Fuzzy Sets and Systems 126 (2002) 177–190www.elsevier.com/locate/fss

A probabilistic de#nition of a nonconvex fuzzy cardinality

Miguel Delgado∗, Daniel S)anchez, Mar)+a J. Mart)+n-Bautista, Mar)+a Amparo VilaDepartment of Computer Science and Arti�cial Intelligence, University of Granada, Avda. Andalucia 38, 18071 Granada, Spain

Received 24 August 1999; received in revised form 30 November 2000; accepted 8 January 2001

Abstract

The existing methods to assess the cardinality of a fuzzy set with #nite support are intended to preserve the properties ofclassical cardinality. In particular, the main objective of researchers in this area has been to ensure the convexity of fuzzycardinalities, in order to preserve some properties based on the addition of cardinalities, such as the additivity property. Wehave found that in order to solve many real-world problems, such as the induction of fuzzy rules in Data Mining, convexcardinalities are not always appropriate. In this paper, we propose a possibilistic and a probabilistic cardinality of a fuzzyset with #nite support. These cardinalities are not convex in general, but they are most suitable for solving problems and,contrary to the generalizing opinion, they are found to be more intuitive for humans. Their suitability relies mainly on thefact that they assume dependency among objects with respect to the property “to be in a fuzzy set”. The cardinality measuresare generalized to relative ones among pairs of fuzzy sets. We also introduce a de#nition of the entropy of a fuzzy set byusing one of our probabilistic measures. Finally, a fuzzy ranking of the cardinality of fuzzy sets is proposed, and a de#nitionof graded equipotency is introduced. c© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy cardinality; Fuzzy relative cardinality; Fuzzy entropy; Equipotency of fuzzy sets

1. Introduction

Measuring the cardinality of a fuzzy set with #nitesupport (fuzzy set from now on) is a necessary task inmany approximate reasoning problems, such as fuzzyquerying in databases, expert systems, evaluation ofnatural language statements, aggregation, decision-

∗ Corresponding author. Tel.: +34-958-244018; fax: +34-958-243317.

E-mail addresses: mdelgado@goliat.ugr.es (M. Delgado),daniel@decsai.ugr.es (D. S)anchez), mbautis@decsai.ugr.es (M.J.Mart)+n-Bautista), vila@decsai.ugr.es (M.A. Vila).

making in fuzzy environment, etc. (see [14,1,3,2]),where natural language sentences are modeled by us-ing a fuzzy representation of imprecise terms.In the #eld of Data Mining, one of the most im-

portant tasks is to discover statistically signi#cant as-sociations among the occurrence of data items in therecords of a database. To do this, counting items isnecessary, i.e., we must #nd the cardinality of the setof records containing each item, together with the rel-ative count of one item with respect to other items.On many occasions, the number of diGerent items

in the database is so high that it is very diHcult to#nd statistically signi#cant associations among them.

0165-0114/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(01)00039 -2

178 M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190

The usual solution is to cluster the items and then #ndassociations among the clusters. Each cluster is a set ofitems that is considered as a “higher-order item” witha clear semantic interpretation. Then, for each clusterwe must perform a count of the number of recordscontaining an item of the cluster, in order to obtainthe cardinality of the cluster.It is widely accepted that in many occasions, fuzzy

clustering is the most natural way to cluster do-mains. A typical example is the fuzzy partitioningof the domain “Age” by the set of fuzzy sets “Veryyoung”, “Young”, “Medium”, “Old”, “Very old”. Insuch cases, the clusters are fuzzy sets of items, andwe have to #nd statistical associations among “fuzzyhigher-order items”. Therefore, measuring the cardi-nality of such fuzzy sets is crucial in order to solvethe problem. When the cardinality is measured, weare able to obtain a solution by calculating the accom-plishment degree among the cardinalities and fuzzyquanti#ers. This #nal step is known as the evaluationof quanti#ed sentences of type I and II.These quanti#ed sentences are assertions about the

number or percentage of objects that verify a certainproperty. We study two kinds of sentences, called typeI and type II sentences, respectively. Examples of suchsentences are “Most of the students are tall” or “Al-most all the intelligent students are tall”. In the #rstcase, the set of objects in which we evaluate the fuzzyproperty is crisp. In the second case, the set is a fuzzyset. The quanti#ers are fuzzy quantities (fuzzy setsover the non-negative integers) or fuzzy percentages(fuzzy sets over the real interval [0, 1] such as “Most”and “Almost all” in the previous examples). The exist-ing methods for evaluating quanti#ed sentences havesome drawbacks that could be avoided with what wehave called the cardinality approach. We evaluate thedegree of ful#llment of type I (resp. type II) sentencesas the compatibility degree between the fuzzy quanti-#er and the fuzzy cardinality (resp. fuzzy relative car-dinality) of the fuzzy set that represents the property.Several authors have proposed ways to measure the

cardinality of a fuzzy set, extending the classic onein diGerent ways (see [13]). The most common ap-proaches are the scalar cardinality and the fuzzy car-dinality of a fuzzy set. The #rst approach claims thatthe cardinality of a fuzzy set is measured by means of ascalar value, either integer or real, whereas the secondapproach assumes the cardinality of a fuzzy set is just

another fuzzy set over the non-negative integers. Thede#nitions by De Luca and Termini [6], Ralescu [9],Dubois and Prade [1] and Wygralak [14,16,17] areto be found in the #rst category. Many authors havesuggested, however, that the second approach is moreappropriate (see [1,14]). In this category are the def-initions by Zadeh [21,22], Dubois and Prade [1] andWygralak [14,18].These extensions of the classical cardinality theory

are intended to preserve as many classical propertiesas possible. Two of the main properties of the classicalcardinality of a set are the valuation, i.e.

