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  1. 2 Mathematica Applicanda

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  1. 2 Heilpern S.

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  1. 1 1983

  2. 1 1980

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Selected problems in the theory of fuzzy sets

100%
Heilpern S.
Mathematica Applicanda
|
1980
|
tom 8
|
nr 16
PL
.
EN
From the introduction: "This paper contains a review of fundamental concepts and theorems of some areas of fuzzy mathematics, and an example of their application to the theory of decision making. Elementary definitions and properties of fuzzy sets are introduced in Chapters 1 and 2 [see L. A. Zadeh, Informat. and Control 8 (1965), 338–353; MR0219427]. Chapter 3 contains rudiments of fuzzy topology as presented by C. L. Chang [J. Math. Anal. Appl. 24 (1968), 182–190; MR0236859] and C. K. Wong [ibid. 43 (1973), 697–704; MR0324613; ibid. 45 (1974), 512–521; MR0341366]. Subsequent chapters deal with fuzzy probabilistic measures on the σ-field of fuzzy sets and contain original results of the author. In the last chapter the author discusses the concepts of fuzzy programming based on papers of R. Bellman and Zadeh [Management Sci. 17 (1970/71), B141-B164; MR0301613], and C. V. Negoiţă and D. A. Ralescu [Applications of fuzzy sets to systems analysis, English translation, Wiley, New York, 1975; MR0490082].''
2
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Fuzzy mappings

100%
Heilpern S.
Mathematica Applicanda
|
1983
|
tom 11
|
nr 22
PL
.
EN
Let X be the class of all fuzzy subsets of a metric space X. A fuzzy subset A is called an approximate value if A is a closed and convex fuzzy subset with supA(x)=1; the class of all such elements is denoted by W(X), and it is a metric space with the distance D(A,B)=sup dist(Aα,Bα), where Aα and Bα denote the α-level of A and B, respectively, and dist( , ) denotes the generalized Hausdorff distance [see, e.g., M. P. Chen and M. H. Shin , J. Math. Anal. Appl. 71 (1979), no. 2, 516–524; MR0548780]. The author is especially concerned with W(R). Algebraic operations in W(R) are defined and basic rules for arithmetic operations on approximate values are proved. Moreover, functions with values in W(R) are also investigated. Finally, a fixed point theorem for fuzzy mappings is stated and an example is given [for the proof see the author, ibid. 83 (1981), no. 2, 566–569; MR0641351].
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