INTUITIONISTIC FUZZY HYPERSOFT SETS

In this paper, a new environment namely, intuitionistic fuzzy hypersoft set (IFHSS) is dened. We introduce some fundamental operators of intuitionistic fuzzy hypersoft sets such as subset, null set, absolute set, complement, union, intersection, equal set etc. Validity and application are presented with appropriate examples. For greater precision and accuracy, in the future, proposed operations in decision making processes such as personal selection, management issues and others will play a vital role.


Introduction
The fuzzy set theory identi…ed in 1965 by Zadeh [26] is one of the most popular theories of recent times. Zadeh speci…ed that there is a considerable amount of ambiguity in most real-life situations and physical problems that the classical set theory and its normal mathematical theories centered on such set theory did not give us the necessary knowledge and inferences. Fuzzy set theory has brought a great paradigmatic change in mathematics, but this theory also has some structural di¢ culties in its nature. Fuzzy set structure is de…ned with the help of membership function. It's di¢ cult to create a membership function for each event, according to Molodtsov, because creating a membership function is too individual.
Molodtsov [15] introduced the soft set theory in 1999 which he felt was more practical. This theory is a relatively new approach to solving problems involving decision making and uncertainty. The major bene…t of soft set theory is that in fuzzy set and other theories it is free from di¢ culties. Soft set theory has become popular among researchers in a short time and many scienti…c studies are carried out on this theory every year [8,16,17,27]. Maji et al. [12,14] proposed an implementation of soft sets focused on parameter reduction to preserve optimum selection artifacts in decision-making problems. Chen [5] investigated a new de…nition and various applications of the reduction of soft sets to parameters. Pei and Miao [19] have shown that soft sets are a special class of information systems. Kong et al. [9] introduced the reduction and algorithm of soft sets to normal parameters. Zou and Xiao [28] discussed the soft data analysis approach. Aktaş and Ça¼ gman [2] de…ned the algebraic structure of soft set theory.
The intuitionistic fuzzy set (IFS) theory [3,4] was created by adding a nonmembership function to the fuzzy set structure. The non-member function makes IFS more functional in decision-making problems. Maji et al. [10,13] combined soft set theory with the theory of the intuitionistic fuzzy sets and called intuitionistic fuzzy soft Set (IFSS). The parameterization and hesitancy acquired by IFSS from this mixture facilitates very accurately the description of real-world situations. IFSS is a valuable method for addressing data uncertainty and vagueness. Many scienti…c paper have shown the suitability of IFSS to issue decision making [7,11].
Smarandache [21] introduced a new technique to deal with uncertainty. By converting the function into multiple decision functions, he generalized the soft to hypersoft set. Many studies have been done recently using hypersoft set structure [1,6,18,20,22,23,24,25].
Multi-criteria decision-making (MCDM) is concerned with coordinating and taking care of matters of preference and preparation, including multi-criteria. Intuitionistic fuzzy soft set environments can not be used to solve certain types of problems if attributes are more than one and further bifurcated. Therefore, there was a serious need to identify a new approach to solve such problems, so a new setting, namely the intuitionistic fuzzy hypersoft Sets (IFHSS), is established for this reason. In the present paper, we introduce intuitionistic fuzzy hypersoft set theory. Intuitionistic fuzzy hypersoft set theory is a mixture of IFS theory and the hypersoft set theory. The complement, subset, equal set, "AND", "OR", intersection, union notions are de…ned on intuitionistic fuzzy hypersoft sets. This paper also is supported by many suitable examples.

