The domain of a convex fuzzy re!asion is convex. The crucial point in the proof given by Chang is the followh~go From sup{R(xl, y ) [ y ~ Y} >~c he deduces the existence of a point yt ~ Y for which R(xt, yt) >>-c holds, claiming in this way the ~upremum is attained. This statement however is not correct since irom sap(, - i/~ i~, e [~*} = 1, one cannot deduce the existence of n ~ N* such that 1 - l/n >~1, where XN* denotes the set ot all strictly positive integers. Nevertheless Property 1 holds, as we show by the following proof. C~_rreeted prcvf of P:~ope_~y !~ Putti.g ~cm(R) ~ A we nave to prove

A ( ~ I + ( 1 - ~.)x2) >~min(A(x~), A(x2)),

V(xt, xe) e X 2, ¥~. ~[0, 1].

Let xt ~ X, x2 ¢ X, ~ ~ [0, 1] and c --- min(A(xt), A(x2)). For every e > 0 we have A ( x t ) > c - e and A(x2) > c - c; hence

~uo(R(xl, y ) [ y ¢ Y} > c - e and

sup{R(x~, y ) [ y e Y} > c - e.

Th~se inequalities imply, by characterization of supremu~,~, that c - e is no longer an upper bound. Hence there exist y~ ¢ Y, y~ ¢ Y such that R(x:, y~) > c - e and R(x.~,y~)>c--e, from which min(R(x,,y~), R ( x : , y ~ ) ) > c - t . Due ,,o the convexity of ~ we find R ( ~ , + (1 - g)x:,, gy[ + (1 - A)y~) > c and hence sup{~(~X 1 + (I

-

-

~)X 2, y) l Y ¢ ~ ) > C *-~E.

02

E. Kerre

Taking limits as e ~ 0, we finally obtain A ( ~ + (1 - ~.)x2) ~ min(A(xl), A(x2)). Properff 2. The direct image o f a convex f u z z y set A on X under a convex f u z z y relation R f r o m X to Y is a convex f u z z y set on Y.

Once again, using a limit procedure as above, one can give a correct proof of this property. From R(dom(R))=ran(R),. i.e. the direct image of the domain of a binary fuzzy relation R under R is equal to its range, we immediately find from Property 1 and 2: ComlimT. A convex f u z z y relation has a convex range as well as a convex domain.

A second example is taken from fuzzy topology. At the ICM 78 [10] I gave a talk concerning the characterization of a Chang fuzzy topology by means of preassigned operations such as the interior and the closure operator. On that occasion I raised the problem concerning the failure of characterization by means of neighbourhoods of fuzzy singletons. A few years later I read a paper of Pu and Liu [11] in which they claim that given a f~rnity of neighbourhoods satisfying the generalized classical properties. (N.1) every fuzzy singleton has at least one neighbourhood and every fuzzy singleton is contained in f:ach of its neighbourhoods; (N.2) the intersectioip of two neighboarhoods of a fuzzy singleton is a neighbourhood; (N.3) each superset c,f a neighbourhood of a fuzzy singleton is a neighbourhood; one can construct a unique fuzzy topology by defining an open fuzzy set as a f~zy set that is a neighbourhood of each of the fuzzy singletons it contains. The proof was indicated as straightforward. My doubt was increasing until we constructed the following counterexample, that contradicts the assertion of Pu and Liu. Counterexamp|e. Let X be a set with at least two elements and s a fuzzy singleton in X. We define the following system: Us =

{A j A ~ [0, II x and s c_A} {X}

if (Vx ~ X ) ( s ( x ) < 1), if (:ix ~ X ) ( s ( x ) = 1).

It is ~ matter of direct verification that the system U~ for each fuzzy singleton s in X, satisfies the conditions (N.1), (N.2) and (N.3). Next, following Pu and Liu, we define ~:h¢class: r = { 0 [ 0 ¢ [0, 1]x and (Vs c_ 0)(0 ~ Us)}. We will now show that z does not constitute a fuzzy Chang topology on X. In

Crispness in fuzzy set theory

63

order to do this, we construct a family of elements of 1: such that its union no longer belongs to T. Let a ¢ X and n e N* \ {1}. we define f! X'-Q,

O,,'X--[O, 1],

x - - J ~'

L1

1

VxeX\{a}.

n

Using Zadeh's definition of union we find c½,

U o . : x - , [ o , q, neN*\{l}

X ~a~

x ~ 1 1, Vx e x \ {a).

One easily verifies

(Vn e ~*\{1})(0. e T),

U

ncN*\{l}

O.~T,

because this union is not a neighbourhood of a fuzzy singleton with support different from (a} and value 1. A paper of Oottwald [14] contains other examples of false propositions on fuzzy Chang topologies. My last example is taken from the L-fuzzy real line as described by Rodabaugh [5]. This paper concerns galley proofs; the published version has been corrected. The L-fuzzy real line R(L), where L is a chain, is defined as the set of all equivalence classes [~.] where S e L a is a decreasing (not necessarily strictly) mapping from R into L, satisfying the boundary conditions: sups = 1 and inf ~. = O. Two s,Jch mappings $,, ~, are called equivalent iff

(Vt e R)(inf ~.,(s) \$
=

inf ~.2(s) and sup ~,(s) S
=

sup ~.2(s)).

