“Imprecision” here is meant in the sense of vagueness rather than the lack of knowledge about the value of a parameter as in tolerance analysis. Fuzzy set theory provides a strict mathematical framework (there is nothing fuzzy about fuzzy set theory!) in which vague conceptual phenomena can be precisely and rigorously studied. It can also be considered as a modeling language well suited for situations in which fuzzy relations, criteria , and phenomena exist. Fuzziness has so far not been defined uniquely semantically, and probably never will. It will mean different things, depending on the application area and the way it is measured. This book entitled “Study on Posbist Reliability Theory and its Applications” present some important aspects Fuzzy Set theory and some recent results in its development. This embodies four chapters . The first chapter consists of an introduction on fuzzy set theory and different ideas with definitions that lead to the growth and extension of Fuzzy Set Theory. In second chapter we describe fuzzy variables. The third chapter consists of study on Posbist systems. At last chapter, we have presented a brief description on application of Fuzzy Set Theory.