An extended characterization of fuzzy bags

An Extended Characterizationof Fuzzy BagsMiguel Delgado,† Marıa-Jose Martın-Bautista,‡ Daniel Sanchez,∗

Marıa-Amparo Vila§

Department of Computer Science and Artificial Intelligence, University ofGranada C/Periodista Daniel Saucedo Aranda s/n, 18071 Granada, Spain

The algebraic structures known as bags were introduced by R. Yager as set-like algebraic structureswhere elements are allowed to be repeated. Since the original papers by Yager, different definitionsof the concept of fuzzy bag, and the corresponding operators, are available in the literature, as wellas some extensions of the union, intersection and difference operators of sets, and new algebraicoperators.

In general, the current definitions of bag pose very interesting issues related to the ontologicalaspects and practical use of bags. In this paper, we introduce a characterization of bags viewingthem as the result of a count operation on the basis of a mathematical correspondence. We alsodiscuss on the extension of our alternative characterization of bags to the fuzzy case. On these basis,we introduce some operators on bags and fuzzy bags, and we compare them to existing approaches.Finally, we deal with the case where no information about the correspondence is available, andonly bounds can be provided for the count of elements of the result of algebraic operators. For thispurpose, the notion of IC-bag (Chakrabarty, in: Proc. Int. Conf. On Computational Intelligencefor Modelling, Control and Automation—CIMCA’2001, Vol. 25, 2001) is employed, and newoperators for these structures are proposed. C© 2009 Wiley Periodicals, Inc.

1. INTRODUCTION

Bags,1 also called multisets,2−4 are set-like algebraical structures where anelement can appear more than once. There are several areas where bags can beuseful, such as statistics,5 but they have been mainly applied in the database area1,6,7

(see Ref. 8 for additional references). Bags have been extended to fuzzy bags inseveral different ways,1,5,9 with similar applications.

To apply bags in practice, it is convenient to study their meaning and theprocesses that could generate them. In the context of computer science, bags canbe useful to model situations of the real world, in particular those related to objectsand their properties.

∗Author to whom all correspondence should be addressed; e-mail: [email protected].†e-mail: [email protected].‡e-mail: [email protected].§e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 24, 706–721 (2009)C© 2009 Wiley Periodicals, Inc. Published online in Wiley InterScience

(www.interscience.wiley.com). • DOI 10.1002/int.20355

AN EXTENDED CHARACTERIZATION OF FUZZY BAGS 707

A usual situation in the real world is to find that several individually differentobjects verify the same properties in a certain context (space, time). In other words,we can say they are instances of the same class. Counting how many objects verifya certain property in a given context seems to be a natural application of bags inpractice.

Following this idea, in this work we propose a characterization of bags based ontwo components: a correspondence that links properties to objects (more generally,instances to classes) and a count of the number of objects that verify each property.The latter corresponds to the classical characterization of bags. In addition, anextension of this characterization to fuzzy bags is introduced. Finally, we proposebounds for the counts of the number of objects that verify a certain property in abag obtained by union, intersection or difference of two bags when only the counts,and not the correspondences, are known.

2. A CHARACTERIZATION OF CRISP BAGS

2.1. Crisp Bags

DEFINITION 1. Let P and O be two finite universes (sets) we call “properties”and “objects,” respectively. A bag Bf is a pair (f, Bf ) where f : P → O is acorrespondence and Bf is the following subset of P × N:

Bf = {(p, card (f (p))), p ∈ P and f (p) �= ∅} (1)

where N is the set of natural numbers and card(X) is the cardinality of the set X.

In this characterization, a bag Bf consists of two parts. The first one is thecorrespondence, f , that can be seen as an information source about the relationbetween objects and properties. The second part, Bf , is a summary of the informationin f obtained by means of a count operation. This summary corresponds to theclassical view of bags. From now on, we shall refer to this kind of summaries assbags.

Example 1. Let O = {John, Mary, Bill, Tom, Sue, Stan, Harry} and P be the set ofpossible ages. Let f1, f2, f3, f4 : P → O be the correspondences in Table I (theexample is from Ref. 1) with fi(p) ⊆ O ∀p ∈ P .

