Available online at www.sciencedirect.com Chaos, Solitons and Fractals 40 (2009) 1356–1360 www.elsevier.com/locate/chaos On the convexity of fuzzy nets Kourosh Nourouzi * Department of Mathematics, K.N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran Accepted 3 September 2007 Abstract In this paper, the convexity properties of fuzzy nets on the Euclidean space Rd is investigated. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Convex sets play a key role in quantum logics and quantum information science [2]. For instance, in quantum mechanics and classical theory, the state of a quantum mechanical system forms a convex set [1]. Also, a range of fuzzy values for an event can be expressed as a convex set. A fuzzy interpretation of convexity is that any mixture of two distributions in a set is also in the set. On the other hand, the notion of fuzzyness has a wide application in many areas of science. In physics, for example, the fuzzy structure of spacetime is followed by the fact that in strong quantum gravity regime spacetime points are determined in a fuzzy manner and therefore the impossibility of determining position of particles gives a fuzzy structure [4,5]. For more applications of fuzzy sets in physics, it is referred to [3,9,12]. In [17], Zadeh paid special attention to the convex fuzzy sets. Studies of convex fuzzy sets were followed by numerous authors [6–8,13,14,16]. In this note, a natural generalization of the concept of fuzzy sets under the name of fuzzy nets is given and then their convexity properties is investigated. 2. Fuzzy nets Let Rd denotes the d-dimensional Euclidean space. A fuzzy set in Rd is a function with domain Rd and values in the closed interval [0, 1]. For a fuzzy set l, the subset of Rd in which l assumes nonzero values, is known as the support of A (see [17]). The definition of a convex fuzzy set can be rewritten as follows: the fuzzy set l : Rd ! ½0; 1 is said to be convex if lðtx þ ð1  tÞyÞ P minflðxÞ; lðyÞg; for all x; y 2 Rd , and t 2 [0, 1]. Equivalently, l is a convex set if and only if the a-level set * Fax: +98 21 22853650. E-mail address: nourouzi@kntu.ac.ir 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.09.013 K. Nourouzi / Chaos, Solitons and Fractals 40 (2009) 1356–1360 1357 ½la ¼ fx 2 Rd : lðxÞ P ag is a convex set for all a 2 [0, 1]. Let J be an index set. A J-tuple of elements [0, 1] is a function t: J ! [0, 1]. If b is an element of J, we denote the value of t at b by tb which is bth coordinate of t. We will denote the set of all J-tuples of elements of [0, 1] by [0, 1]J. Definition 1. For an index set J, a fuzzy net l : Rd ! ½0; 1J on Rd is defined by l(x) = (li(x))i2J, where each li is a fuzzy set on Rd , and (li(x))i2J is a J-tuple. In case of J is a singleton set, the notions of fuzzy net and fuzzy set are the same, and if J ¼ N we have a fuzzy sequence (½0; 1N is the countably infinite cartesian product of [0, 1] with itself). The set of all fuzzy nets on Rd will be denoted by FNðRd Þ. Throughout of this section J stands for any index set, and # will be a fuzzy set valued function d # : FNðRd Þ ! ½0; 1R which maps each element l in FNðRd Þ to a fuzzy set #(l) on Rd . Example 1. In each the following cases # is a function which maps every element l ¼ ðli Þi2J 2 FNðRd Þ to a fuzzy set on Rd . (i) #(l) = supili, #(l) = infili, #(l) = lim supili, and #(l) = lim infili. (ii) (The projection