On the level convergence of a sequence of fuzzy numbers

Abstract

In this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space E 1 established by Wu and Wu (J. Math. Anal. Appl. 210 (1997) 499) and improve the expressions of the supremum and infimum. As a straightforward corollary of this result, we obtain a necessary and sufficient condition under which, for a bounded sequence of fuzzy numbers { u n }, the pair of functions sup n u n −( λ) and sup n u n +( λ) can determine a fuzzy number. Secondly, we give a necessary and sufficient condition for a sequence of fuzzy numbers { u n } to be levelwise convergent in E 1, and generalize some important theorems in real number spaces to fuzzy number spaces. Finally, we prove the existence of supremum and infimum for the level-continuous fuzzy-valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained.