Theory of Fuzzy Soft Sets from a New Perspective

Abstract

The theory of fuzzy sets introduced by Zadehshould actually have been a generalization of the classicaltheory of sets in the sense that the theory of sets shouldhave been a special case of the theory of fuzzy sets.Unfortunately, this is not the case. It has been acceptedthat for a fuzzy set A and its complement A c , neither A  A c is the null set, nor A Ac is the universal set.Whereas the operations of union and intersection of twocrisp sets are indeed special cases of the correspondingoperations of two fuzzy sets, they end up giving peculiarresults while defining A Ac and A Ac . In thisregard, H. K. Baruah proposed that in the currentdefinition of the complement of a fuzzy set, fuzzymembership function and fuzzy membership value hadbeen taken to be the same, which led to the conclusionthat the fuzzy sets do not follow the set theoretic axiomsof exclusion and contradiction. For the complement of anormal fuzzy set, fuzzy membership function and fuzzymembership value are two different things, and thecomplement of a normal fuzzy set has to be definedaccordingly. H. K. Baruah has reintroduced thedefinition of fuzzy set and redefined the complement of afuzzy set accordingly. In 1999, Molodstov introduced thetheory of soft sets, which can be seen as a newmathematical approach to vagueness. The theory offuzzy soft set, a more generalized concept, initiated byMaji, is a combination of fuzzy set and soft set. In thispaper, we improve the notion of union and intersection offuzzy sets proposed by Baruah and generalize theconcept of complement of a fuzzy set when the fuzzyreference function is not zero. We are defining arbitraryfuzzy union and intersection using the definition of fuzzysets given by Baruah and then we are using thisdefinition of fuzzy set and newly defined complement inthe fuzzy sets occurring in a fuzzy soft set so that fuzzysoft sets, too, follow the axioms of exclusion andcontradiction. Accordingly we propose some new notionsregarding fuzzy subset, null fuzzy soft set, absolute fuzzysoft set, complement of a fuzzy soft set. We put forwarddefinitions of “AND” and “OR” operations for anarbitrary collection of fuzzy soft sets and finallyDeMorgan inclusions and DeMorgan laws proved byAhmad and Kharal in fuzzy soft set theory have beenverified according to our new notions.