Wygralak, Maciej

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KybernetikaMaciej WygralakA supplement to Gottwald's note on fuzzy cardinalsKybernetika, Vol. 20 (1984), No. 3, 240--243Persistent URL: http://dml.cz/dmlcz/125583Terms of use:© Institute of Information Theory and Automation AS CR, 1984Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A - V O L U M E 20 (1984), N U M B E R 3 A SUPPLEMENT TO GOTTWALD S NOTE ON FUZZY CARDINALS MACIEJ WYGRALAK We supplement the review of fuzzy cardinality definitions placed in [3]. To be exact, we pre-sent approaches in which cardinality of a finite fuzzy subset is expressed by a fuzzy natural number and indicate the most appropriate one. S. Gottwald placed in [3] a comparative review of approaches to the problem how to define fuzzy cardinality, i.e. how to count elements of a universe which are in its fuzzy subset. In accordance with the concepts presented in [3], cardinality of a fuzzy subset was defined either as a non-negative real number or as a family of usual cardinals. In this note we shall present and compare such approaches in which cardinality of finite fuzzy subset is expressed by means of a fuzzy number. To this end, we must introduce some notation and terminology. Throughout this note, by a fuzzy subset A of some fixed universal set U we shall mean a function A : U -> / , where I := [0, l ] with := standing for "equals by the definition". Membership grade of an element x e U in A will be denoted by A(x). The classical subset {x : A(x) > 0} will be called support of A and denoted supp (A). If support of a fuzzy subset is finite, then that subset is called finite, too. Throughout the paper we shall assume that A is finite and card (supp (A)) = n, where card (M) denotes the usual cardinality of a classical subset M of U. The subset A, : = := {x : A(x) S; t], where t el0 and I0 := (0, l ] , is called «-level set of A. The sequence a0 — fll = a2 = ••• = °n > an + l ~ an + 2 = °n + 3 = ••• is defined in the following way: a0 := 1, a( ( l _ i <[ n) denotes the ith element in descending sequence consisting of positive membership grades in A, at := 0 for i > n. Let N := {0, 1, 2,...}. If F : N - » I (i.e. U := N), the F will be called fuzzy-natural number (in short, fn-number). F is said to be convex iff F(j) ^ min (F(i), 240 F(k)) for each triplet i = j g k (cf. [4]). Let © denote addition of fn-numbers. Then the fn-number F © G is defined by membership grades (F © G) (k) : = sup min (F(i), G(j)) . i + j=k As a chronologically first fuzzy approach to cardinality of finite fuzzy subsets, we shall consider the fn-number FGCount° (see [1, 7]) with membership grades F G C o u n t ° ( / c ) : = ( m a X ^ G / ^ : ^ d , ( ' 4 , Ay ' [0 if card (A,) =t= k fc or each / The values FGCount^(fc) may be considered degrees to which cardinality of A equals k. One can easy notice (see [ l ]) that FGCount" (a) is always normalized, i.e. there exists a natural number h such that FGCount°1(h) = = 1, (b) is strictly decreasing on its support, (c) is a non-convex fn-number, (d) does not fulfil the additivity property FGCount° © FGCount^ = FGCount°nB © FGCount°uB , where A n B and A u B denote (resp.) intersection and union of A and B, i.e. (A n B) (x) : = min (A(x), B(xj), (A u B) (x) : = max (A(x), B(x)). In order to avoid the lack of convexity, an important modification of the definition of FGCount° was proposed in [2] and [8]. As a consequence, we get then a new fn-number defining fuzzy cardinality, namely the FGCount^ where FGCount (k) - JmaX {' e / ° : Card M = fc} • ,(/c) . - | 0 . f ^ ^ ^ < k f Q r e a c h f Let T be a finite fn-number such that T(0j = g0, T(l) = gv ..., T(s) = gs and T(r) = 0 for r = s+1, s + 2 , . . . . In such a case we shall use the following "vectorial" notation T= (g0, gu ••-, gs). It is easy to prove (see e.g. [2], [6], [8]) that the following propositions are valid: (a) FGCount^A;) = max FGCount°(j). jzk (b) FGCount^ = (a0, au ..., a„). Hence FGCountA is convex. (c) If A <=• B, then FGCount^ c FGCountB (monotonicity). Remark. Y <= Z := (Y(x) ^ Z(x) for all x e U). (d) FGCount^ © FGCountB = FGCount^nB © FGCount^uB (additivity). Let D denote a classical n-element subset of U. Then, contrary to expectation, we get FGCountD = (1, 1, . . .A ) with support of FGCountD consisting of n + 1 elements. This result is sensible provided that FGCountA(/c) defines degree to which A has at least rather than exactly k elements. Thus FGCount, as definition of fuzzy cardinality, is unsatisfactory. Namely, for classical subsets it does not collapse to 241 usual cardinal number. That is why a next definition of fuzzy cardinality was introdu-ced in [2]. To be exact, the new definition is again a simple modification of the pre-vious one. Let S£k(A) denote the family of fe-element classical subsets (of U) containing At. Then fuzzy cardinality of A will be defined by the finite fn-number Crd^ with member-ship grades r max min A(x), Crd^(fe) : = )zej?k(A) x<=z 10 if Sek(A) is empty (if Aj is