map) #(l) = #i(l) = li. Example 2. Special implication nets on [0, 1]J Let a = (ai), b = (bi) 2 [0, 1]J. 1. Zadeh implication net N Z ða; bÞ ¼ ðð1  ai Þ _ ðai ^ bi ÞÞi2J : 2. Lukasiewicz implication net N Lu ða; bÞ ¼ ðð1  ai þ bi Þ ^ 1Þi2J : 3. Mamdani implication net N M ða; bÞ ¼ ðai ^ bi Þi2J : 4. Gaines–Rescher implication net If  1 ai 6 bi ; ci ¼ 0 ai > bi ; then N GR ða; bÞ ¼ ðci Þi2J : 5. Gödel implication net If  1; ai 6 bi ; ci ¼ bi ; ai > bi ; then N G ða; bÞ ¼ ðci Þi2J : 6. Goguen implication operator If ( 1; ai ¼ 0; ci ¼ bi ; ai > bi ; ai then 1358 K. Nourouzi / Chaos, Solitons and Fractals 40 (2009) 1356–1360 N G0 ða; bÞ ¼ ðci Þi2J : 7. Kleene–Dienes implication net N KD ða; bÞ ¼ ðð1  ai Þ _ bi Þi2J : 8. N0 implication net If  1; ci ¼ ð1  ai Þ _ bi ; ai 6 bi ; ai > bi ; then N 0 ða; bÞ ¼ ðci Þi2J :  implication net 9. N If  1; ai 6 bi ci ¼ 1  ai ; ai > bi ; then  ða; bÞ ¼ ðci Þ N i2J : Definition 2. For a 2 [0, 1], the #a-level set ½l#a of a fuzzy net l : Rd ! ½0; 1J , is defined by ½l#a ¼ fx 2 Rd : #ðlÞðxÞ P ag: Definition 3. A fuzzy net l : Rd ! ½0; 1J is called #-convex if #(l) is a convex fuzzy set on Rd . Definition 4. The fuzzy #-hypograph of a fuzzy net l : Rd ! ½0; 1J denoted by H #l is defined as H #l ¼ fðx; tÞ : x 2 Rd ; t 2 ð0; #ðlÞðxÞg: Theorem 1. A fuzzy net l : Rd ! ½0; 1J is #-convex if its fuzzy #-hypograph H #l is convex. Proof. Suppose that H #l is convex and a 2 [0, 1]. Choose x; y 2 ½l#a , and k 2 [0, 1]. Then ðx; aÞ 2 H #l , and ðy; aÞ 2 H #l . Since H #l is convex, we have ðkx þ ð1  kÞy; aÞ 2 H #l . Thus #(l)(kx + (1  k)y) P a, i.e., kx þ ð1  kÞyÞ 2 ½l#a . But ½l#a ¼ ½#ðlÞ#a . Now the fact that #(l) is convex readily implies that l is convex. h Definition 5. A fuzzy net l : Rd ! ½0; 1J is said to be closed if #a-level set of l is a closed set in Rd for each a 2 [0, 1]. Theorem 2. The fuzzy net l : Rd ! ½0; 1J is closed if and only if #-hypograph H #l is closed. # Proof. Suppose that ½la is closed for each a. Let {(xn, tn)} be a sequence of points in H #l which is convergent to (x,t). If (x,t) does not belong to H #l , then #(l)(x) < t. Choose k such that #(l)(x) < k < t. Then x lies in the complement of ½l#k . The openness of the complement of ½l#k implies that there is a positive integer N such that xn belongs to the complement of [l]k for all n P N. Thus #(l)(xn) < k for all n P N. Hence tn 6 #(l)(xn) < k for all n P N, which is a contradiction. Conversely, suppose that H #l is closed and {xn} is a convergent sequence in ½l#k to x. So #(l)(xn) P a, or ðxn ; aÞ 2 H #l . Now the closedness of H #l completes the proof. h Corollary 1. Suppose that l = (li) and k = (ki) are two fuzzy nets. Then, the fuzzy net l ^ k = (li ^ ki)i2I is a #-convex fuzzy net provided that #(l ^ k) = #(l) ^ #(k). Proof. The proof will be a simple result of the fact that ½l ^ k#a ¼ ½#ðl ^ kÞa ¼ ½#ðlÞ ^ #ðkÞa ¼ ½#ðlÞa \ ½#ðkÞa ; for all a 2 [0, 1]. h K. Nourouzi / Chaos, Solitons and Fractals 40 (2009) 1356–1360 1359 The #-support of a fuzzy net l is defined by the set S# ðlÞ ¼ fx 2 Rd : #ðlÞðxÞ > 0g: It is clear that S# ðlÞ ¼ ; if and only if #(l) = 0. Corollary 2. If l is a #-convex fuzzy net, then S# ðlÞ is a convex set. Proof. Since the fuzzy net l is #-convex if and only if #(l) is a convex fuzzy set, it implies that S# ðlÞ is convex. h Definition 6. A fuzzy net k is said to be #-dominated by the fuzzy net l and denoted by l P #k if #(l)(x) P #(k)(x) for all x 2 Rd . Definition 