Card(A) + Card(B) = Card(A ∪ B) + Card(A ∩ B)

(1)

and the additivity property, de#ned as

A ∩ B = ∅ ⇒ Card(A) + Card(B) = Card(A ∪ B):

(2)

Wygralak and Pilarski have shown that additivity,based on the extension principle, is veri#ed by at-norm-based generalized fuzzy cardinality if it isconvex [18]. Convexity of a fuzzy measure M can bede#ned as

M is convex

⇔ (x6y6 z ⇒ M (y)¿ min(M (x); M (z)):

(3)

Also in [18] it is shown that t-norm-based generalizedfuzzy cardinalities verify the valuation property (basedon the extension principle) if and only if standardintersection and union are used.One of the claims we make in this paper is that

in general convexity is not a natural property for thefuzzy cardinality of a fuzzy set. We noticed this forthe #rst time when we were using convex cardinali-ties to evaluate quanti#ed sentences on real cases. Inmany experiments we realized that the results obtainedusing methods based on convex cardinalities werenot always coherent with the intuitively expected cor-rect results. These conclusions motivated us to searchfor cardinalities that were better suited to real-lifeproblems.A very simple example can help us to clarify our

claims. Let X = {John; Mike; Peter} be a set of

M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190 179

friends, and let us suppose that John is blonde, andMike and Peter are fairly blonde. We could say thatJohn is in the set of blonde people with degree 1 andMike and Peter are in the same set with degree 0.5.Suppose we want to buy a cap for every blonde

person in X . Deciding how many caps we shall buyis equivalent to obtaining the cardinality of the fuzzyset XBlonde = 1=John+0:5=Mike+0:5=Peter. If we arestrict with the concept of blonde, we shall buy onlyone cap (for John). But if we relax our criterion wecould think of buying three caps, one for each friendin X . What should be clear is that buying two capsis not a solution for the problem, because as John iscertainly blonde one of the caps should be for him.What can we do then with the other cap, given thatMike and Peter are equally blonde? In other words, thecardinality of XBlonde can be one or three, but not two,because Mike is in XBlonde if and only if Peter also is.Hence, the possibility that the cardinality of XBlonde

is two is 0. But convex cardinalities do not agree withthis. Any convex fuzzy cardinality will give a possi-bility greater than 0 that “two” is the cardinality ofXBlonde, because clearly the possibility that the cardi-nality is “one” or “three” are both greater than 0, and aconvex fuzzy cardinality veri#es (3). For instance, theconvex cardinality FECount(XBlonde)= 0=0 + 0:5=1 +0:5=2 + 0:5=3. In this sense, we #nd convexity to becounterintuitive in some cases.This problem aGects the evaluation of quanti#ed

sentences by means of the cardinality approach. Forinstance, the evaluation of the sentence “Q2=3 of peopleinX are blonde” (whereQ2=3 = {1=2=3} is the quanti#er“exactly 2=3”) should be 0, as the cardinality of XBlondecannot be two. But the result of the evaluation of thissentence by using the cardinality approach with anyfuzzy cardinality C is⊕

{C(0)� Q2=3(0); C(1)� Q2=3(1=3);

C(2)� Q2=3(2=3); C(3)� Q2=3(1)} = C(2);

where ⊕ and � are a t-conorm and a t-norm, respec-tively (usually, maximum andminimum for possibilis-tic cardinalities). Hence, if the fuzzy cardinality C isconvex, the evaluation of the sentence will not be theintuitively expected zero value.Related to the problem of the cardinality of a fuzzy

set is that of the relative cardinality of a fuzzy setF with respect to a fuzzy set G (i.e. the percent-

age of objects in the fuzzy set G that are also in thefuzzy set F). A scalar measure was described in [20].Fuzzy measures are de#ned in [22,1]. Similar exam-ples of non-intuitive results can be detailed for typeII sentences with relative cardinalities if convexity isrequested.In the #rst part of the paper (Sections 2 and 3), we

brieQy review the existing cardinality measures. Wealso discuss the convexity and its relation to the addi-tivity and valuation properties. In the second part, wepropose both a probabilistic and a possibilistic mea-sure of the relative cardinality of fuzzy sets, and wetackle two related problems: the entropy of a fuzzy setand the equipotency of fuzzy sets.

2. Some existing measures of the absolute andrelative cardinality of a fuzzy set

Let F and G be two fuzzy sets de#ned overX = {x1; : : : ; xn}. Letfj and gj be the jth greatest valueof the multisets {F(xi) | xi ∈X } and {G(xi) | xi ∈X },respectively, and let f0 = g0 = 1. Also let fm= gm=0∀m¿n.

2.1. Scalar measures

2.1.1. Power of a fuzzy setThis measure was introduced by De Luca and

Termini in [6] to be

P(F) =n∑

i=1

F(xi): (4)

The power is also called the �-count, and it is anexample of an energy measure of a fuzzy set (see [7]).The authors propose to use P(F) as a scalar cardinalfor fuzzy subsets. Themain drawback of this use is thatfor fuzzy sets with high support when the values F(xi)are very low, P(F) may be great enough to ensure Fis “big”, which is somewhat counter-intuitive.

2.1.2. Ralescu’s cardinality measureThe following method was proposed by Ralescu in

[9]:

ncard F =

0 F = ∅;j F = ∅ and fj ¿ 0:5;

j − 1 F = ∅ and fj ¡ 0:5;

(5)

180 M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190

where j= max{16 s6 n |fs−1+fs¿1}. Ralescu’sidea is to provide an integer value as the scalar car-dinality of a fuzzy set (the power is in general a realnumber), and to avoid the problem of accumulatinglow values to give a high cardinality. Ralescu alsoshows that

ncard F = |F0:5|; (6)

where F� is the �-cut of F .

