Complex Fuzzy Set-Valued Complex Fuzzy Measures and Their Properties

Hindawi Publishing Corporation e Scientiﬁc World Journal Volume 2014, Article ID 493703, 7 pages http://dx.doi.org/10.1155/2014/493703 Research Article Complex Fuzzy Set-Valued Complex Fuzzy Measures and Their Properties Shengquan Ma1,2 and Shenggang Li1 1 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China 2 School of Information and Technology, Hainan Normal University, Haikou 571158, China Correspondence should be addressed to Shengquan Ma; [email protected] Received 21 April 2014; Revised 27 May 2014; Accepted 29 May 2014; Published 24 June 2014 Academic Editor: Jianming Zhan Copyright © 2014 S. Ma and S. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ∗ Let () be the set of all fuzzy complex numbers. In this paper some classical and measure-theoretical notions are extended to the ∗ case of complex fuzzy sets. They are fuzzy complex number-valued distance on (), fuzzy complex number-valued measure ∗ on (), and some related notions, such as null-additivity, pseudo-null-additivity, null-subtraction, pseudo-null-subtraction, autocontionuous from above, autocontionuous from below, and autocontinuity of the defined fuzzy complex number-valued measures. Properties of fuzzy complex number-valued measures are studied in detail. 1. Introduction fuzzy measure to fuzzy set [10]. In 1989, Buckley presented the concept of fuzzy complex number [11] which inspired that It is well known that additivity of a classical measure primly people needed to consider the measure and integral problem depicted measure problems under error-free condition. But about fuzzy complex number. when measure error was unavoidable, additivity could not At the beginning of the 90s, Guang-Quan [12–21]intro- fully depict the measure problems under certain condition. duced fuzzy real distance and discussed the fuzzy real To overcome such difficulties, fuzzy measure has been devel- measure based on fuzzy sets and then gave the fuzzy real oped. Research on fuzzy measures was very deep in those value fuzzy integral and established fuzzy real valued measure aspects: research based on a certain number of subsets of theory on fuzzy set space. During 1991-1992, Buckley and Qu a classic set and a real value nonaddable measure (such as [22, 23] studied the problems of fuzzy complex analysis: fuzzy Choquet’s content theory [1], Sugeno’s measure theory [2]), complex function differential and fuzzy complex function research based on fuzzy sets and a real value measure (e.g., integral. During 1996–2001, Qiu et al. studied serially basic Zadeh’s addable measure [3]), and especially the research on problems of fuzzy complex analysis theory, including the fuzzy value measures which generalizes the set value measure continuity of fuzzy complex numbers and fuzzy complex theory. valued series [24], fuzzy complex valued functions and their Being a newly developing theory developed in the later differentiability25 [ ], and fuzzy complex valued measure and 1960s, set value measure had been applied in many fields [4– fuzzy complex valued integral function [26, 27]. Wang and 6]. After the appearance of fuzzy numbers, people naturally