A Special Class Of Fuzzy Sets On The Differential And Integral Calculus

The recognition and knowledge of human to the objective world deepen with the time and time again. With the endless supply of information, man's recognition and understanding of the world also become endless. In 1965, a American expert of control theory named Zadeh wrote a paper titled "Fuzzy Set" which results in the birth of fuzzy mathematics. It also means the people's recognition has been leveled up. After the development in recent years, fuzzy mathematics has become a very profound science which is widely used in various fields such as biology, the control theory, medicine, and the theory of systems.In this paper, we mainly associate fuzzy mathematics with analytical mathematics and study a special fuzzy set named the fuzzy set with fuzzy borders of real line.In Introduction Chapter, we give a brief illustration of the background and the central idea of this paper. We also introduction some essential basic knowledge.In Chapter Two, we give and prove the mid-valued differential theorem on fuzzy set with fuzzy borders of real line. With it, we will generalize the Taylor's formula and L'Hospital's rule. We also research the cuts of this kind of fuzzy set and give new concept called the quotient of the cut of two fuzzy numbers and give some properties of this kind of quotient.In Chapter Three, we will give some basic theorems on the fuzzy set with fuzzy borders of real line. Otherwise, the meaning of the integration on this kind of set will be given.