Revisiting separation properties of convex fuzzy sets

Article type: Research Article

Authors: Gómez-Acevedo, Horacio^*

Affiliations: Arkansas Children’s Nutrition Center and Department of Pediatrics, University of Arkansas for Medical Sciences, Little Rock, AR, USA

Correspondence: [*] Correspondence to: Horacio Gómez-Acevedo, Arkansas Children’s Nutrition Center and Department of Pediatrics, University of Arkansas for Medical Sciences, 15 Children’s Way, Little Rock, AR 72202, USA. Tel.: +1 501 364 2792; gomezacevedohoracio@uams.edu

Abstract: Separation of convex sets by hyperplanes has been extensively studied on crisp sets. In a seminal paper from L. A. Zadeh [1] separability and convexity are investigated, however there is a flaw on the definition of degree of separation. We revisited separation on convex fuzzy sets that have level-wise (crisp) disjointness with non-empty interior at certain level and introduced the concept of minimal level of separation for such fuzzy sets. On this context, the smallest level in which a separation by a hyperplane occurs coincides with the maximal degree of the (fuzzy) intersection. Moreover, this property suggests an algorithm for finding the maximal grade of a (fuzzy) intersection based on hyperplane separability level-wise of fuzzy sets.

Keywords: Basic concepts, fuzzy convexity, separation of convex fuzzy sets

DOI: 10.3233/IFS-151613

Journal: Journal of Intelligent & Fuzzy Systems, vol. 29, no. 2, pp. 845-849, 2015