Li [28]gavethefuzzycomplexvaluedmeasurebasedonthe thought of related measure and integral. In 1986 Zhang [7] fuzzy complex number concept of Buckley, studied Lebesgue defined a kind of fuzzy set measure on ,in1998Wu integraloffuzzycomplexvaluedfunction,andobtainedsome et al. generalized the codomain of fuzzy measure to fuzzy important results. real number field and defined the Sugeno integral of fuzzy Asforapplicationsoffuzzycomplexnumbertheory, number fuzzy measure [8], and Guo et al. also defined the Ramotetal.[29, 30]studiedcomplexfuzzysetsandcomplex fuzzy value measure integral of fuzzy value function [9] fuzzy logic, Dick [31] studied fuzzy complex logic more pro- which generalized the Sugeno integral about fuzzy value foundly, Ha et al. [32] applied fuzzy complex set in statistical 2 The Scientific World Journal ∗ learning theory and obtained a key theorem of statistical ∈() is said to be a fuzzy infinity [21](writtenas∞̃) ̃ learning theory, Fu and Shen [33] studied modeling problems if one of the supports of =̃ Re and =Im is an ∗ of fuzzy complex number, and Fu and Shen [34]applied unbound set. For any 1,2 ∈ (),onemakesthefollowing fuzzy complex in pattern recognition and classification and appointments: obtained important results. Please see [35–37]forother applications. 1 ≤2 if and only if Re 1 ≤ Re 2 and Im 1 ≤ Im 2; This paper will extend the classical measure to fuzzy = if and only if ≤ and ≤; complex number-valued measure, which can better express 1 2 1 2 2 1 the interactions among the attributes (cf. [32, 34–37]) and, 1 <2 if and only if 1 ≤2 and Re 1 ≤ Re 2 or thus, is expected to have extensive applications in informa- Im 1 < Im 2; tion fusion technology, classification technology, machine ≥0 ≥0 ≥0 learning, pattern recognition, and other fields. Section 2 is if and only if Re ,Im . ∗ some preliminary notions (including fuzzy complex number, One uses A to denote a family (which is obviously non- ∗ real fuzzy distance between two fuzzy real numbers, and empty) of subsets of () that satisfies the following two fuzzy complex numbers) and some basic operations conditions: and order relation of fuzzy complex. Section 3 is prepared ̃ ∗ ̃ ̃ ̃ ̃ for the next section. We defined, based on Ha’s work (1) for each ∈A ,if={inf 0 | 0 ⊂ } has upper ∗ [38], the concepts of fuzzy complex distance and complex bound, then sup ∈̃ (); fuzzy set value complex fuzzy measure (an extension of ̃ ∗ ̃ ̃ ̃ ̃ (2) for each ∈A ,if={sup 0 | 0 ⊂ } has lower fuzzy measure) on fuzzy complex number field. We also ̃ ∗ present, based on Zhang’s work [21], the concepts of null- bound, then sup ∈ (). additivity, pseudo-null-additivity, null-subtraction, pseudo- null-subtraction, autocontinuous from above, autocontinu- 2.2. Fuzzy Distance of Fuzzy Numbers ous from below, and autocontinuity of fuzzy complex value ∗ ∗ fuzzy complex measure on complex fuzzy number set (this Definition 2 (see cf. [21]). A mapping :̃ () × () → ∗ measurehasthepropertiesPGPandSA/SB).InSection 4, () satisfying the following conditions is called a fuzzy ∗ we deduced some important properties on complex fuzzy set metric or a fuzzy distance on (): value complex fuzzy measure which are generalizations of the ̃ ∗ ̃ ̃ corresponding results in measure theory; we also obtain some (1) for any ,̃ ∈(), (̃ ,̃ ) ≥,and 0 (̃ ,̃ ) = 0 if ̃ results on related integral theory. and only if =̃ ; ̃ ∗ ̃ ̃ (2) for any ,̃ ∈ (), (̃ ,̃ ) = (̃ , )̃ ; 2. Preliminaries ̃ ∗ ̃ ̃ (3) for any ,̃ , ∈̃ (), (̃ ,̃ ) ≤ (̃ ,̃ )̃ + (̃ ,̃ ). 