Cardinal

Fuzzy Sets and Systems 118 (2001) 479–487www.elsevier.com/locate/fss

Cardinality concepts for type-two fuzzy sets

Lee-Chae Jang∗;1, Dan RalescuDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA

Received April 1998; received in revised form January 1999

Abstract

In this paper, we de�ne generalized concepts of cardinality of a fuzzy-valued function and obtained some propertiesof these new concepts. Also, we investigate examples for the calculation of two generalized cardinality of fuzzy-valuedfunctions and compared with concepts of cardinality of a fuzzy set and a fuzzy-valued function. c© 2001 Elsevier ScienceB.V. All rights reserved.

Keywords: Fuzzy set; Fuzzy number; Fuzzy cardinality; Sigma count; Fuzzy-valued function; Fuzzy-valued cardinality;Fuzzy-valued sigma count

1. Preliminaries

A fuzzy set is de�ned by a function f :X → [0; 1], where X is an ordinary set. Zadeh [7], Ralescu [2],Ralescu [3], Wygralak [5], Dubois and Prade [1], Yager [6] investigated concepts of cardinality of a fuzzyset and obtained some properties of these new concepts. A statement such as “most students are smart” hasa truth value between 0 and 1; such a statement is of the general form “Qx’s are A” where Q is a fuzzyquanti�er and A is a fuzzy subset of a (�nite) universe of discourse X = {x1; x2; : : : ; xn}. An explicit formulafor the truth-value of such a statement was given for the �rst time in [2].In this paper, we consider a fuzzy-valued function F :X →F(R) instead of a fuzzy set A, that is, instead

of a real-valued function on X . We denote such a fuzzy-valued function by

F =(

x1 x2 · · · xn�F; a(1) �F;a(2) · · · �F;a(n)

):

This notation will be made clear in Section 2. We will use the following concept of fuzzy number. Let R bethe real numbers and let [0, 1] be the unit interval in R. Let F(R) denote the set of all fuzzy sets in R.

∗ Corresponding author.1On leave from Department of Applied Mathematics, KonKuk University, Chungju, South Korea.

0165-0114/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(99)00052 -4

480 L.-C. Jang, D. Ralescu / Fuzzy Sets and Systems 118 (2001) 479–487

De�nition 1.1. A fuzzy set A∈F(R) is called a fuzzy number, if and only if(1) A is normal, i.e. there exists x0 ∈R such that �A(x0)= 1.(2) A is convex, i.e. �A(�x + (1− �)y)¿min(�A(x); �A(y)) for all x; y∈R and �∈ [0; 1].(3) For any �∈ (0; 1]; A�= {x: �A(x)¿�} is a closed interval and cl(A0)= cl({x: �A(x)¿0}) is compact.

Here, cl(A0) is the closure of A0.

It follows from the de�nition above that A∈F(R) is a fuzzy number if and only if there exists a closedinterval [r; s] such that

�A(x)=

1; x∈ [r; s] 6= ∅;L(x); x¡ r;R(x); x¿s

where L(x) is a right continuous function and R(x) is a left continuous function (see [6]).

2. Cardinality and f-cardinality

Let X = {x1; x2; : : : ; xn} be a �nite set, and let A be a fuzzy subset of X , represented by its membershipfunction �A :X → [0; 1]. We denote the �-level set of A by L�(A)= {x∈X | �A(x)¿�} for 06�61. Thecomplement �A of A has membership function � �A =1− �A.Ralescu [3] and Ralescu [2] discussed some properties of cardA, the cardinality of fuzzy set A. This

should be a fuzzy subset of {0; 1; : : : ; n}, with cardA(k) being interpreted as the possibility that A has exactlyk elements (06k6n). Also, Ralescu [2] investigated that cardA was k to the extent to which exactly kelements of X belong to A while the other (n− k) elements do not belong to A and obtained the followingexplicit formular for the fuzzy cardinality:

card A(k)= �(k) ∧ (1− �(k+1)); k =0; 1; : : : ; n

where �(1); �(2); : : : ; �(n) are the values of �A(x1); : : : ; �A(xn) arranged in decreasing order of magnitude, and�(0) = 1; �(n+1) = 0.Now we will consider a new de�nition of f-cardF , the fuzzy-valued cardinality of a fuzzy set-valued func-

tion F :X →F(R) with the following membership functions �F(xi)(x)=�F;ai(x) of each F(xi), for i=1; 2; : : : ; n;