2.1.3. Wygralak’s cardinality measureThis cardinality measure is described in [14]. The

motivation for this de#nition is the same as Ralescu’smethod. Wygralak proposes one of the values of thefollowing integer interval as the scalar cardinality

WS(F) = [|F0:5|; |F0:5|]; (7)

where

F0:5 = {x ∈ X |F(x)¿ 0:5}is the strong �-cut of F at level 0.5, and

F0:5 = {x ∈ X |F(x)¿ 0:5}is the �-cut of F at level 0.5. Wygralak also givesmathematical and intuitive reasons for why the val-ues |F0:5| and |F0:5| are preferred as the best integerapproximations to the cardinality of F .Recently, Wygralak has proposed an axiomatic ap-

proach for the de#nition of scalar measures [16,17].Following this approach, a function sc is a scalar car-dinality iG

sc(F) =∑x∈X

f(F(x))

with f : [0; 1]→ [0; 1] such that f(0)= 0; f(1)= 1and f(a)6 f(b) if a6 b. Some examples of scalarcardinalities of this kind are provided in [16,17]. Par-ticular cases are |F0:5| and |F0:5|, and in general any�-cut or strong �-cut of F .

2.1.4. Dubois and Prade’s cardinality measureDubois and Prade [1] propose a real interval of val-

ues as possible scalar cardinalities of F , the boundsof the interval being the lower and upper expec-tation for the cardinality. The proposed interval isDPs(F)= [|F1|; P(F)].

2.2. Fuzzy cardinality measures

2.2.1. Zadeh’s �rst de�nitionIn [21], Zadeh introduces a fuzzy measure for the

cardinality of a fuzzy set F to be

Z(F; k) = sup{� | |F�| = k}: (8)

This de#nition is based on the representation theoremof fuzzy sets based on �-cuts. The main problem thathas been attributed to this method is that the additivityproperty of classical cardinality (based on the additionby means of the extension principle) is lost, as Z(F)is not a convex fuzzy set.

2.2.2. Zadeh’s FECount(F)To avoid the nonconvexity of Z(·), Zadeh proposed

the fuzzy cardinality measure FECount(F) in [22],de#ned as

FECount(F) = FGCount(F) ∩ FLCount(F); (9)

where FGCount(F; k)= sup{� | |F�| ¿ k} can be in-terpreted as the possibility that the cardinality of F isat least k. Also

FLCount(F; k) = FGCount(F; k)− 1;

where the bar stands for the standard complement, canbe interpreted as the possibility that the cardinality ofF is at most k. It is easy to see that the possibility of thecardinality of F being at least k is FGCount(F; k)=fk

with f0 = 1.An equivalent expression for the FECount measure

was introduced by Wygralak in [11,12,15]. It is easyto show that FECount(F) can be expressed as

FECount(F; k) = min(fk; fk+1); (10)

where fk is the kth greatest value of F(xi) and the barrepresents the standard fuzzy negation. Starting froma diGerent approach, Ralescu [9] has reached the samede#nition.

2.2.3. Dubois and Prade’s fuzzy cardinalityThis measure is de#ned in [1] as

DP(F) = sup{*F(S) | S ⊆ X and |S| = k}; (11)

M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190 181

where

*F(S) =

{inf{F(x) | x ∈ S} if S ∈�(F);0 if S ∈ �(F) (12)

and �(F)= {S |F1⊆ S ⊆Support(F)}. It is easy tosee that

DP(F; k) =

{0 if k¡|F1|;fk otherwise:

(13)

SoDP(F) can be obtained from FGCount(F) by turn-ing the values that are below the lower expectationfor the cardinality (in the sense of 2:1:4) to be 0. Themotivation behind this method was that FLCount(F)was obtained from FGCount(F) in a way that is notcoherent with the possibility theory. This cardinalityis also convex.

2.3. Scalar relative cardinality

The relative cardinality of a set F with respect toa set G is de#ned as the percentage of elements in Gthat are also in F . When F and G are fuzzy sets, bothscalar and fuzzy measures for the relative cardinalityhave been introduced, although the most used measureis the scalar measure de#ned in the next subsection.

2.3.1. Zadeh’s scalar relative cardinalityThe relative cardinality of F with respect to G is

de#ned in [20] as

Card(F=G) =P(F ∩ G)

P(G): (14)

2.4. Fuzzy relative cardinality

As is the case of the cardinality of a fuzzy set, therelative cardinality of fuzzy sets is considered by mostauthors as being a fuzzy set over [0, 1] rather than ascalar value.

2.4.1. Zadeh’s FGCount(F=G)The fuzzy measure called FGCount(F=G) was de-

#ned by Zadeh in [22] as a fuzzy multiset over [0, 1]as follows:

FGCount(F=G) =∑

�∈+(F)∪+(G)

�/

P(F� ∩ G�)P(G�)

;

(15)

where +(F) and +(G) are the level set of F and G,respectively.

2.4.2. Dubois and Prade’s fuzzy relative cardinalityThis measure was proposed in [1] following the

approach of (11). It is de#ned for every q∈ [0; 1]∩Qto be

DP(F=G; q)

= sup{min(*F(S); *G(T ))

∣∣∣∣ |S ∩ T ||T | = q

}(16)

with S ∈�(F); T ∈�(G).