Table I. Several correspondences: age-people.

p 21 27 17 35

f1(p) {John, Tom} ∅ {Bill, Sue} ∅f2(p) {John, Tom, Stan} ∅ {Bill, Sue} {Harry}f3(p) {Stan} {Mary} ∅ {Harry}f4(p) {John, Stan} ∅ {Bill} ∅

International Journal of Intelligent Systems DOI 10.1002/int

708 DELGADO ET AL.

Examples of bags are Bfi = (fi, Bfi ) 1 ≤ i ≤ 4, where

Bf1 = {(21, 2), (17, 2)}

Bf2 = {(21, 3), (17, 2), (35, 1)}

Bf3 = {(21, 1), (27, 1), (35, 1)}

Bf4 = {(21, 2), (17, 1)}

Following Ref. 1, we note CountBf (p) = card(f (p)) the number of objectsthat verify property p in the bag Bf . For instance, CountBf2 (21) = 3 andCountBf3 (35) = 1.

2.2. Operations and Relations

In the following, let f, g : P → O be two correspondences.

DEFINITION 2 (Union, intersection, difference). Let ∗ ∈ {∩, ∪, \}. Then

Bf ∗ Bg = Bf ∗g = (f ∗ g, Bf ∗g

)(2)

where f ∗ g : P → O such that (f ∗ g)(p) = f (p) ∗ g(p) ∀p ∈ P .

Definition 2 is coherent with the characterization of bags given in Definition 1.The union, intersection, or difference operation, is applied to the correspondencesthat define the bag. If we see the correspondences f and g as two informationsources, these operations are different ways to aggregate that information. Finally,the aggregation is summarized as an sbag by means of a count operation.

Example 2. This is a continuation of Example 1. Table II shows some operationsbetween correspondences in Example 1:

Table II. Operations on correspondences from Example 1.

r 21 27 17 35

(f1 ∪ f2)(p) {John, Tom, Stan} ∅ {Bill, Sue} {Harry}(f1 ∪ f3)(p) {John, Tom, Stan} {Mary} {Bill, Sue} {Harry}(f2 ∪ f3)(p) {John, Tom, Stan} {Mary} {Bill, Sue} {Harry}(f1 ∩ f2)(p) {John, Tom} ∅ {Bill, Sue} ∅(f1 ∩ f3)(p) ∅ ∅ ∅ ∅(f2 ∩ f3)(p) {Stan} ∅ ∅ {Harry}(f1\f2)(p) ∅ ∅ ∅ ∅(f1\f3)(p) {John, Tom} ∅ {Bill, Sue} ∅(f2\f3)(p) {John, Tom} ∅ {Bill, Sue} ∅(f3\f2)(p) ∅ {Mary} ∅ ∅



The corresponding sbags are

Bf1∪f2 = {(21, 3), (17, 2), (35, 1)}

Bf1∪f3 = {(21, 3), (27, 1), (17, 2), (35, 1)}

Bf2∪f3 = {(21, 3), (27, 1), (17, 2), (35, 1)}

Bf1∩f2 = {(21, 2), (17, 2)}

Bf1∩f3 = ∅

Bf2∩f3 = {(21, 1), (35, 1)}

Bf1\f2 = ∅

Bf1\f3 = {(21, 2), (17, 2)}

Bf2\f3 = {(21, 2), (17, 2)}

Bf3\f2 = {(27, 1)}

Regarding relations between bags, we can consider different definitions bylooking at the whole information or the count summary. From the point of view ofthe information (correspondences) from which a bag is obtained, we introduce thefollowing definitions:

DEFINITION 3 (Subbag). A bag Bf is a subbag of Bg , noted Bf ⊆ Bg when f (p) ⊆g(p) ∀p ∈ P .

DEFINITION 4 (Equality). Two bags Bf and Bg are equal, noted Bf = Bg whenBf ⊆ Bg and Bg ⊆ Bf .

The following definition introduces a relation between bags based on theircorresponding sbags:

DEFINITION 5 (Ranking by cardinality). A bag Bf is

– smaller than Bg , noted Bf < Bg , when countBf (p) < countBg (p) ∀p ∈ P .– equipotent to Bg , noted Bf ≡ Bg , when countBf (p) = countBg (p) ∀p ∈ P .– greater than Bg , noted Bf > Bg, when countBf (p) > countBg (p) ∀p ∈ P .

The relations “smaller” and “equipotent” in Definition 5 are equivalent to thenotions of inclusion (subbag) and equality of sbags as introduced in Ref. 1. Thefollowing results are easy to show.


710 DELGADO ET AL.

PROPOSITION 1. If Bf ⊆ Bg then Bf ≤ Bg .