�F;ai ; ci ; di(x)=

1; x∈ [ri; si] 6= ∅;Li(x); x¡ ri;Ri(x); x¿si

where ai (i=1; 2; : : : ; n) are some real numbers, si=((m−1)=m)ai and ti=((m+1)=m)ai, the functions Li arestraight lines through points (1; si); (ci; 0) and the functions Ri are straight lines through points (1; ti); (di; 0),for some numbers ci; di and i=1; 2; : : : ; n. We denote the set of such fuzzy numbers by F0.

De�nition 2.1. The f-cardinality of a fuzzy-valued function F is the fuzzy-valued function f-cardF : {0; 1; : : : ; n}→F0(R) de�ned for every k by

f-cardF(k)= �F;a(k) ; c(k) ; d(k) ∧ �F;1−a(k+1) ;1−d(k+1) ;1−c(k+1)where a(0)¿a(1)¿ · · ·¿a(n) are the ordered values of the ai’s, a(0) = 1 and a(n+1) = 0, and �F;0(x)= 1;�F;1(x)= 0 for all x∈R. Furthermore, we de�ne the operation ∧ by

�F;a; c; d ∧ �F;b; e;f = �F;a∧b; c∧e; d∧f:

L.-C. Jang, D. Ralescu / Fuzzy Sets and Systems 118 (2001) 479–487 481

Notes 2.2. (1) Let a1 = · · · = ar =1 and ar+1 = · · · = an=0 and let Li=Ri=0, we de�ne the membershipfunctions �F;ai ; ai ; ai of xi by

�F;ai ; ai ; ai(x)={1; x= ai;0 otherwise:

In this case, F is a non-fuzzy set with r elements. Then by a simple calculation we obtain

f-cardF(k)={1; k = r;0 otherwise:

Thus this equation agrees with the cardinality (number of elements) of non-fuzzy sets.(2) We consider a fuzzy set A and a fuzzy-valued function F by

A=(x1 x2 · · · xna(1) a(2) · · · a(n)

);

F =(

x1 x2 · · · xn�F; a(1) ; a(1) ; a(1) �F;a(2) ; a(2) ; a(2) · · · �F;a(n) ; a(n) ; a(n)

)

and then, we obtain

f-cardF(k) = �F;a(k) ; a(k) ; a(k) ∧ �F;1−a(k+1) ;1−a(k+1) ;1−a(k+1) = �F;a(k)∧1−a(k+1) ; a(k)∧1−a(k+1) ; a(k)∧1−a(k+1)= �F; card A(k); card A(k); card A(k) = X{card A(k)}:

Here, we denote X{z} by

X{z}(x)={1 if x= z;0 otherwise

(see Ralescu [3]).We now investigate the following simple example of f-cardF

(x1 x2 x3 x4

�F;0:9; c1 ; d1 �F;0:8; c2 ; d2 �F;0:1; c3 ; d3 �F;0; c4 ; d4

)

where c1; c2; c3 ∈R; c4 = 0 and d1; d2; d3 ∈R; d4 = 0 and m=5. We calculate only one f-cardF(0)f-cardF(0) = �F;1; c0 ; d0 ∧ �F;1−0:9;1−d1 ;1−c1 = �F;1∧0:1; c0∧1−d1 ; d0∧1−c1 = �F;0:1; c0∧1−d1 ; d0∧1−c1

and we omit the other calculations. We obtain

f-cardF =(

0 1 2 3 4�F;0:1; f0 ; g0 �F;0:2; f1 ; g1 �F;0:8; f2 ; g2 �F;0:1; f3 ; g3 �F;0; f4 ; g4

)

where fi= ci ∧ 1− di+1 and gi=di ∧ 1− ci+1 for i=0; 1; 2; : : : ; n.