3. Convexity, additivity and valuation

3.1. The convexity problem

We agree with the idea that the cardinality of afuzzy set is also a fuzzy set, but as we explained in theintroduction, we think that this fuzzy set ought not tobe convex in the general case. The example describedin Section 1 illustrates our claim. The aHrmation thatthe cardinality of XBlonde can be one or three but nottwo could seem strange in a #rst view. But all thepeople we asked about “how many caps would youbuy?” have answered “one or three”. All of them found“two” not to be a valid answer. In this sense, andat least from a practical point of view, it is actuallyconvexity that seems to be counterintuitive.When extending scalar operations on classical sets

to fuzzy operations on fuzzy sets, we usually repre-sent the fuzzy set by means of a weighted set of clas-sical sets. Once this step has been made, we performthe operations over the elements of the representationand we obtain a weighted collection of results, whichcan be interpreted as a fuzzy set. One example of thiskind of procedure is the well-known extension princi-ple (see [22], appendix). A further step can be to ob-tain a scalar value that summarizes the information ofthe fuzzy result, following some criterion (see [2] forexample). In this procedure, one of the most importantissues is the choice of the classical sets that will repre-sent the original fuzzy set, together with their weightin the representation.There are several representation methods, the most

well-known being Zadeh’s representation theorem.

182 M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190

This is a possibilistic representation where the fuzzyset F is represented as the set {F�} of �-cuts of F , �being the weight of F� in the representation. Zadeh’s#rst method is based on this representation. Duboisand Prade [1] de#ne another possibilistic representa-tion of F as the set �(F) with *F(S) (see (12)) beingthe weight for every S ∈�(F). They use this rep-resentation to obtain their fuzzy cardinality measuredescribed in Section 2.2.3. A probabilistic representa-tion is also proposed by Dubois and Prade in [2] usingthe set �(F) and assuming F is normalized. Finally,Wygralak uses the set P(X ) of subsets of X as therepresentation, *F(A) again being the weight for everyA∈P(F). Following this representation, the possiblecardinalities for XBlonde are Ws(XBlonde)∈ [1; 3].In our opinion, the representations must take into

account the dependency among objects with respectto the property “to be in F”. Regarding the examplein Section 1 and following both �(XBlonde) and P(X )approaches, the set {John, Mike} is a valid represen-tative of XBlonde, but with respect to the property “xbelongs to XBlonde” we think it is not, because if Mike∈XBlonde then Peter ∈XBlonde. We can see that the �-cut representation of XBlonde is {{John}, {John, Mike,Peter}}. We think that only representations based on�-cuts preserve the dependency among objects fromthe point of view of cardinality. DiGerent weightingof the �-cuts are possible. As a consequence, the onlyvalid values for the cardinality of a fuzzy set arethe cardinalities of its �-cuts. However, this approachturns convexity to be unneeded.

3.2. Convexity and the extension principle

Given two fuzzy sets F and G over N; F + G canbe obtained by using the well-known extensionprinciple as

(F + G)(i) = maxi=a+b

min(F(a); G(b)): (17)

The extension principle ensures that the addition offuzzy subsets of nonnegative integers, whether con-vex or not, is a convex fuzzy integer. Hence, addingnonconvex fuzzy cardinalities by means of the exten-sion principle, we obtain a convex cardinality, andtherefore it is not possible in general for a noncon-vex fuzzy cardinality to verify the additivity property.Let us consider the following simple example. Let

Y = {Mary; Susan; Katy} and let YBlonde = 1=Mary+0:5=Susan+0:5=Katy. Obviously, XBlonde ∩YBlonde = ∅,so for any fuzzy cardinality C, ifC veri#es the additiv-ity then C(XBlonde)+C(YBlonde)=C(XBlonde ∪YBlonde).If C were not convex (in the sense that only car-

dinalities of �-cuts were allowed) then C(XBlonde),C(YBlonde) and C(XBlonde ∪YBlonde) should not beconvex (the only possible values for the cardinalityare 1 and 3 for XBlonde and YBlonde, and 2 and 6 forXBlonde∪YBlonde). ButC(XBlonde)+C(YBlonde) obtainedby means of the extension principle is convex, andhence C(XBlonde) + C(YBlonde) =C(XBlonde ∪YBlonde).Therefore, if C is not convex then the additivity (andhence the valuation property), as based in the ex-tension principle, does not hold. Moreover, it couldhappen that C(XBlonde)+C(YBlonde) does not representthe cardinality C(H) for any fuzzy set H , so that itcould seem that C is not well-de#ned from the opera-tional point of view. However, all these assertions arebased on the use of the extension principle for addingfuzzy cardinalities. If we consider the extension prin-ciple to be the axiomatic reference for the addition,we must agree that C neither verify additivity andvaluation nor it is well formed.Even for convex cardinalities, using the exten-

sion principle under Zadeh’s original formulation(17) can oGer counterintuitive results. For example,the cardinality FECount(XBlonde)= 0=0 + 0:5=1 +0:5=2 + 0:5=3, and obviously FECount(XBlonde)=FECount(YBlonde). Hence, one would expect thefollowing: FECount(XBlonde) + FECount(YBlonde)= 2FECount(XBlonde)= 2FECount(YBlonde) (here2FECount(F)= 1{2}FECount(F), where 1{i}, i∈N,is a fuzzy set over N such that 1{i}(x)= 1 iG x= i,and 0 otherwise). But by using the extension principlefor both the addition and product (the latter can beperformed by replacing + with × in (17)), we obtain

FECount(XBlonde) + FECount(YBlonde)= 0=0 + 0=1 + 0:5=2 + 0:5=3+0:5=4 + 0:5=5 + 0:5=6

and

2FECount(XBlonde) = 2FECount(YBlonde)= 0=0 + 0=1 + 0:5=2 + 0=3+0:5=4 + 0=5 + 0:5=6

and they are diGerent!