COROLLARY 1. If Bf = Bg then Bf ≡ Bg .

That means the concepts of subbag and equality of bags given by Definitions 3and 4 are more restrictive than those proposed for sbags in Ref. 1 (where sbags arecalled bags as usual). The reciprocal is not true in general. For instance, in Example 1we can see that Bf1 ⊆ Bf2 and consequently Bf1 ≤ Bf2 . However, Bf4 ≤ Bf1 butBf4 �⊆ Bf1 .

2.3. Discussion

The operations introduced in Definition 2 verify the following properties:

PROPOSITION 2. Let p ∈ P with f (p) ∩ g(p) = ∅. Then

1. countBf ∩g (p) = 02. countBf ∪g (p) = countBf (p) + countBg (p)3. countBf \g (p) = countBf (p)

Proof. Trivial. �

COROLLARY 2. Let f (p) ∩ g(p) = ∅ ∀p ∈ P . Then

1. Bf ∩g = ∅2. Bf ∪g = Bf ⊕ Bg

3. Bf \g = Bf

where ⊕ is the addition of bags introduced in Ref. 1 to be

countA⊕B(p) = countA(p) + countB(p) (3)

A particular case of the condition in Corollary 2 is f (P ) ∩ g(P ) = ∅. Forinstance, in Example 2 we have (f1 ∩ f3)(p) = ∅. We can see that Bf1∩f3 = ∅,Bf1∪f3 = Bf1 ⊕ Bf3 , and Bf1\f3 = Bf1 .

PROPOSITION 3. Let p ∈ P with f (p) ⊆ g(p). Then

1. countBf ∩g (p) = countBf (p)2. countBf ∪g (p) = countBg (p)3. countBg\f (p) = countBg (p) − countBf (p)4. countBf \g (p) = 0



Proof. Trivial. �

COROLLARY 3. Let f (p) ⊆ g(p) ∀p ∈ P . Then

1. Bf ∩g = Bf

2. Bf ∪g = Bg

3. Bg\f = Bg � Bf

4. Bf \g = ∅

where A � B is the removal of bag B from A introduced in Ref. 1 to be

countA�B(p) = max{countA(p) − countB(p), 0} (4)

The results of the first two cases in Proposition 3 correspond to the usualdefinitions of intersection and union of bags.1 However, the following propositionshows there is a more general case where this property holds:

PROPOSITION 4. Let f, g : P → O such that f (p) ⊆ g(p) or g(p) ⊆ f (p) ∀p ∈ P .Then

1. Bf ∩g = Bf ©∩ Bg

2. Bf ∪g = Bf ©∪ Bg

where ©∩ and ©∪ are, respectively, the intersection and union of bags introducedin Ref. 1 to be

countA©∩ B(p) = min{countA(p), countB(p)} (5)

countA©∪ B(p) = max{countA(p), countB(p)} (6)

Proof. Let p ∈ P such that f (p) ⊆ g(p). Then min{countBf (p), countBg (p)} =countBf (p) = countBf ∩g (p) and max{countBf (p), countBg (p)} = countBg (p) =countBf ∪g (p). The proof is similar for the case g(p) ⊆ f (p). �

As we have seen, the operations in Definition 2 reduce to different usualoperations defined on sbags in some particular cases. However, in the general case,these operations may have no counterpart in the existing literature.

Example 3. In Example 2, we have

Bf1∪f4 = {(21, 3), (17, 2)}

Bf1 ⊕ Bf4 = {(21, 4), (17, 3)}International Journal of Intelligent Systems DOI 10.1002/int

712 DELGADO ET AL.

Bf1 ©∪ Bf4 = {(21, 2), (17, 2)}

Bf1∩f4 = {(21, 1), (17, 1)}

Bf1 ©∩ Bf4 = {(21, 2), (17, 1)}

Bf1\f4 = {(21, 1), (17, 1)}

Bf1 � Bf4 = {(17, 1)}

It is easy to verify that Bf1∪f4 �= Bf1 ⊕ Bf4 and Bf1∪f4 �= Bf1 ©∪ Bf4 . In additionBf1∩f4 �= Bf1 ©∩ Bf4 and Bf1\f4 �= Bf1 � Bf4 .