Notes 2.3. If we consider the following fuzzy set A (see in [3, Example 1]):

A=(x1 x2 x3 x40:9 0:8 0:1 0

);


then the cardinality of A is

card A=(0 1 2 3 40:1 0:2 0:8 0:1 0

):

This fact shows that the fuzzy cardinality of a fuzzy-valued function is a generalization of cardinality offuzzy sets. In other words, we consider the fuzzy numbers (0:9; �A;0:9; c1 ; d1); (0:8; �A;0:8; c2 ; d2); (0:1; �A;0:1; c3 ; d3);(0; �A;0;0;0) instead of the non-fuzzy numbers 0.9, 0.8, 0.1, 0. We now proceed to general results about ourconcepts of cardinality; some are non-trivial extensions of corresponding results in [3].

Proposition 2.4. f-cardF(k)=X{1} if and only if F is a non-fuzzy set with k elements.

Proof. Suppose f-cardF(k)=X{1}. Then

f-cardF(k)(x)= �F;a(k) ; c(k) ; d(k) (x)∧ �F;1−a(k+1) ;1−d(k+1) ;1−c(k+1) (x)=X{1}(x); x∈R:Thus we obtain �F;a(k) ; c(k) ; d(k) (x)=X{1}(x) and �F;1−a(k+1) ;1−d(k+1) ;1−c(k+1) (x)=X{1}(x). So, a(k) = 1 and

a(k+1) = 0. This implies that a(1) = · · · = a(k) = 1 and a(k+1) = · · · = a(n) = 0. Therefore F is a non-fuzzy setwith k elements. Conversely, the assumption implies

�F;a(i) (x)={1; x= a(i);0 otherwise

for i=1; : : : ; k and

�F;a(k) (x)= 0 for all x

for i= k+1; : : : ; n. So, a(i) = c(i) =d(i) = 1 for i=1; 2; : : : ; k and a(i) = c(i) =d(i) = 0 for i= k+1; : : : ; n. Thus, wehave that f-cardF(k)= �F;a(k) ; c(k) ; d(k)∧�F;1−a(k+1) ;1−d(k+1) ;1−c(k+1) = �F;1;1;1 =X{1}. Therefore, f-cardF(k)=X{1}.

From the de�nition of the cardinality of fuzzy-valued functions, we obtain the following proposition.

Proposition 2.5. For every k =0; 1; : : : ; n; the fuzzy cardinality f-cardF(k) of a fuzzy-valued function F liesin F0.

Now we will give a representation of f-cardF by using the de�nition. Since

a(i) ∧ [1− a(i+1)]={a(i) if a(i) + a(i+1)61;1− a(i+1) otherwise

there is a unique value j such that the following inequalities hold:

1 + a(0)¿a(1) + a(2)¿ · · ·¿a( j−1) + a( j)¿1¿a( j) + a( j+1)¿ · · ·¿a(n−1) + a(n)¿a(n)¿0:Thus, De�nition 2.1 gives

f-cardF(i) ={�F;1−a(i+1) ; f(i) ; g(i) for 06i6j − 1;�F; a(i) ; f(i) ; g(t) for j6i6n

and we obtain the following representation of f-cardF , i.e., fuzzy-valued cardinality of fuzzy-valued function:(0 1 · · · j − 1 j j + 1 · · · n

�F;1−a(1) ; f(0) ; g(0) �F;1−a(2) ; f(1) ; g(1) · · · �F;1−a( j) ; f( j−1) ; g( j−1) �F;a( j) ; f( j) ; g( j) �F;a( j+1) ; f( j+1) ; g( j+1) · · · �F;a(n) ; f(n) ; g(n)

)


where f(i) = c(i) ∧ [1− d(i+1)] and g(i) =d(i) ∧ [1− c(i+1)] for i=0; 1; : : : ; j − 1 and f(i) = c(i) ∧ c(i+1) and g(i)=d(i) ∧d(i+1) for i= j; j+1; : : : ; n. Here, Fa(i) ; c(i) ; d(i) means that a(0)¿a(1)¿ · · ·¿a(n). But, it is not necessarythat {c(i)} and {d(i)} are nonincreasing.

