The n-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets

  • Authors:

  • You-guang Shang;Xue-hai Yuan;E. Stanley Lee

  • Affiliations:

  • School of Mathematical Sciences, Dalian University of Technology, Dalian,116024, PR China;Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, PR China;Department of Industrial and Manufacturing Systems Engineering, Kansas State University, Manhattan, KS66506, USA

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2010

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Abstract

The connections among the n-dimensional fuzzy set, Zadeh fuzzy set and the finite-valued fuzzy set are established in this paper. The n-dimensional fuzzy set, a special L-fuzzy set, is first defined. It is pointed out that the n-dimensional fuzzy set is a generalization of the Zadeh fuzzy set, the interval-valued fuzzy set, the intuitionistic fuzzy set, the interval-valued intuitionistic fuzzy set and the three dimensional fuzzy set. Then, the definitions of cut set on n-dimensional fuzzy set and n-dimensional vector level cut set of Zadeh fuzzy set are presented. The cut set of the n-dimensional fuzzy set and n-dimensional vector level set of the Zadeh fuzzy set are both defined as n+1-valued fuzzy sets. It is shown that a cut set defined in this way has the same properties as a normal cut set of the Zadeh fuzzy set. Finally, by the use of these cut sets, decomposition and representation theorems of the n-dimensional fuzzy set and new decomposition and representation theorems of the Zadeh fuzzy set are constructed.