3. A CHARACTERIZATION OF FUZZY BAGS

3.1. Fuzzy Bags

DEFINITION 6. A fuzzy bag Bf is a pair (f , Bf ) where f : P → O is a fuzzycorrespondence (i.e., f (p) ∈ P(O), with P(O) being the set of fuzzy subsets of O)and Bf is the following subset of P × P(N ):

Bf = {(p, ED (f (p))), p ∈ P and f (p) �= ∅} (7)

where ED(f (p)) is the fuzzy cardinality of the fuzzy set f (p) introduced in Ref. 10to be

ED(f (p)) =∑

αi∈�(f (p))∪{1}(αi − αi+1)/|f (p)αi

| (8)

with �(F ) = {α1, . . . , αk}, αi > αi+1, and considering αk+1 = 0.

From Definition 6, a crisp bag is a particular case of fuzzy bag where f is crispand the cardinality of f (p) is the fuzzy set ED(f (p)) = 1/|f (p)|.

3.2. Operations and Relations

DEFINITION 7 (Union). The union of fuzzy bags Bf and Bg is the fuzzy bag

Bf ∪ Bg = Bf ∪g = (f ∪ g, Bf ∪g) (9)



where (f ∪ g)(p)(o) = max{f (p)(o), g(p)(o)} ∀p ∈ P, o ∈ O.

DEFINITION 8 (Intersection). The intersection of fuzzy bags Bf and Bg is the fuzzybag

Bf ∩ Bg = Bf ∩g = (f ∩ g, Bf ∩g) (10)

where (f ∩ g)(p)(o) = min{f (p)(o), g(p)(o)} ∀p ∈ P, o ∈ O.

DEFINITION 9 (Difference). The difference Bf \Bg is the fuzzy bag

Bf \Bg = Bf \g = (f \g, Bf \g) (11)

where (f \g)(p)(o) = min{f (p)(o), (1 − g(p)(o))} ∀p ∈ P, o ∈ O.

The following are some relations between fuzzy bags:

DEFINITION 10 (Subbag). A fuzzy bag Bf is a subbag of Bg , noted Bf ⊆ Bg whenf (p) ⊆ g(p) ∀p ∈ P , i.e., f (p)(o) ≤ g(p)(o) ∀p ∈ P, o ∈ O.

DEFINITION 11 (Equality). Two fuzzy bags Bf and Bg are equal, noted Bf = Bg

when Bf ⊆ Bg and Bg ⊆ Bf .

In Ref. 10, a fuzzy ranking of the cardinalities of two fuzzy sets was introduced.Since cardinalities are fuzzy, it seems natural to provide a fuzzy ranking that wedefine as a fuzzy subset of R = {<, =, >}. The fuzzy ranking of the cardinalitiesof fuzzy sets F and G is defined for every ∗ ∈ R to be

RProb(F, G)(∗) =∑

αi :|Fαi |∗|Gαi |(αi − αi+1) (12)

with αi ∈ �(F ) ∪ �(G) ∪ {1}. Let us remark that

∑∗∈R

RProb(F, G)(∗) = 1 (13)

DEFINITION 12 (Ranking by cardinality). A fuzzy bag Bf is

– smaller than Bg , noted Bf < Bg , when RProb(f (p), g(p))(<) = 1 ∀p ∈ P

– equipotent to Bg , noted Bf ≡ Bg , when RProb(f (p), g(p))(=) = 1 ∀p ∈ P

– greater than Bg , noted Bf ≥ Bg , when RProb(f (p), g(p))(>) = 1 ∀p ∈ P


714 DELGADO ET AL.

Note that by Equation 13 if RProb(f (p), g(p))(<) = 1 then

RProb(f (p), g(p))(=) = RProb(f (p), g(p))(>) = 0

so we are sure that the cardinality of f (p) is smaller than the cardinality of g(p). Asimilar conclusion holds for the cases “equipotent” and “greater.”

4. USING IC-BAGS TO DEAL WITH MISSING INFORMATION

Suppose three bags Bf , Bg , and Bh defined on the same set of objects andproperties, and such that Bg = Bh. As we have seen in previous sections, it ispossible in general that Bf ∗g �= Bf ∗h for any ∗ ∈ {∩, ∪, \}, something that does notoccur in the ordinary definitions and operations of bags existing in the literature.

To illustrate this fact, consider the correspondences in Table III (a modificationof Table I).