In this paper, we will de�ne the following concepts of complement (denoted by �F) of a fuzzy-valuedfunction F

�F;a(i) ; c(i) ; d(i) = �F;1−a(n−i+1) ;1−d(n−i+1) ;1−c(n−i+1)

for i=1; 2; : : : ; n. The reason for this de�nition is made clear next.

Proposition 2.6. For every fuzzy-valued function F; we have f-card �F(i)= f-cardF(n− i).

Proof.

f-card �F(i) = � �F; a(i) ; c(i) ; d(i) ∧ � �F;1−a(i+1) ;1−d(i+1) ;1−c(i+1)= �F;1−a(n−i+1) ;1−d(n−i+1) ;1−c(n−i+1) ∧ �F;a(n−i) ; c(n−i) ; d(n−i)

= f-cardF(n− i):

The following result will be useful in the aggregation of fuzzy-valued function. More details on the aggre-gation of fuzzy sets can be found in Ralescu and Ralescu [4]. First, we de�ne the maximum ∨ of elementsof F0.

De�nition 2.7. Let F :X →F0 be a fuzzy-valued function. The maximum of fuzzy sets F(xi) and F(xj), (thatis, �F;ai ; ci ; di and �F;aj ; cj ; dj) is de�ned by

�F;ai ; ci ; di ∨ �F;aj ; cj ; dj = �F;ai∨aj ; ci∨cj ; di∨dj :

3. Non-fuzzy f-cardinality

Let 06�61 and let F :X →F0 be a fuzzy-valued function. Then the �-level set L�(�F;a; c; d) of a fuzzy set{(x; �F; a; c; d(x)) | x∈R} is denoted by

L�(�F;a; c; d)= {x∈R | �F;a; c; d(x)¿�}:In this section, we discuss the concept of non-fuzzy f-cardinality for a fuzzy-valued function.

De�nition 3.1. The non-fuzzy f-cardinality of F , denoted by nf-cardF is the integer

nf-card�0 F ={0; F = ∅,[nf]; F 6= ∅,

where 06�061, and [nF ] is the greatest integer less than nF and

nF ={j + l{L�0 (�F;a( j) ; c( j) ; d( j) )} if a(i)¿0:5,(j − 1) + l{L�0 (�F;a(j−1) ; c(j−1) ; d(j−1) )} if a(i)¡0:5.

Here l(B) denotes the length of an interval B.We note that the length of level sets are well-de�ned, because they are closed intervals. The following

simple procedure can be used to calculate nf-cardF .


Algorithm 3.2. (1) If F = ∅, set nf-card�0 F =0. Stop.(2) If F 6= ∅, �nd j= max{16s6n | a(s−1) + a(s)¿1}.(3) Calculate a( j). If a( j)60:5; set nF = j + l{L�0 (�F;a( j) ; c( j) ; d( j) )}.If a( j)¡0:5, set nF =(j − 1) + l{L�0 (�F;a( j−1) ; c( j−1) ; d( j−1) )}.(4) Calculate [nF ]. Set nf-card F = [nF ]. Stop.

For the next result, we need to introduce an ordering for fuzzy-valued sets on X . We de�ne F6G by

a(k)6b(k) and �F;a(k) ; c(k) ; d(k) (x)6�G;b(k) ; f(k) ; g(k) (x − (b(k) − a(k))); for all x∈Rfor k =1; 2; : : : ; n. Using this de�nition of ordering of fuzzy-valued functions, we can prove the followingproposition, which gives an important property of the non-fuzzy f-cardinality.

Proposition 3.3. Let F and G be a fuzzy-valued functions from X into F0. If F6G; then

nf-card�0 F6nf-card�0 G:

Proof. Since F6G; a(k)6b(k) for k =1; 2; : : : ; n. We put k = max{s | b(s−1) + b(s)¿1}. From the proof ofProposition 5 in [3], we have k6j. If F = ∅, our result becomes trivial. But if F 6= ∅, we also have G 6= ∅.In this case, if a(k)¿0:5, then we have b( k)¿0:5; b( j)¿0:5 and