2.1. Fuzzy Complex Numbers. In this paper, is the set of all ̃ ̃ ̃ ̃ real numbers set, is the set of all complex numbers, is an (, ) is called the fuzzy distance of fuzzy real numbers and ∗ ̃ ordinary set, () isthesetofallrealfuzzynumberson, . Δ() is the set of all interval numbers, (, A) is a measurable ∗ ∗ ∗ ∗ space (thus A is a -algbra), and () is the set of all fuzzy Example 3. The mapping :̃ () × () → () ∗ ∗ complex numbers on .Let+ () = {:̃ ≥0,̃ ∈̃ ()} defined by ∗ ̃ ̃ ̃ ̃ ∗ √ and let + () = {+|, ∈+ (), = −1}. ̃(,̃ ̃) = ⋃ [− −− , − −− ∨ + −+] ̃ ∗ 1 1 sup Definition 1 (see [12]). Let ,̃ ∈().Thenthemapping ∈[0,1] ≤≤1 ̃ ̃ ̃ (,̃ ) : → [0, 1] defined by (,̃ )(+)=()̃ ∧ () is (1) called a fuzzy complex number, where ̃ is called the real part ̃ ̃ ̃ ∗ of (,̃ ) (written as Re(,̃ ))and is called the imaginary is a fuzzy distance on (). ̃ √ part (written as Im(,̃ )), and = −1. One will identify ∗ ∗ ̃ Remark 4. Analogously, a mapping :̃ () × () → (,̃ 0) with ̃ and, thus, think fuzzy complex numbers are an ∗ () satisfying the following conditions is called a fuzzy extension of fuzzy real numbers. The set of all fuzzy complex ∗ ∗ () numbers on is denoted by (). metric or a fuzzy distance on : , (, ) ≜ + = {+ | For any subsets of ,write ,̃ ̃∈∗() (̃ ,̃ ̃) ≥ 0 (̃ ,̃ ̃) = 0 ∈ , ∈} ∘ ∈ {+,−,⋅} (1) for any , ,and if .Theoperation is described as =̃ ̃ follows: and only if ; ,̃ ̃∈∗() (̃ ,̃ ̃) = (̃ ̃, )̃ (2) for any , ; (1) 1 ∘2 =(Re 1 ∘ Re 2, Im 1 ∘ Im 2) for any 1,2 ∈ ∗ ,̃ ̃, ∈̃ ∗() (̃ ,̃ ̃) ≤ (̃ ,̃ )̃ + (̃ ,̃ ̃) (); (3) for any , . ∗ ̃ ̃ ̃ (2) ⋅1 =(Re 1,Im 1) for any 1 ∈() and = (, ) is called the fuzzy distance of fuzzy complex numbers ̃ (, ) ∈ (=+,,∈,=√−1). ̃ and . The Scientific World Journal 3 ∗ ̃ Example 5. Let ̃be a fuzzy distance on ().Thenthemap- to be convergent to according to distance ̃, denoted by :∗() × ∗() → ∗() ̃ ̃ ping defined by ()̃ lim→∞ = . (1,2)=(̃ Re 1, Re 2)∨(̃ Im 1, Im 2) (2) Definition 11. Let be a nonempty complex number set, let () be the set of all complex fuzzy sets on , and let ̃ be a ∗ is a fuzzy distance on (). fuzzy complex value metric on (). A complex fuzzy set- value complex fuzzy measure is a mapping ̃ : () → {}⊂∗() ∈ ∗ ∗ ̃ ̃ ̃ ̃ ∗ √ Definition 6 (see cf. [12]). Let and let + () (where + () =+ { |, ∈+ ()}, = −1) ∗() {} . is said to converge to according to a fuzzy metric which satisfies the following conditions: ∗ on () (written as lim→∞ =) if, for each >0, (1) (⌀)̃ =; 0 there exists a positive integer such that (, )<for all ≥ . (2) for any ,̃ ̃ ∈ () satisfying ⊂̃ ̃, (̃ )̃ ≤ (̃ )̃ (i.e., Re (̃ )̃ ≤ Re (̃ )̃ and Im (̃ )̃ ≤ Im (̃ )̃ ); 3. Complex Fuzzy Set-Valued Complex ̃ (3) (lower semicontinuous) if {} ⊂ () with ⊂ Fuzzy Measures ̃ ̃ +1, ( = 1,2,...),then()̃ lim→∞(̃ )= (⋃̃ ∞ ̃ ) The notion of complex fuzzy measure on family of classical =1 ; ̃ ̃ sets was given in [26].

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