Research On Cauchy Inequalities In Fuzzy Mathematics And Their Applications

Fuzzy theory has been widely present in every corner of society, serving as the mathematical tools that deal with the uncertainty of things, which has a very wide range of applications in the information age. This paper has introduced Cauchy inequalities and their application in mathematics, which based on mathematics theory, and explored the conditions and expression of Cauchy inequality in fuzzy math, which started from the fuzzy probability and variety of fuzzy integrals. This paper mainly includes the following:Part Ⅰ:The development of Cauchy inequality and fuzzy math at home and abroad has been introduced. At the same time, the purpose and significance of this paper has been given.Part Ⅱ:Classic fuzzy decomposition has been introduced, simultaneously a new fuzzy decomposition has been proposed.Part Ⅲ:The fuzzy integrals on fuzzy sets have been given. Based on the characteristics of these integrations, the Cauchy inequalities in this fuzzy integration have been discussed.Part Ⅳ:Sugeno integral, Choquet integral, seminormed fuzzy integral and semiconormed fuzzy integral have been proposed. Although the Cauchy inequality can not establish on these integration, this paper has given the condition in which the Cauchy type inequalities satisfy. Finally, this paper has proposed the new proofs.Part Ⅴ:This paper has introduced fuzzy probability random variable and fuzzy random variable, discussed the Cauchy inequality in fuzzy probability, and given the Cauchy inequality and its application in random fuzzy variable.