�F;a(k) ; c(k) ; d(k) (x)6�G;b(k) ; f(k) ; g(k) (x − (b(k) − a(k))); for all x∈R:Using this inequality, it follows that

l{L�0 (�F;a(k) ; c(k) ; d(k) )}6l{L�0 (�G;b(k) ; f(k) ; g(k) )}and

l{L�0 (�G;b(k) ; c(k) ; d(k) )}6l{L�0 (�G;b( j) ; f( j) ; g( j) )}:Hence, [nF ]6[nG]. If a(k)¡0:5 and b( j)¿0:5, the result becomes trivial. Finally, in the case a(k)¡0:5 andb( j)¡0:5, the calculations are similar to the case a(k)60:5. Therefore, the proof is completed.

We note that if F is a fuzzy set, the above proposition implies Proposition 5 of [3]. Let us consider nowsome simple examples.

Example 3.4. Let F be the fuzzy-valued function:

F =(

x1 x2 x3 x4�F;0:9;−1:1;2:4 �F;0:8;−1:0;2:3 �F;0:1;−0:9;1:8 �F;0;0;0

)

where m=5. Then we calculate f-cardF as in Section 2 with c0 = 1; d0 = 1 and c5 = 0; d5 = 0. First, we �ndfi and gi where fi= ci ∧ 1− di+1 and gi=di ∧ 1− ci+1 for i=0; 1; 2; 3; 4; 5,

f0 = c0 ∧ 1− d1 = 1 ∧ (1− 2:4)= − 1:4 and g0 =d0 ∧ 1− c1 = 1 ∧ {1− (−1:1)}=2:1:We obtain f0 = − 1:4; g0 = 2:1 and calculate similarly the other terms. Hence, we have

f-card�0 F =(

0 1 2 3 4�F;0:1;−1:4;2:1 �F;0:2;−1:3;2:4 �F;0:8;−0:8;1:9 �F;0:1;−0:9;1 �F;0;0;0

):

In this example, j=2 and a(2)¿0:5. If we put �0 = 1, then l{L1(�F;0:8;−0:8;1:9)}=0:32. So, nF =2:32 andhence nf-card1 F =2. But if �0 = 0:5, then we have l{L0:5(�F;0:8;−0:8;1:9)}=1:57 and nf-card0:5 F =3.


We will now compare these results with Example 1 in [3]. The fuzzy set is

A=(x1 x2 x3 x40:9 0:8 0:1 0

):

If m=5 and �0 = 0:5, we obtain that the cardinality of A is 2: the same answer as in [3]. But if m=5 and�0 = 0:5, we obtain that the cardinality of A is 3. Our de�nition then is a useful tool, because we can dealwith some fuzziness in the degree of membership of a fuzzy set.

4. Fuzzy-valued sigma count

In this section, we discuss the concepts of sigma count for fuzzy-valued functions. Zadeh [7] investigatedthe concept of sigma count for fuzzy sets and its applications. We will consider here this concept in the caseof fuzzy-valued functions. Let A be, as usual, a fuzzy set with membership function �A :X → [0; 1]. Then

�-countA=n∑k=1

�A(xk)

where X = {x1; : : : ; xn} (see [7], for example). In order to introduce the concept of sigma count for fuzzy-valued functions, we must de�ne a new concept addition in F0.

De�nition 4.1. Let F be a fuzzy-valued function and let �F;a; c; d; �F; b; e;f be in F0. The we de�ne the additionby the formula

�F;a; c; d + �F;b; e;f = �F;a+b; c+e; d+f:

De�nition 4.2. Let X = {x1; : : : ; xn} and let F :X →F0 be a fuzzy-valued function. The fuzzy-valued sigmacount of F is

f�-countA=n∑k=1

�F;ak ; ck ; dk :

The following proposition is easy to prove:

Proposition 4.3. If F is a fuzzy set; then �-countF = f�-countF .

From this proposition, we can know that our de�nition of f�-count is a generalization of �-count.

Example 4.4. Let F be the following fuzzy-valued function:

F =(

x1 x2 x3 x4�F;0:9; c1 ; d1 �F;0:8; c2 ; d2 �F;0:1; c3 ; d3 �F;0; c4 ; d4

)

where c1; c2; c3 ∈R; c4 = 0 and d1; d2; d3 ∈R; d4 = 0 and m=5. If we calculate the f�-count of F , we obtain

f�-countF = �F;1:8; f; g

where f=∑4

k=1 ck and g=∑4

k=1 dk .