7. The #-convex hull of a fuzzy net k is defined to be inffl 2 FNðRd Þ : lP# k; l is #-convexg; and will be denoted by co#(k). Theorem 3. Suppose that # is surjective. Then #(co#(k)) = co#(#(k)). Proof. By definition, the convexity of fuzzy net co#(k) implies that #(co#(k)) is a convex fuzzy set. On the other hand co#(k) P k, so #(co#(k) P #(k). Hence #ðco# ðkÞÞ P co# ð#ðkÞÞ: Conversely, let l be a convex fuzzy set with l P #(k). Since # is onto, there exists a fuzzy net j such that l = #(j). It is clear that j is a convex fuzzy net and j P k. Thus j P co#(k), i.e., l = #(j) P #(co#(k)). Because l was arbitrary, we have co# ð#ðkÞÞ P #ðco# ðkÞÞ; and the proof is complete. h 3. Some applications (a) The theory of falling shadow that relates probability concepts with the membership function of fuzzy sets was introduced by Wang and Sanchez [15]. As a statistical experiment, they determined the membership function of the fuzzy concept young by three groups of students. For three groups, the membership functions were almost identical. It showed that the stability of the membership function of fuzzy notions exists in the theory of fuzzy sets. According to this, a refinement of the theory of falling shadow can be given on the basis of fuzzy nets as follows: Let P ðXÞ be the power set of the universe discourse X. Set x ¼ fAjx 2 Ag; and  ¼ fxjx 2 Ag A where x 2 X, and A 2 P ðXÞ.  # B. Now, if (D, A, P) is a A hypermeasurable structure ððP ðXÞ; BÞ on X is so that it is a r-field in P ðXÞ and X probability space and ðP ðXÞ; BÞ a hypermeasurable structure on X, then a net (ci)i is a random net on X if each ci : D ! P ðXÞ is A  B measurable. Consider the fuzzy net (li)i, where li ðxÞ ¼ Pðff 2 Djf 2 ci ðfÞgÞ: Then the fuzzy net (li)i is the falling shadow of the random net (ci)i and (ci)i is a cloud on (li)i. Example 3. Suppose that A is the Borel field on [0, 1] and m is the Lebesgue measure. Associated with the fuzzy net (Ai)i we define Aik ¼ fx 2 X jAi ðxÞ P kg; 1360 K. Nourouzi / Chaos, Solitons and Fractals 40 (2009) 1356–1360 for every k 2 [0, 1]. Now, the net (ci)i with ci : ½0; 1 ! P ðXÞ; k ! Aik ; is a random net and (ci) is a cloud on (Ai). (b) In quantum mechanics, the state f of a physical entity S is represented by a unit vector xf in some Hilbert space H. Also, an experiment r on S is represented by a self-adjoint operator Tr on H. In the case that the experiment r has only two possible outcomes, then the self-adjoint operator Tr is an orthogonal projection. The probability P to get an outcome for the experiment r is given by kTr(xf)k2. This quantum mechanical probability leads to fuzzy sets related to a quantum mechanical measurement situation [10,11]. Let CðSÞ be the set of all states of the entity S. Each family of experiments {ri}i2I defines a fuzzy net l ¼ ðlri Þ on CðSÞ via lri ðf Þ ¼ kT ri ðxf Þk2 . If X is the set of all two possible experiments, then every family of states {fi}i2I also defines a fuzzy net u = (ui)i2I on X as ui ðrÞ ¼ kT r ðxfi Þk2 . 4. Conclusion A generalization of fuzzy sets, that is fuzzy nets has been given. According to this concept, some results related to convexity of fuzzy sets has been extended. References [1] Bloore FJ. Geometrical description of the convex sets of states for systems with spin-1/2 and spin-1. J Phys A: Math Gen 1976;9:2059–67. [2] Bunce LJ, Maitland Wright JD. 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