Transforms of Undefined Sets, Undefined Structures, and Undefined β-Systems

The concept proposed by Zadeh.L.A.[82] defining a fuzzy subset A of a given universe X characterizing the membership of an element x of X belonging to A by means of a membership function µA etc., in handling the imprecise real life situations mathematically. Now, several branches of fuzzy mathematics like fuzzy algebra, fuzzy topology, fuzzy control theory, fuzzy measure the- ory etc. have emerged. But in the decision making, the fuzzy theory takes care of membership of an element x only, that is the evidence of x belonging to A. It does not take care of the evidence against x belonging to A. It is felt by several decision makers and researchers that in proper decision making, the evidence of x belonging to A and evidence not belonging to A are both necessary, and how much x belongs to A or how much x does not belong to A are necessary. This is the problem with fuzzy sets. To counter this problem Gau.W.L. and Buehrer.D.J[28] introduced the concept of Vague Set Theory.