5. Non-fuzzy fuzzy-valued sigma count

Let F be a fuzzy-valued function from X = {x1; x2; : : : ; xn} into F0. We consider now a concept of a non-fuzzy f�-count of F . Such a concept can be used in the calculation of the possibility distribution and theaggregation of fuzzy-valued functions (in a manner similar to [2,4]).

De�nition 5.1. The non-fuzzy fuzzy-valued sigma count of F , denoted by nf�-countF , is de�ned by

nf�-countF = [mF ]

where mF = max{x∈R | �F;a; c; d(x)= 1} and [mF ] is the greatest integer less than mF .

Since �F;a; c; d is a fuzzy number, the set {x∈R | �F;a; c; d(x)= 1} is a closed set in R. So, we obtain thatthere is only one real number mF . That is, the nf�-count is well de�ned. Using similar methods to those in[3], the following simple procedure can be used to calculate nf�-countF .

Algorithm 5.2. (1) Calculate S =∑n

k=1 ak .(2) Calculate mF = S + S × 1=m.(3) Calculate nf�-countF = [mF ]. Stop.

Using our de�nition of ordering of fuzzy-valued function in Section 3 and the de�nition of F0, we obtainthe following proposition which is an important property of the non-fuzzy f-cardinality:

Proposition 5.3. Let F and G be fuzzy-valued functions from X into F0. If F6G; then

nf�-count F6nf�-countG:

Proof. Since F6G, clearly ak6bk for k =1; 2; : : : ; n. Then we have

mF =n∑k=1

{ak +

1mak

}6

n∑k=1

{bk +

1mbk

}=mG:

That is,

nf�-countF = [mF ]6[mG] = nf�-countG:

Let us consider some examples and explain the di�erence between n-card and nf�-count.

Example 5.4. If m=5 and

F =(

x1 x2 x3�F;0:7; c1 ; d1 �F;0:5; c2 ; d2 �F;0:5; c3 ; d3

);

then mF =(0:7+ 15 × 0:7) + (0:5+ 1

5 × 0:5) + (0:5+ 15 × 0:5)=2:04. We obtain nf�-countF = [2:04]= 2. This

value is equal to the value of n-cardA in Example 2 of [3]. But if m=10, by similar calculations, we havemF =1:87 and hence nf�-cardF =1. In this case, the two results are di�erent. This means that if X is theset of members of some recreational football team, we take m=5 and if X is the set of members of someprofessional football team, we take m=10. In other words, the degree of each membership function for theprofessional football team is greater than that for the recreational football team. So, if F is a professionalfootball team, we consider the cardinality as nf�-cardF =1 and if F is a recreational team, we consider thecardinality as nf�-countF =2.


References

[1] D. Dubois, H. Prade, Fuzzy cardinality and the modeling of imprecise quanti�cation, Fuzzy Sets and Systems 16 (1985) 199–230.[2] A.L. Ralescu, A note on rule representation in expert systems, Inform. Sci. 38 (1986) 193–203.[3] D. Ralescu, Cardinality, quanti�ers, and the aggregation of fuzzy criteria, Fuzzy Sets and Systems 69 (1995) 355–365.[4] A.L. Ralescu, D. Ralescu, Extensions of fuzzy aggregation, Fuzzy Sets and Systems 86 (1997) 321–330.[5] M. Wygralak, Fuzzy cardinals based on the generalized equality of fuzzy subsets, Fuzzy Sets and Systems 18 (1986) 143–158.[6] C. Yu, Correlation of fuzzy numbers, Fuzzy Sets and Systems 55 (1993) 303–307.[7] R.R. Yager, Counting the number of classes in a fuzzy set, IEEE Trans. Systems Man Cybernet. 23 (1993) 257–264.[8] L.A. Zadeh, A computational approach to fuzzy quanti�ers in natural languages, Comput. Math. 9 (1983) 149–184.

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