Intersection and union of non-normal fuzzy sets using aggregation functions

Article type: Research Article

Authors: Wu, Hsien-Chung^{; *}

Affiliations: Department of Mathematics, National Kaohsiung Normal University, Taiwan

Correspondence: [*] Corresponding author. Hsien-Chung Wu, Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan. E-mail: hcwu@nknucc.nknu.edu.tw.

Abstract: The intersection and union of non-normal fuzzy sets using general aggregation functions are studied in this paper. The conventional intersection and union of fuzzy sets are based on the membership functions using the max and min functions, or t-norm and s-norm in general in which the fuzzy sets may assume to be normal in order to follow the boundary conditions that include 0 and 1 in the unit interval [0, 1]. When the fuzzy sets are taken to be non-normal, the α-level sets may be empty for some α ∈ [0, 1]. In this paper, we shall consider the aggregation functions, instead of t-norm and s-norm, satisfying the concepts of compatibility that are based on the α-level sets of intersection and union. In this case, the emptiness of α-level sets should be avoided, which is the main issue of this paper.

Keywords: Aggregation functions; Interval range; Normal fuzzy sets; Set intersection; Set union.

DOI: 10.3233/JIFS-190270

Journal: Journal of Intelligent & Fuzzy Systems, vol. 37, no. 3, pp. 4113-4132, 2019