empty, what implies Sf0(A) = {0}, we additionally put min A(x) := 1). xe0 One can consider Crd^(fe) to be degree to which cardinality of A equals fc. It is easy to verify that (cf. [2], [6]) (a) Crd^ = (0, 0, ..., 0, 1, am+1, am+2, ••-, a,), where m := card (Ax) and the con-stant sequence composed of zeros is m-element one. Thus Crd̂ , is always convex. (b) CrdD = (0 , . . . , 0, 1) with the figure one placed at the (n + l)th position and D as previously. (c) Crd^ 0 CrdB = Crd^nB ® C r d ^ B . (d) Crd^ = FGCounty iff card (A,) = 0. Unfortunately, the monotonicity does not hold for Crd-cardinality. But it is quite obvious that property (b) excludes, in principle, monotonicity. On the other hand, property (b) is, from the practical as well set-theoretical points of view, more import-ant than monotonicity. This is well-known that the theory of fuzzy subsets is closely connected with the Lukasiewicz many-valued logic (see e.g. [5]). Indeed, it suffices to interpret each membership grade A(x) as representing the truth-value of the statement "x is in A". Therefore, the next approach is based on that logic. Let 3?k(A) denote the family of all the fe-element classical subsets of supp (A). Moreover, let p -* q := min( l , 1 — p + q) (Lukasiewicz implication operator) and p*-* q : = min (p ~» q, q -> p) for p,qel. Then deg (R, S) : = inf (R(x) <-» S(xj) xeU for arbitrary fuzzy subsets R and S of U. One can consider deg (R, S) to be degree to which R equals S. Let us define finite fn-number Cd^ by means of membership g r a d C S C d (k) . = /max {deg (A, Y) : Ye 0>k(A)} , A } ' [0 if 3Pk(A) is empty. Then Cd^(fe) will be considered degree to which A has exactly fe elements.'This is, in fact, a quality of the best (using the criterion deg (A, Y)) approximation of A by elements from SPk(A). One can easy verify (see [6]) that (a) Cd^(fe) = mm (ak, 1 - ak+1) for k = 0,1,2, .... (b) For the classical n-element subset D of U we get CdD(n) = 1 and CdD(/) = 0 fory # n. 242 (c) Cd^, = (1 - ai, 1 — a2, •••, 1 — ap, ap, ap+l, ..., a„), where p := min {': a, + + fl, + i ^ 1}. Hence Cd ,̂ is always convex. (d) At most one cardinal number is "favoured", i.e. there exists at most one natural number kf such that CdA(kf) > 0-5. (e) FGCount^ = 2Cd0.5X, where membership grades in 0-5A and 2Cd0.5/< are defined as follows: (0-5.4) (x): = 0-5A(x) and (2Cd0.5j4) (k) : = min (1,2Cd0.5A(kj). (f) Cd^ © CdB = Cd^nB © Cd^uB. (g) Let Ac denote the complement of A, i.e. Ac(x) := 1 — A(x). If U is finite and card (U) = m, then C d ^ j ) = Cd^(m - j) for ;' = 0, 1, . . . , m. One can easy give counterexamples that both the important properties (d) and (g) do not hold for FGCount^ and Crd^. Obviously (g) is a counterpart of the elementary law card (Dc) = m — card (D), where D denotes now a classical subset of m-element universe. To summarize the discussion, it seems to be more suitable to define cardinality of a finite fuzzy subset as a fuzzy natural rather than positive real number (or a family consisting of usual cardinals). Then the fn-number Cd^ is, from the set-theoretical point of view, defined in a most natural way and fulfills many natural postulates (see e.g. properties (b), (f), (gJJ except the monotonicity (what is, however, explicable). (Received March 3, 1983.) R E F E R E N C E S [1] D. Dubois: Proprietes de la cardinality floue d'un ensemble flou fini. BUSEFAL 5 (1981), P. Sabatier Univ., Toulouse, France, 11—12. [2] D. Dubois: A new definition of the fuzzy cardinality of finite fuzzy sets preserving the classical additivity property. BUSEFAL 8 (1981), P. Sabatier Univ., Toulouse, France, 65—67. [3] S. Gottwald: A note on fuzzy cardinals. Kybernetika 16 (1980), 156—158. [4] M. Mizumoto and K. Tanaka: Some properties of fuzzy numbers. In: Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, R. R. Yager, Eds.), North-Holland, Amsterdam 1979, 153-164. [5] M. Wygralak: A few words on the importance of Jan Lukasiewicz works for fuzzy subsets theory. In: Proc. 8th Symp. on Numerical Methods and Appl. of Math., Academy of Economy Poznan, Poland (Sept., 1982), in print. [6] M. Wygralak: On fuzzy cardinalities and fuzzy binomial coefficient. Fuzzy Sets and Systems (submitted). , [7] L. A. Zadeh: A theory of approximate reasoning. In: Machine Intelligence, Vol. 9. (J. E. Hayes, D. Michie, L. I. Mikulich, Eds.), John Wiley and Sons, New York 1979, 1 4 9 - 194. [8] L. A. Zadeh: Fuzzy probabilities and their role in decision analysis. In: Proc. IFAC Symp. on Theory and Appl. of Digital Control, New Delhi (January, 1982). Dr. Maciej Wygralak, Institute of Mathematics, A. Mickiewicz University, Matejki 48/49, 60—769 Poznan. Poland. 243 

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A supplement to Gottwald's note on fuzzy cardinals

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A supplement to Gottwald's note on fuzzy cardinals

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Institute of Mathematics AS CR, v. v. i.

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