Soft Fuzzy and fuzzy soft set theory With applications

we classify the fuzzy sets families which produced seven different families and two new definitions for fuzzy points. A definition of families which is called soft sets and by using the composition technique between fuzzy points and soft points we get new three different fuzzy soft points also via nine combinations between soft points and fuzzy points, new five soft fuzzy points are extracted .For the new three different fuzzy soft points, fuzzy soft topological space also for the new five different soft fuzzy points, soft fuzzy topological space is studied respectively. It worth to say that it is possible to define adherence, accumulation, exterior, interior and boundary points according to fuzzy soft sets and on the soft fuzzy sets with studying the separation axioms on them added to the above as well .Soft probability and soft fuzzy reliability in topological spaces are presented in light of concepts of soft sets and probability theory, to show the relation between soft probability and topology is presented via an illustrative example. Evaluation of reliability for different types of systems as an application concern different kind of uncertainty by using distinct mathematical tools as vague sets, intuitionistic fuzzy sets, fuzzy soft sets and soft fuzzy set is presented and added to this applications to clarify the relation between the topology and the confidence interval through an example.