M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190 183

With an alternative and suitable de#nition of ad-dition for integer fuzzy cardinalities, this problemcan be solved. Wygralak [15] has proposed a mod-i#ed extension principle that works for the convexcardinality FECount, ensuring

∑ni=1 FECount(F)=

nFECount(F) for any F . In our example and follow-ing [15]

2FECount(XBlonde)

= FECount(XBlonde) + FECount(XBlonde)

= 0=0 + 0=1 + 0:5=2 + 0:5=3 + 0:5=4

+0:5=5 + 0:5=6: (18)

However, we think that the addition of a (fuzzy) car-dinal number with itself should have no odd naturalnumber in its support. But regarding (18), there is anonzero possibility that 2FECount(XBlonde) is 3 or 5.In our opinion, that is counterintuitive, and it is an-other reason why nonconvex cardinalities make sense.Now we are studying the addition for nonconvex car-dinalities. Some proposals to perform a suitable addi-tion will be dealt with in future papers.

4. A new de!nition for a cardinality measure

We start this section by de#ning a family of fuzzycardinality measures based on the evaluation of fuzzylogic sentences. This approach is followed, for exam-ple, by Wygralak [13–15].

4.1. The family E of measures

De!nition 4.1. The possibility that at least k elementsof X belong to F , L(F; k), is the evaluation of thelogical sentence “∃X1⊆X | |X1|= k and X1⊆F”, withX1 a crisp set

L(F; k)=1 if k = 0;0 if k ¿n;⊕

Ik{⊗

{F(xi1 ); : : : ; F(xik )}} if 16 k6 n;

(19)

where Ik is the set of k-tuples of indexes ij withij ∈{1; : : : ; n} ∀j∈{1; : : : ; k} de#ned byIk = {(i1; : : : ; ik) | i1¡i2¡ · · · ¡ik} (20)

and⊕ and⊗ are a t-conorm and a t-norm, respectively.We note L to be the family of functions L(F; k) foreach pair of t-conorm and t-norm.

Proposition 4.2. Let ⊕ be the maximum and ⊗be the minimum. Then L(F; k)=fk and thereforeL(F)=FGCount(F).

Proposition 4.3. Every measure of the family L iswell-de�ned in the crisp case; i.e. if F is crisp thenL(F; k)= 1 i= |F |¿k.

De!nition 4.4. The possibility that exactly k elementsof X belong to F; E(F; k); is

E(F; k) = L(F; k)⊗ L(F; k + 1) ; (21)

where ⊗ is any t-norm (not necessarily the same oneused in L) and the bar stands for a fuzzy complement.

For each set of t-conorm, t-norms and complement,the expression (21) de#nes a family of fuzzy cardinal-ities that we denote E. A direct interpretation of (21)is that “the cardinality of F is k if it is “at least k” andit is not “at least k+1”. (21) is also the evaluation ofthe logic sentence

“∃X1⊆X | |X1|= k and X1⊆F and F ⊆X1”

i.e. the evaluation of the sentence “∃X1⊆X | |X1|= kand X1 =F”. It can be seen that in the evaluation ofthese sentences, every crisp subset X1⊆X with car-dinality k is considered. We might #nd that some ofthese subsets are not �-cuts of F , and this can be aproblem on account of that which we discussed inSection 3. As we shall see in Section 4.2, at least onemeasure in the family E exists such that E(F; k) =0if and only if there exists �∈ [0; 1] such that |F�|= k.

Proposition 4.5. Every measure of the family E iswell-de�ned in the crisp case; i.e. if F is crisp thenE(F; k)= 1 if |F |= k.

Proposition 4.6. The measure FECount(F) de�nedby (9) is a member of the family E.

Proof. We only need to de#ne L as in Proposition4.2 by using max and min, and using the standard

184 M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190

complement (n(x)= 1− x) and the t-norm min in theexpression (21). We then obtain E(F; k)=fk ∧fk+1.

Proposition 4.7. The measure DP(F) de�ned by(11) is a member of the family E.

Proof. Let us de#ne L as in Proposition 4.2 by usingmax and min, and using the fuzzy complement

c(x) ={1 if x¡ 1;0 if x = 1

and the t-norm min in expression (21). Let |F1|= h.We shall consider two cases:(1) Let k¡h. Then by (13), we have DP(F; k)= 0.

We also have L(F; k)=L(F; k + 1)=1 and thenL(F; k + 1)=0, and henceE(F; k)= 0=DP(F; k).

(2) Let k¿h. Then DP(F; k)=fk and L(F; k+1)¡1so L(F; k + 1)=1 and hence E(F; k)=L(F; k)=DP(F; k)=fk .

Let us remark that our objective is not to deepen inthe properties of every cardinality in the family, butto de#ne a general framework in which we could in-tegrate new and existing de#nitions. In Section 3 weclaimed that fuzzy cardinalities should not be convexin general, and that it is intuitively true that the onlypossible values for the cardinality for a fuzzy set arethe cardinality values of its �-cuts, from which weconstruct the fuzzy cardinal. The study of the cardi-nalities of the family E that verify these requirements,together with the discussion of their properties, willbe an object of future research.

4.2. The measure ED

De!nition 4.8. We introduce the fuzzy cardinalitymeasure ED to be

ED(F; k) = fk − fk+1: (22)

Proposition 4.9. The measure ED(F) is the basicprobability assignment of Z(F).

Proof. Let us consider the set of �-cuts of F plus ∅.The possibility that a given F� is a subset of F is �,and an integer k exists with 16k6n such that �=fk .Obviously, the possibility that ∅ is a subset of F is

1=f0. If we consider this representation as a pos-sibility distribution over a collection of nested sets,then the diGerences fk − fk+1 can be interpreted asthe basic probability assignment to every �-cut of F ,in the sense of Dempster–Shafer’s theory of evidence.Hence, ED can be obtained from a representation of Fas the collection of its �-cuts, weighted with the basicprobability. It is only necessary to assign the proba-bility that it is a valid representation of F to the car-dinality of every �-cut.

