Some Properties in Fuzzy Convex Sets

Liu Ying-Ming
Additional contact information
Liu Ying-Ming: Sichuan University, Department of Mathematics

A chapter in Advances in Fuzzy Sets, Possibility Theory, and Applications, 1983, pp 37-47 from Springer

Abstract: Abstract In the basic and classical paper [10], where the important concept of fuzzy set was first introduced, L. A. Zadeh developed a basic framework to treat mathematically the fuzzy phenomena or systems which, due to intrinsic indefiniteness, cannot themselves be characterized precisely. He pays special attention to the investigation on the fuzzy convex sets which consists of nearly the second half of the space of his paper. The main results on fuzzy convex sets given in [10] are summarized as follows: (1) The separation theorem; and (2) The theorem on the shadows of fuzzy convex sets. The revised correct version of the separation theorem has been given in [9] by employing induced fuzzy topology. Using the concept of fuzzy hyperplane, Lowen has established some further separation theorem for fuzzy convex sets [6]. Concerning the theorem of shadow of fuzzy convex sets, Zadeh has further investigated in [11]. But in this respect, there exists still some drawbacks which will be shown via a counterexample in the present paper. Perhaps the lack of fuzzy topological assumption in the above mentioned results leads to the appearance of these shortcomings. Such a situation seems to be only natural in the early stage of development of the fuzzy set theory. Adding some assumptions about fuzzy topology, we are able to yield several positive results on the shadows of fuzzy convex sets. Finally we shall give some simple and direct proofs of two theorems that describe the relationships between the fuzzy convex cones and the fuzzy linear subspaces and that originally appeared in [6]. The present proofs do not appeal to the representation theorem established in [6] by R. Lowen.

Keywords: Separation Theorem; Present Proof; Fuzzy Topology; Fuzzy Topological Space; Lower Case Greek Letter (search for similar items in EconPapers)
Date: 1983
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4613-3754-6_4

Ordering information: This item can be ordered from
http://www.springer.com/9781461337546

DOI: 10.1007/978-1-4613-3754-6_4

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().