Preliminary
denote the degree of membership and nonmembership of u to H; respectively. The set of all intuitionistic fuzzy sets over U will be denoted by IF P (U ): De…nition 2. [15] Let U be an initial universe and E be a set of parameters. A pair (H; E) is called a soft set over U; where H is a mapping H : E ! P(U ): In other words, the soft set is a parameterized family of subsets of the set U: De…nition 3. [13] Let U be an initial universe and E be a set of parameters: A pair (H; E) is called an intuitionistic fuzzy soft set over U; where H is a mapping given by, H : E ! IF P (U ): In general, for every e 2 E; H(e) is an intuitionistic fuzzy set of U and it is called intuitionistic fuzzy value set of parameter e: Clearly, H(e) can be written as a intuitionistic fuzzy set such that H(e) = f(u; H (u); H (u)) : u 2 U g: De…nition 4. [21] Let U be the universal set and P (U ) be the power set of U . Consider e 1 ; e 2 ; e 3 ; :::; e n for n 1, be n well-de…ned attributes, whose corresponding attribute values are resspectively the sets E 1 ; E 2 ; :::; E n with E i \ E j = ;; for i 6 = j and i; j 2 f1; 2; :::; ng; then the pair (H; E 1 E 2 ::: E n ) is said to be Hypersoft set over U where H : E 1 E 2 ::: E n ! P (U ):

Intuitionistic Fuzzy Hypersoft Sets
De…nition 5. Let U be the universal set and IF P (U ) be the intuitionistic fuzzy power set of U . Consider e 1 ; e 2 ; e 3 ; :::; e n for n 1, be n well-de…ned attributes, whose corresponding attribute values are respectively the sets E 1 ; E 2 ; :::; E n with E i \ E j = ;; for i 6 = j and i; j 2 f1; 2; :::; ng: Let A i be the nonempty subset of E i for each i = 1; 2; :::; n: An intuitionistic fuzzy hypersoft set de…ned as the pair (H; A 1 A 2 ::: A n ) where; H : A 1 A 2 ::: A n ! IF P (U ) and where and are the membership and non-membership value, respectively such that For sake of simplicity, we write the symbols for E 1 E 2 ::: E n ; for A 1 A 2 ::: A n and for an element of the set : De…nition 6. i) An intutionistic fuzzy hypersoft set (H; ) over the universe U is said to be null intuitionistic fuzzy hypersoft set and denoted by 0 (U IF H ; ) if for all u 2 U and 2 ; H( ) (u) = 0 and H( ) (u) = 1: ii) An intutionistic fuzzy hypersoft set (H; ) over the universe U is said to be absolute intuitionistic fuzzy hypersoft set and denoted by 1 (U IF H ; ) if for all u 2 U and 2 ; H( ) ( ) = 1 and H( ) (u) = 0: Example 7. Let U be the set of cars given as U = fu 1 ; u 2 ; u 3 g also consider the set of attributes as E 1 = Fuel, E 2 =Transmission, E 3 = Color and their respective attributes are given as are subset of E i for each i = 1; 2; 3: Then the intuitionistic fuzzy hypersofts (H; 1 ) and (G; 2 ) de…ned as follows;     Corollary 8. It is clear that each intuitionistic fuzzy hypersoft set is also intuitionistic fuzzy soft set. An example of this situation is provided below.
Example 9. We consider that Example-7. If we select the parameters from a single attribute set such as E 1 while creating the intuitionistic fuzzy hypersoft set, then the resulting set becomes the intuitionistic fuzzy soft set. Therefore, it is clear that each intuitionistic fuzzy hypersoft set is also intuitionistic fuzzy soft set.That is, the intuitionistic fuzzy hypersoft set structure is the generalized version of the intuitionistic fuzzy soft sets.
Theorem 27. Let U be an initial universe set, 1 ; 2 and (H; 1 ); (G; 2 ) be two intuitionistic fuzzy hypersoft sets over the universe U: Proof. We only prove (i). The other properties can be similarly proved. For all ( 1 ; 2 ) 2 1 2 and u 2 U; On the other hand, Then, Then the "AND" and "OR" operations of these sets are given as below.

Conclusion
The aim of this paper is to overcome the uncertainty trouble in more particular way by way of combing Intuitionistic fuzzy set with Hypersoft set. Some operations of Intuitionistic Fuzzy Hypersoft set such as subset, equal set, union, intersection, complement, AND, OR operations are presented. By de…ning these notions, the foundation of the intuitionistic fuzzy hypersoft set structure was built.  The validity and implementation of the proposed operations and de…nitions are validated through presenting suitable instance. Matrices, similarity measure, single and multi-valued, interval valued, functions, distance measures, algorithms: score function, VIKOR, TOPSIS, AHP of Intutionistic Fuzzy Hypersoft sets will be future work. We hope that, this study will play a critical function in future decision-making research.