From Table III, we have the bags Bfi = (fi, Bfi ) 1 ≤ i ≤ 4 where

Bf1 = {(21, 2)}

Bf2 = {(21, 3), (35, 1)}

Bf3 = {(21, 1), (27, 1), (35, 1)}

Bf4 = {(21, 2)}

Consider the bags Bf3 , Bf1 , and Bf4 . The three bags are different becausethe correspondences are different. However, Bf1 = Bf4 . The ordinary definitions ofbags existing in the literature yield

Bf3 ©∪ Bf1 = Bf3 ©∪ Bf4 = {(21, 2), (27, 1), (35, 1)}However, if we calculate the bags by applying the operations on the correspondencesas in Table IV and then we calculate the sbags, we obtain

Bf3∪f1 = {(21, 3), (27, 1), (35, 1)}Bf3∪f4 = {(21, 2), (27, 1), (35, 1)}

where obviously Bf3∪f1 (21) �= Bf3∪f4 (21) and hence Bf3∪f1 �= Bf3∪f4 .

Table III. Other correspondences: age-people.

p 21 27 35

f1(p) {John, Tom} ∅ ∅f2(p) {John, Tom, Stan} ∅ {Harry}f3(p) {Stan} {Mary} {Harry}f4(p) {John, Stan} ∅ ∅



Table IV. Some operations on correspondencesfrom Table III.

p 21 27 35

(f3 ∪ f1)(p) {John, Tom, Stan} {Mary} {Harry}(f3 ∪ f4)(p) {John, Stan} {Mary} {Harry}

This situation appears because in our bags we deal with more informationthan simply the summary (sbags, counts). Operations on different functions havingthe same counts could yield different functions with different counts. However, itcould be the case in some occasions that only the sbag for a given bag is known,i.e., correspondences exist and are not available, but we know the summary. Insuch cases, we can only provide bounds for the counts of the union, intersectionand difference of bags. We think that IC-bags11−13 are a very useful tool for thispurpose.

An IC-bag is a bag for which the exact amount of elements from a set P is notwell known, but lower and upper bounds are available. One IC-bag β drawn fromP is characterized by two functions C

β

l and Cβu such that

Cβ

l : P → N and Cβu : P → N

and such that Cβ

l (p) ≤ Cβu (p) ∀p ∈ P . The values C

β

l (p) and Cβu (p) are the lower

and upper bound of the count for element p in bag β .

4.1. Crisp Case

An sbag can be seen as a particular case of IC-bag where Cβ

l (p) = Cβu (p) ∀p ∈

P . Hence, to calculate possible bounds for the union, intersection, and difference ofsbags (considering that there is a correspondence that is not known), a natural wayto proceed could be to employ the corresponding operations on IC-bags.

However, the usual operations on IC-bags do not consider correspondences,but start from the usual concept of bag as a count only. For example, in a IC-bag wehave

Cβ1∪β2

u (p) = max{Cβ1

u (p), Cβ2

u (p)}.

But in our approach, this corresponds to a case where the (unknown) correspon-dences that generate the IC-bags, say fβ1 and fβ2 , verify fβ1 (p) ⊆ fβ2 (p) orfβ2 (p) ⊆ fβ1 (p) (equivalently, fβ1 (p) ∩ fβ2 (p) ∈ {fβ1 (p), fβ2 (p)}), whereas themaximum possible value corresponds to a situation where fβ1 (p) ∩ fβ2 (p) = ∅,and in this case

Cβ1∪β2

u (p) = Cβ1

u (p) + Cβ2

u (p).


716 DELGADO ET AL.

Table V. New operations on IC-bags and conditions on thecorrespondences.

Operations Correspondences

Cβ1∪β2

l (p) = max{Cβ1

l (p), Cβ2

l (p)} fβ1 (p) ∩ fβ2 (p) ∈ {fβ1 (p), fβ2 (p)}C

β1∪β2

u (p) = Cβ1

u (p) + Cβ2

u (p) fβ1 (p) ∩ fβ2 (p) = ∅C

β1∩β2

l (p) = 0 fβ1 (p) ∩ fβ2 (p) = ∅C

β1∩β2

u (p) = min{Cβ1

u (p), Cβ2

u (p)} fβ1 (p) ∩ fβ2 (p) ∈ {fβ1 (p), fβ2 (p)}C

β1\β2

l (p) = 0 fβ1 (p) ⊆ fβ2 (p)

Cβ1\β2

u (p) = Cβ1

u (p) fβ1 (p) ∩ fβ2 (p) = ∅

Hence, we need to define different operators for union, intersection, and difference.These are shown in Table V, including the possible situations between the (unknown)correspondences.