Corollary 4.10. ED(F; k)¿0 if and only if ∃�∈[0; 1] such that |F�|= k.

Corollary 4.11. ED(F) is not a convex fuzzy set ingeneral.

A very similar measure is de#ned by Dubois andPrade [3,2]. The main diGerence is that Dubois andPrade’s measure requires the set F to be normalized.We do not impose this requirement because if F is notnormalized, the possibility that the set Ff0 =F1 = ∅ isa subset of F is known to be 1, so its basic probabil-ity assignment is f0−f1 = 1−f1. Another diGerenceis that this representation only considers �-cuts as theset of crisp representatives of F , while Dubois andPrade’s representation considers that the set of repre-sentatives of F is �(F). In any case, the basic proba-bility assignment of a set S ∈�(F) such that S is notan �-cut is 0.

Proposition 4.12. The measure ED is a member ofthe family E.

Proof. Using max and min in L and standard negationand Lukasiewicz’s t-norm t(a; b)= max{0; a+b−1}in (21).

Proposition 4.13. Let Fc be the standard comple-ment of the set F with respect to X . The measure EDveri�es

ED(Fc; k) = ED(F; n− k): (23)

Proof. It is easy to see that fck =1 − fn−k+1 ∀k∈{0; : : : ; n+ 1}. Then ED(Fc; k)=fck − fck+1 = (1−fn−k+1)−(1−fn−k)=fn−k−fn−k+1 =ED(F; n−k).

M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190 185

A scalar value for the cardinality of a fuzzy set canbe obtained as a summary of the information containedin a fuzzy measure. In the case of the probabilisticmeasure ED, the expected value for the cardinalitycan be de#ned as the usual centroid defuzzi#cationapproach of fuzzy control

Ex(F) =n∑

i=0

i ∗ ED(F; i): (24)

A formula similar to (24) was faced in [4]. Also, thesame formula was proposed by Dubois and Prade [2]with the sum starting in i=1. This is due to the factthat they give a probability 0 that the cardinality of Fcould be 0, because they force F to be normalized. Inany case, the term for i=0 is 0 and does not aGectthe #nal result. Moreover, for every subset A∈�(F)that is not an �-cut of F , it has been shown in [2] thatthe probability of A is 0, so the expected result is thesame as Ex(F).

Proposition 4.14. The expected scalar value for thecardinality of F; Ex(F); is the power of F de�ned in(4).

Notice that although Z(F) and ED(F) are not con-vex in general, the summary of ED(F) is P(F), anadditive measure, and ED(F) can be interpreted as thebasic probability assignment of Z(F).

5. New de!nitions for the relative cardinality

Following the representation approach by means of�-cuts, either possibilistic or probabilistic, we are go-ing to propose both a possibilistic and a probabilisticmeasure of the relative cardinality of a fuzzy set Fwith respect to a fuzzy set G.

5.1. Preliminaries

We require the fuzzy set G to be normalized, be-cause otherwise there is a non-zero probability that thecardinality of G is 0, and then the relative cardinalityof F with respect to G is not de#ned. If G is not nor-malized, then we can normalize it in the usual way, bymultiplying by a factor 1=max{G(x)}. We must ap-ply the same factor to the set F ∩G to maintain therelative cardinality of F with respect to G.

The following are some preliminary de#nitions:

De!nition 5.1. We introduce M (F) to be the follow-ing subset of [0; 1]

M (F) = {F(xi) | xi ∈ X }: (25)

De!nition 5.2. We introduce M (F=G) to be the fol-lowing subset of [0; 1]

M (F=G) = M (F ∩ G) ∪M (G): (26)

We will use the following notation: M (F=G)={�1; : : : ; �m} where we assume 1= �1¿�2¿ · · ·¿�m¿�m+1 =0.

De!nition 5.3. We introduce C(F=G; �i) to be the ra-tional value

C(F=G; �i) =|(F ∩ G)�i |

|G�i |: (27)

De!nition 5.4. We introduce CR(F=G) to be the fol-lowing subset of [0; 1]∩QCR(F=G) = {C(F=G; �i) | �i ∈ M (F=G)}: (28)

5.2. A new possibilistic measure of the relativecardinality

De!nition 5.5. We introduce the fuzzy relative cardi-nality ES(F=G) for all q∈Q to be

ES(F=G; q) = max{�i ∈ M (F=G) | q = C(F=G; �i)}:(29)

Proposition 5.6. The measure ES(F=G) is well-de�ned in the crisp case; i.e. if F and G are crisp thenES(F=G)={1=q} where q=C(F=G; 1)= |F ∩G|=|G|.

Proposition 5.7. Let G=X . Then

ES(F=X; k=n) = Z(F; k): (30)

Proof. First, we have M (F=X )=M (F)∪{1}. Asthe �-cut f0 = 1 is always used in the representa-tion of F , then {F� | �∈M (F=X )} is the represen-tation of F in terms of its �-cuts. On the otherhand, we consider that |X�|= n ∀�∈M (F=X ). HenceCR(A=X )={|F�|=n | �∈M (F=X )} and from (29) itfollows ES(F=X; k=n)= �=Z(F; k).

186 M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190

5.3. A new probabilistic measure of the relativecardinality

De!nition 5.8. We introduce the fuzzy relative cardi-nality ER(F=G), q∈Q, to be the random setER(F=G; q) =

∑C(F=G;�i)=q

(�i − �i+1): (31)

Proposition 5.9. The measure ER(F=G) is well-de�ned in the crisp case; i.e. if F and G are crisp thenER(F=G)={1=q} where q=C(F=G; 1)= |F ∩G|=|G|.

Proposition 5.10. Let G=X . Then

ER(F=X; k=n) = ED(F; k): (32)

Proof. Analogous to the proof of Proposition 5.7.

6. Related problems

6.1. Entropy of a fuzzy set

The concept of entropy of a fuzzy set was #rst de-#ned by De Luca and Termini in [6] as a measure ofthe degree of “fuzziness” of a fuzzy set. This de#nitionmakes use of the function S(x) de#ned by Shannonfor every x∈ [0; 1] asS(x)=− x ln(x)− (1− x) ln(1− x) (33)

and assuming 0 ln(0)= 0. The entropy of a fuzzy setF is de#ned in [6] as

d(F) =n∑

i=1

S(F(xi)): (34)

Since then, other de#nitions of the entropy of a fuzzyset can be found in papers, such as the measures pro-posed by Yager [19], Kosko [5], Shang and Jiang [10],Pal and Pal [8] among others. All these measures ver-ify the following properties, proposed in [6], for anentropy measure:(1) The entropy of a fuzzy set F is 0 if and only if F

is a crisp set.(2) The maximum value of the entropy is reached if

and only if F(x)= 0:5 ∀x∈X .(3) For every sharpened version G of F (i.e. G is

a fuzzy set verifying G(x)6F(x) if F(x)60:5

and G(x)¿F(x) if F(x)¿0:5) the entropy of Fis greater than the entropy of G.

A reasonable relation between fuzzy cardinality andentropy of a fuzzy set F is that “the crisper thefuzzy set, the crisper the fuzzy cardinality”, i.e.“the lower the entropy, the crisper (the clearer)the fuzzy cardinality”. Obviously, the reverse re-lation “the crisper the fuzzy cardinality, the lowerthe entropy” must also hold. This could allow usto measure the entropy of a fuzzy set by measuringthe entropy of its fuzzy cardinality. The measuresFECount(F) and DP(F) seems to verify this re-lation, since if F is crisp then both FECount(F)and DP(F) are crisp, and if F(x)= 0:5 ∀x∈X thenFECount(F; k)=DP(F; k)= 0:5 ∀k ∈{0; : : : ; n}.We think that the discussion in Section 3 can be ex-

tended to the relation between fuzzy cardinality mea-sures and entropy measures. It is easy to see that therelation just described between entropy and cardinal-ity is not veri#ed by the measures Z(F) and ED(F).First we propose the following de#nition of the en-tropy of a fuzzy set.

De!nition 6.1. We introduce the entropy of a fuzzyset F to be

Ent(F) = −n∑

k=0

ED(F; k) ln(ED(F; k)): (35)

That is to say, the well-known functionalH (similar toShannon entropy, see [6]) with K =1 over the prob-ability distribution ED(F). As pointed out in [6], thisde#nition does not verify all the properties proposedfor the entropy. But we think that some discussioncould arise over the interpretation of these properties.The entropy of a fuzzy set has also been interpretedby De Luca and Termini in [6,7] as a measure of in-formation in the sense of “the amount of informationwe need to turn the fuzzy set into a crisp one”, i.e. theamount of information we need to decide, for everyx∈X , whether x∈F or not. This is clearly related tothe problem of the fuzzy cardinality of a fuzzy set,and the dependency among objects. Let us considerthe following example:

Example 6.2. Let F =0:5=x1+0:5=x2+0:5=x3+0:5=x4be a fuzzy set and let G=1=x1 + 0:75=x2 + 0:5=x3+

M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190 187

Fig. 1. Fuzzy sets F and G over X .

Fig. 2. Fuzzy cardinalities ED(F) and ED(G).

0:25=x4. Fig. 1 shows graphical representations of thesets F and G:

Following the property (2), F is the fuzziest set overfour objects. But the question is, how much informa-tion do we need to turn F into a crisp set? We onlyneed to decide whether an object with a degree 0.5 isin F or not (we can apply this information to every ob-ject with a degree 0.5). Although we agree that the in-formation needed is the maximum for one object (anyother degree is nearer to 1 or 0 than 0.5), we considerthat in the case of G we need the same information (todecide for x3) and some additional information (to de-cide for x4, etc.). We think that the problem with prop-erty 2 is that we must assume independence amongthe objects with respect to the problem of deciding ifone object is in F or not. Regarding the fuzzy sets Fand G in Fig. 1 we could ask, which set makes us feelmore comfortable when deciding its cardinality? andhence, which set seems to be “crisper” in that sense?In our opinion, “F” is the answer. In fact, the cardi-nalities ED(F) and ED(G) of Fig. 2 tell us that thereare less integer candidates with a high probability ofbeing the cardinality of F .

Following this approach, which is the “fuzziest”fuzzy set? By the properties of function H , the high-est entropy value is reached when the probability isequidistributed. Therefore, if we have n elements, thehighest entropy is reached when ED(F; k)= 1=(n+1)for every k (taking into account the probabilityED(F; 0)). The following proposition de#nes thefuzziest fuzzy set over n objects.

Proposition 6.3. The fuzziest set over n objects; A(n);can be de�ned as

A(n)(xi) = i=(n+ 1): (36)

As an example, if n=4 then A(4) = 0:2=x1+0:4=x2+0:6=x3 + 0:8=x4.

Proposition 6.4. The entropy Ent(F) is 0 if and onlyif F is a crisp set.

Proof. It is trivial in the respect that if F is crisp thenED(F) is also crisp.

Proposition 6.5. The entropy Ent(F) veri�es Ent(F)=Ent( WF).

188 M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190

Proof. Trivial regarding (35) and (23).