$>t

$>t

One easily verifies the following properties: (1) Every element [~] of the L-fuzzy real line contains a left continuous element and a right continuous one. However, since every k is decreasing we have: left continuity is equivalent with upper-semicontinuity and right-continuity with lower-semicontinuJty. These special elements are obtained as:

~q(t) = inf ~(s),

Vt e R,

~.,(t) = sup ~(s),

$
Vt e R.

$>t

(2) Let o~e L, [~] e R(L). Then we have:

( 3 . a ( ~ . . ) e ~)(Vt e R)(t > a ( ~ . . ) ¢~ ~.(t) < .'), (3,b()~, oc) e R )(Vt e ff~)(t < b()~, ol) ¢:~)~,(t) > oO, where ' indicates an order-reversing involution on L and/~ denotes the extended real numbers.

64

E. Kerre

(3) Equality of two L-fuzzy real numbers: [A] = [u] ~

('¢0~e L)(a(A, o0 =

a(u, u) and b(A, o:)= b(u, u)).

In Rodabaugh [5] the proof cf the sufficiency conditions for (3) ~,uns as follows. From o~=inf,<,~,(r) and the left continuity of A, i.e. A(t)= ¢, Rodabaugh

deduces a(A, o~')= t. The following simple counterexample contradicts this assertion

1],

I,

Vte ]-oo, -I],

t 1 -~+~,

V t e ] - 1, 0],

1

t--,, -

Vt e ]0, 1], t

-~+I,

V t e ] l , 2l,

0,

vt ]2, + %

For t = 0 we find inf A(r)

r
-

½- ~.

However a(A, o r ' ) - a(A, ½)- 1, hence a(A, a:')~ t. Nevertheless, the result as stated is valid, the proof however being more complicated. For a correct proof we refer to [12]. 5. Co,tielusion Education will play a central role in making fuzzy set theory a more popular and widely known instrument to solve problems. Good education needs wellfounded, clear-cut b~sic notions. After twenty years of fuzzy research, time has e ' ~ e to dra~, up an inventory, to trim up the great amount of fuzzy material in order t~: create a stTovg unified framework. The fuzzy community has to think about ~ e material available, obtaining a consensus about the labels that should be use J in the future: should we label a convex fuzzy set on I~ as a fuzzy real number or should we take another definition for this basic concept? Maybe it would be a good idea to have some kind of pedagogical working group within the IFSA community. For pedagogical reasons the leading journal Ft:~.zy Sets and Systems needs papers describing the state of art on different domains, for example on the p:omising field oi expert systems, as well as papers of a comparative or synthesizing nature. Finally I wish to thank my numerous students, especially P. Ottoy, whose enthusiasm forced me to deepen these fuzzy domains that fell outside my research area.

Crispness in fuzzy set theory

65

[1] M. Mizomoto and K. Tanaka, Some properties of fa,7.zynumbers, in: M. Gupta, R. Ragade, R. Yager, Eds., Advances in Fuzzy set Theory and Applications (North-Holland, Amsterdam, 1979) 153-164. [2] D. Dubois and H. Prade, Operations on fuzzy numbers, lnternat. J. System Sci. 9 (1978) 613-626. [3] D. Dubo'_o and H. Prade, Fuzzy real algebra: some results, Fuzzy Sets and Systems 2 (1979) 327-~48. [4] B. Hutton, ~ugmality in fuzzy topological spaces, J. Math. Anal. Appl. 50 (1975) 74-79. [5] S. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fu::zy Sets and Systems (1981) galley proofs. [6] R. Lowen, On (8~(L), ~), Fuzzy Sets and Systems 10 (1983) 203-209. [7] A. Kaufmann and M. Gupta, lnt.,oduction to Fuzzy Arithmetic (Van Nostrand Rheinhold. New York, 1986). [8] ~. Kerre and P. Ottoy, On the different notions of neighbourhood in Chang-Goguen fuzzy t¢,~pological spaces, Simon Stevin 61 (1987) 131-146. [9] S.S.U. Chang, On risk and decision making in a fuzzy environment, in: L. Zadeh, Ed., Fuzzy Sets ~,~ndtheir Application to Cognitive and Decision Processes (Academic Press, New York, 1975). [10] E. Kerre, Fuzzy topologizing with preassigned operatiow,, International Congress of Mathematicians, Helsinki (1978). [11] P.M. Pu and Y.M. Liu, Fuzzy topology I, J. Math. Anal. Appl. 76 (1980) 571-599. [121 S. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Sy~tems 8 (1982) 39-52. 131 L.A. Zadeh, Fuzzy Sets, Inform. and Control $ (1965) 338-353. [:4] S. Gottwald, Fuzz)' points and local properties of fuzzy topological spaces, Fuzzy Sets and Systems $ (1981) 199-202. [151 C.L. Chang, Fuzzy topc~|ogical spaces, J. Math. Anal. Appl. 24 (1968) 182-190