4.2. Fuzzy Case

We can extend the operations defined in Table V to the fuzzy case by extendingthe operators of addition and calculation of maximum and minimum to the fuzzycardinalities given by ED.

4.2.1. Extending Addition

To define addition of fuzzy ED cardinalities, we must take into account thatthe measure ED is a probability distribution on N. The following proposition holds:

PROPOSITION 5. Let R be a probability distribution on N, then there exists at leastone fuzzy set FR over a large enough universe X such that R = ED(FR).

Proof. Let supp(R) = {i1, . . . , im} with i1 < i2 < · · · < im and let i0 = 0. FR canbe obtained by the following algorithm:

1. FR ← ∅ ; j ← 1; μ = 12. If i1 = 0 then j ← 2; μ ← 1 − R(i1)3. While j ≤ m

(a) add ij − ij−1 new elements with degree μ to FR

(b) μ ← μ − R(ij+1)(c) j ← j + 1

It is easy to check that

�(FR) = {1} ∪ {FR(xj )|xj ∈ supp(FR)} = {1} ∪ {1 − rp | 1 ≤ p < m}



with rp = ∑1≤k≤p R(ik). Also, let �(FR) = {α1, . . . , αm} with α1 > α2 > · · · >

αm. Then∣∣(FR)αj

∣∣ = ij and αj − αj+1 = R(ij ). Hence

ED(FR) =∑

1≤j≤m

R(ij )/ij = R. (14)

�

Example 4. Let R = 0.12/1 + 0.6/2 + 0.28/4, then we have

supp(R) = {i1 = 1, i2 = 2, i3 = 4}

and �(FR) = {1, 0.88, 0.28}, so the algorithm will add i1 − i0 = 1 new element toFR with degree μ = 1, i2 − i1 = 1 new element with degree 0.88, and i3 − i2 = 2new elements with degree 0.28. Hence, FR will be of the form

FR = 1/x1 + 0.88/x2 + 0.28/x3 + 0.28/x4 (15)

The basic idea for the addition of fuzzy ED cardinalities is the following: Giventwo fuzzy cardinalities, find two disjoint fuzzy sets such that their respective cardi-nalities are those we are considering (always possible by proposition 5), calculatethe union, and then obtain the cardinality of the resulting set.

DEFINITION 13. Let N1 and N2 be fuzzy cardinalities. Let FN1 and FN2 be two fuzzysets such that FN1 ∩ FN2 = ∅, ED(FN1 ) = N1 and ED(FN2 ) = N2. Then

N1 + N2 = ED(FN1 ∪ FN2 ) (16)

Example 5. Let

N1 = 0.5/1 + 0.5/3 (17)

N2 = 0.2/0 + 0.6/2 + 0.2/4. (18)

Then

FN1 = 1/x1 + 0.5/x2 + 0.5/x3 (19)

FN2 = 0.8/y1 + 0.8/y2 + 0.2/y3 + 0.2/y4 (20)


718 DELGADO ET AL.

Table VI. Calculation of N1 + N2 by α-cuts.

αi αi − αi+1 x = |(FN1 )αi| y = |(FN2 )αi

| x + y

1 0.2 1 0 10.8 0.3 1 2 30.5 0.3 3 2 50.2 0.2 3 4 7

and finally

N1 + N2 = ED(FN1 ∪ FN2 ) = 0.2/1 + 0.3/3 + 0.3/5 + 0.2/7. (21)

A useful way to calculate the addition of N1 and N2 is to calculate the addition ofthe crisp cardinalities of α-cuts of FN1 and FN2 (i.e., the crisp cardinalities of α-cutsof FN1 ∪ FN2 ), and then obtaining the fuzzy cardinality representing the addition.Table VI shows this procedure for N1 and N2 of Example 5, and the second and fifthcolumns yield N1 + N2 given in Equation 21.

As a final remark, it is easy to show that this addition is commutative, associa-tive, and has as neutral element the fuzzy cardinality 1/0.

