The Bound Of Fuzzy Number Set And Approximation Of Fuzzy Number

Fuzzy phenomena exist widely in engineering and real life.However,the traditional classical set theory can only describe the phenomenon that the boundary of concepts has clear boundaries,and can not describe its fuzzy phenomena.In order to describe the fuzzy phenomena with uncertain boundary attributes of concepts more accurately by mathematical methods,Zadeh put forward the concept of fuzzy sets in 1965.Fuzzy number is a special kind of fuzzy set,which is a kind of fuzzy subset with special properties in real number field.Fuzzy numbers can be used as a tool to describe uncertainties or inaccuracies.Just as the upper and lower bounds of real number set are important contents of real number theory,the upper and lower bounds of fuzzy number set are also important research contents of fuzzy number theory.On the other hand,it is well known that approximation theory is an important research field in mathematics.Its importance is not only reflected in mathematical theory,but also has a very wide range of applications.Similarly,exploring the problem of approximating general fuzzy numbers with simple fuzzy numbers is also an important content in the study of fuzzy set theory.The main work of this paper is divided into two parts: Firstly,based on the partial order defined by the inclusion relation of fuzzy sets,we discuss the upper and lower bounds of bounded sets in High-dimensional Fuzzy ellipsoid space,prove the existence of upper and lower bounds and give their concrete expressions.Secondly,by using weighted distance as a measure,we discuss the problem of approximating general fuzzy numbers with simple fuzzy numbers under the condition that the level set of fuzzy numbers is known and the abscissa of fuzzy numbers is known,and give the expression of its best approximation,and obtain some properties of two kinds of approximation operators.The structure of this paper is as follows:1.In the first chapter,we summarize the development history of fuzzy numbers,the research status of the upper bound,upper bound,lower bound,lower bound of fuzzy numbers and the problem of approximating general fuzzy numbers with some special fuzzy numbers.2.In the second chapter,we summarize some concepts in the theory of fuzzy sets,and introduce some results and notations needed in the following chapters.3.In the third chapter,we give a partial order relation of fuzzy numbers based on the inclusion relation of fuzzy sets.Under this partial order relation,we give the definitions of upper and lower bounds of sets in High-dimensional Fuzzy ellipsoid number space,prove that there must be upper and lower bounds of bounded sets,and obtain the concrete expressions of upper and lower bounds.4.In the fourth chapter,we use weighted distance as metric to prove the existence of the best approximation of general fuzzy numbers by simple fuzzy numbers under the conditions of given level set of fuzzy numbers and abscissa of given fuzzy numbers,and give the expression of its best approximation,and some properties of the best approximation.5.In the fifth chapter,we make a summary and outlook of the article.