Abstract Fuzzy Sets and Systems 118 (2001) 479–487 Cardinality concepts for type-two fuzzy

Abstract In this paper, we deÿne generalized concepts of cardinality of a fuzzy-valued function and obtained some properties of these new concepts. Also, we investigate examples for the calculation of two generalized cardinality of fuzzy-valued functions and compared with concepts of cardinality of a fuzzy set and a fuzzy-valued function. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy set; Fuzzy number; Fuzzy cardinality; Sigma count; Fuzzy-valued function; Fuzzy-valued cardinality; Fuzzy-valued sigma count
F; a(1)
x2
F; a(2)
··· ···
xn
F; a(n)
:
This notation will be made clear in Section 2. We will use the following concept of fuzzy number. Let R be the real numbers and let [0, 1] be the unit interval in R. Let F(R) denote the set of all fuzzy sets in R.
F; a(k ) ; c(k ) ; d(k )
∧
F; 1−a(k +1) ; 1−d(k +1) ; 1−c(k +1) F; 0 (x ) = 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; 1 (x ) = 0 for all x ∈ R. Furthermore, we deÿne the operation ∧ by
Fuzzy Sets and Systems 118 (2001) 479–487
/locate/fss
ቤተ መጻሕፍቲ ባይዱ
Cardinality concepts for type-two fuzzy sets
Lee-Chae Jang ∗; 1 , Dan Ralescu
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA Received April 1998; received in revised form January 1999
480
L.-C. Jang, D. Ralescu / Fuzzy Sets and Systems 118 (2001) 479–487
Deÿnition 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 deÿnition above that A ∈ F(R) is a fuzzy number if and only if there exists a closed interval [r; s] such that x ∈ [r; s] = ∅; 1; L(x); x¡ r; A (x ) = 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 ÿnite set, and let A be a fuzzy subset of X , represented by its membership function A : X → [0; 1]. We denote the -level set of A by L (A) = {x ∈ X | A (x)¿ } for 06 61. The complement A of A has membership function A = 1 − A . Ralescu [3] and Ralescu [2] discussed some properties of card A, the cardinality of fuzzy set A. This should be a fuzzy subset of {0; 1; : : : ; n}, with card A(k ) being interpreted as the possibility that A has exactly k elements (06k 6n). Also, Ralescu [2] investigated that card A was k to the extent to which exactly k elements of X belong to A while the other (n − k ) elements do not belong to A and obtained the following explicit 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 deÿnition of f-card F , the fuzzy-valued cardinality of a fuzzy set-valued function F : X → F(R) with the following membership functions F (xi ) (x)= F; ai (x) of each F (xi ), for i = 1; 2; : : : ; n; x ∈ [ri ; si ] = ∅; 1; Li (x); x¡ ri ; F; ai ; ci ; di (x ) = 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 are straight 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 F 0. Deÿnition 2.1. The f-cardinality of a fuzzy-valued function F is the fuzzy-valued function f-card F : {0; 1; : : : ; n} →F 0 (R) deÿned for every k by f-card F (k ) =
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 deÿne the membership functions F; ai ; ai ; ai of xi by
1. Preliminaries A fuzzy set is deÿned 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 fuzzy set and obtained some properties of these new concepts. A statement such as “most students are smart” has a truth value between 0 and 1; such a statement is of the general form “Qx’s are A” where Q is a fuzzy quantiÿer and A is a fuzzy subset of a (ÿnite) universe of discourse X = {x1 ; x2 ; : : : ; xn }. An explicit formula for the truth-value of such a statement was given for the ÿrst 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