4.2.2. Extending Max and Min

In Equation 12, we proposed a fuzzy ranking of fuzzy cardinalities that can beemployed to determine to what extent one fuzzy cardinality is greater than, equal,or lower than another. The main problem that we find is that in some occasions, thedegree to which one cardinality is greater than, or lesser than another one is not 1,i.e., it is possible that given a pair of cardinalities, none of them is the maximumor minimum of both. An example is the cardinalities N1 and N2 of the previousexample, whose fuzzy ranking yields

RProb

(FN1, FN2

) = 0.5/〈+0/ = +0.5/〉

To solve that problem, we propose to obtain the lesser (resp. greater) fuzzy cardinalitythat is greater (resp. lower) than a given pair of fuzzy cardinalities, i.e., the supremum(resp. infimum) of both. We shall replace the maximum and minimum calculationin Table V by the calculation of such cardinalities.

These fuzzy cardinalities can be obtained following the same procedure byalpha-cuts that we suggested for the addition, but replacing the addition by max andmin, respectively. Tables VII and VIII show the calculations for the cardinalities N1

and N2, where the second and the last columns are the degrees and elements of thesupport of the corresponding fuzzy sets, i.e.,

sup(N1, N2) = 0.2/1 + 0.3/2 + 0.3/3 + 0.2/4

inf (N1, N2) = 0.2/0 + 0.3/1 + 0.3/2 + 0.2/3



Table VII. Calculation of the lesser cardinality greaterthan N1 and N2 by α-cuts.


| max{x, y}

1 0.2 1 0 10.8 0.3 1 2 20.5 0.3 3 2 30.2 0.2 3 4 4

Table VIII. Calculation of the greater cardinality lesserthan N1 and N2 by α-cuts.


| min{x, y}

1 0.2 1 0 00.8 0.3 1 2 10.5 0.3 3 2 20.2 0.2 3 4 3

In the case that the maximum or minimum yield the same value for different(always consecutive) alpha-cuts, we add the corresponding differences that repre-sent the basic probability assignment for those alpha-cuts. The result is always acardinality, since the cardinality values are always non-decreasing as the value ofalpha decreases. This is easy to show since

∣∣∣(FN1

)αi

∣∣∣ <

∣∣∣(FN1

)αi+1

∣∣∣∀i ∈ �(FN1 ) and

∣∣∣(FN2

)αi

∣∣∣ <

∣∣∣(FN2

)αi+1

∣∣∣∀i ∈ �(FN2 ), hence

max{∣∣∣(FN1

)αi

∣∣∣ , ∣∣∣(FN2

)αi

∣∣∣} ≤ max{∣∣∣(FN1

)αi+1

∣∣∣ , ∣∣∣(FN2

)αi+1

∣∣∣}

and

min{∣∣∣(FN1

)αi

∣∣∣ , ∣∣∣(FN2

)αi

∣∣∣} ≤ min{∣∣∣(FN1

)αi+1

∣∣∣ , ∣∣∣(FN2

)αi+1

∣∣∣}

∀i ∈ �(FN1 ) ∪ �(FN2 ).By definition, the ranking of fuzzy cardinalities is performed by alpha-cuts.

Hence, it is trivial to show that sup(N1, N2) (resp. inf (N1, N2)) is the lower (resp.greater) fuzzy cardinality verifying

RProb

(FN1, Fsup(N1,N2)

) = RProb

(FN2, Fsup(N1,N2)

) = 1/ <


720 DELGADO ET AL.

(resp.)

RProb

(FN1, Finf (N1,N2)

) = RProb

(FN2, Finf (N1,N2)

) = 1/ >

5. CONCLUSIONS

We have proposed a new characterization of bags and their operations on thebasis of a mathematical correspondence between properties and objects, from whichclassical bags (that we call sbags) are obtained as counts. This view enrich the clas-sical view of bags, allowing to define algebraic operations that reduce to classicalalgebraic and numerical operations on classical bags, and even providing new op-erations without classical counterpart. This characterization has been extended tofuzzy bags on the basis of fuzzy correspondences.

The new approach can be very useful in practice when we are dealing withcounts of the number of objects that verify a certain property. That is the case forinstance of queries in relational databases, where projections on nonkey attributesyield a bag, since it is possible to define a correspondence between tuples of theoriginal table and tuples in the result. Another example is the set of answers to acertain question in a poll, where different people can give the same answer. Again,it is possible to define a correspondence between people and the correspondinganswers.

Some questions will be dealt with in the future, such as the study of additionalproperties of operations on fuzzy bags, and the extension of them to fuzzy shadows.9

Acknowledgment

This work has been supported by the Spanish Ministry of Science and Technology undergrant TIC2002-04021-C02-02.

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An extended characterization of fuzzy bags

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