Proposition 6.6. Let F be a fuzzy set. We de�nen fuzzy sets {F1; : : : ; Fn} as Fi=F(xi)=xi. HenceF =

⋃{Fi} and i = j ⇒ Fi ∩Fj = ∅. Then

d(F) =n∑

i=1

Ent(Fi): (37)

Proof. We only need to show that Ent(Fi) =S(F(xi)). The entropy of Fi is

Ent(Fi) =−ED(F; 0) ln(ED(F; 0))

−ED(F; 1) ln(ED(F; 1));

where ED(F; 0)= b0−b1 = 1−F(xi), and ED(F; 1)=b1− b2 = b1 =F(xi), so Ent(Fi)= S(F(xi)) andhence d(F)=

∑ni=1 Ent(Fi).

We think that entropy measures which agree withproperty (2) assume independence among objects, andtherefore they obtain the total entropy of the set asthe sum of the entropy of every set Fi. De Luca andTermini agree in [7] that additivity is a strong require-ment which is only valid when independence amongobjects holds. They also agree in the same paper thatsome interesting classes of entropy measures do notverify property 2. However, we agree with them inthat for some problems, additivity can hold or it canbe a reasonable approximation.

6.2. A cardinality-based ranking for fuzzy sets

One of the problems related to the cardinality isthat of the equipotency of fuzzy sets, i.e. decidingwhether two given fuzzy sets F and G have the samecardinality. We can extend this problem to that ofranking the cardinality of two fuzzy sets using therepresentation of a fuzzy set F as a set of �-cuts.We think that the ranking of the cardinalities of

fuzzy sets should be fuzzy, since the cardinality iswidely considered to be fuzzy. In particular, the sameargument can be claimed with respect to the equipo-tency of fuzzy sets, so we should give a fuzzy degreeof equipotency of two fuzzy sets. We will de#ne thefuzzy ranking of the cardinalities of two fuzzy sets as

a fuzzy set over R= {¡;=;¿}. Following our gen-eral criterion, we should defuzzify the fuzzy rankingwhen providing it as a #nal result. Nevertheless, thesize of the set R (|R|=3) allows us to give it as aresult to the user.

De!nition 6.7. We introduce the possibilistic rankingof the cardinalities of F and G for every ∗∈R to be

RPoss(|F | ∗ |G|) = max{�i ∈ M (F=G) | |F�i | ∗ |G�i |}:(38)

De!nition 6.8. We introduce the probabilistic rank-ing of the cardinalities of F and G for every ∗∈Rto be

RProb(|F | ∗ |G|) =∑

�i :|F�i |∗|G�i |(�i − �i+1): (39)

The following de#nitions of the degree of equipo-tency of two fuzzy sets follow from De#nitions 6.7and 6.8.

De!nition 6.9. We introduce the possibilistic degreeof equipotency among two fuzzy sets F and G to be

EqPoss(F;G) = RPoss(|F | = |G|): (40)

De!nition 6.10. We introduce the probabilistic de-gree of equipotency among two fuzzy sets F and Gto be

EqProb(F;G) = RProb(|F | = |G|): (41)

Proposition 6.11. The possibilistic equipotencyEqPoss veri�es

EqPoss(F;G) = 1 ⇔ |F1| = |G1|: (42)

Proof. EqPoss(F;G)= 1 ⇔ max{�i ∈M (F=G) | |F�i |= |G�i |}=1⇔|F1|= |G1|.

Proposition 6.12. The probabilistic equipotencyEqProb veri�es

EqProb(F;G) = 1 ⇔ |F�i | = |G�i | ∀�i ∈ M (F=G):

(43)

M. Delgado et al. / Fuzzy Sets and Systems 126 (2002) 177–190 189

Proof. EqProb(F;G)= 1 ⇔ ∑{�i ∈M (F=G) | |F�i |= |G�i |}

(�i − �i+1)= 1⇔|F�i |= |G�i | ∀�i ∈M (F=G).

Proposition 6.13. The probabilistic equipotencyEqProb veri�es

EqProb(F;G) = 1 ⇔ Z(F) = Z(G)

⇔ ED(F) = ED(G): (44)

Proof. The cardinality Z(F) is based on the cardinal-ity of the �-cuts of F , and there is a one to one relationbetween a set of cardinalities of �-cuts and a fuzzycardinality Z , so the �-cuts of G are equal to the �-cutsof F if and only if Z(F)=Z(G). By Proposition 4.9,Z(F)=Z(G) if and only if ED(F)=ED(G).

This last property is not veri#ed by EqPoss, andhence we prefer EqProb for measuring the degree ofequipotency of fuzzy sets.

7. Concluding remarks

We have discussed the fuzzy cardinality and fuzzyrelative cardinality of fuzzy sets. We have justi#edwhy only the representation of a fuzzy set by meansof its �-cuts is intuitively well-suited for the pur-poses of measuring cardinality, if we take into accountthe dependence among objects. We have proposed aprobabilistic and nonconvex measure ED for the fuzzycardinality. Also we have introduced a family E ofcardinalities where ED is included, as well as otherexisting fuzzy cardinalities. The study of the cardinal-ities of the family E that verify the dependence amongobjects, together with the discussion of their proper-ties, will be an object of future research. We havealso proposed both a probabilistic (ER(F=G)) and apossibilistic (ES(F=G)) fuzzy measure of the relativecardinality of fuzzy sets that generalize the measuresof cardinality ED(F) and Z(F), respectively, in thecase G=X . We have applied some of these conceptsto the problem of evaluating quanti#ed sentences, andwe have obtained methods with better properties thanexisting ones. We have proposed a de#nition of theentropy of a fuzzy set based on our probabilistic mea-sure ED, relating the development of entropymeasuresto the independence problem. Finally, we have pro-

posed a fuzzy ranking of fuzzy sets based on the fuzzycardinality measures ED and Z . We expect to applythe method ER to the evaluation of type II quanti#edsentences to obtain a probabilistic method in furtherresearch.

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