Some aspects on hesitant fuzzy soft set

Abstract In this paper, we introduce some operations on hesitant fuzzy soft sets and discuss some of their properties.

Enginoğlu (2010) studied several basic notions of the soft set theory. Maji, Biswas, and Roy (2001) introduced the concepts of fuzzy soft set theory. The hesitant fuzzy set, as one of the extension of Zadeh (1965) fuzzy set, allows the membership degree that an element to a set presented by several possible values, and it can express the hesitant information more comprehensively than other extensions of fuzzy set. In Torra and Narukawa (2009) introduced the concept of hesitant fuzzy set. In Xu and Xia (2011) defined the concept of hesitant fuzzy element, which can be considered as the basic unit of a hesitant fuzzy set, and is a simple and effective tool used to express the decisionmakers hesitant preferences in the process of decision-making. So many researchers (see Liao & Xu, 2014; has done lots of research work on aggregation, distance, similarity and correlation measures, clustering analysis, and decision-making with hesitant fuzzy information. In Babitha and John (2013) defined another important soft set hesitant fuzzy soft sets. They introduced basic operations such as intersection, union, compliment, and De Morgan's law was proved. Broumi and Smarandache (2014) introduced the operations over interval-valued intuitionistic hesitant fuzzy sets and proved some basic reaults. In Wang, Li, and Chen (2014) applied hesitant fuzzy soft sets in multicriteria group decision-making problems. Torra (2010), Torra andNarukawa (2009), andVerma andSharma (2013) discussed the relationship between hesitant fuzzy set and showed that the envelope of hesitant fuzzy set is an intuitionistic fuzzy set. A lot of work has been done about hesitant fuzzy sets, however, little has been done about the hesitant fuzzy soft sets.
In this paper, we study some operations on hesitant fuzzy soft set. We also establish some interesting properties of this notion.

Preliminary results
In this section, we recall some basic concepts and definitions regarding fuzzy soft sets, hesitant fuzzy set, and hesitant fuzzy soft set.
Definition 2.1 Maji et al. (2001) Let U be an initial universe and F be a set of parameters. Let P (U) denote the power set of U and A be a non-empty subset of F. Then, F A is called a fuzzy soft set over U, where F:A →P(U) is a mapping from A into P (U). Molodstov (1999) F E is called a soft set over U if and only if F is a mapping of E into the set of all subsets of the set U.

Definition 2.2
In other words, the soft set is a parameterized family of subsets of the set U. Every set F( ),∈E, from this family may be considered as the set of -element of the soft set F E or as the set of -approximate elements of the soft set.
Definition 2.3 Torra (2010) Given a fixed set X, then a hesitant fuzzy set (shortly HFS) in X is in terms of a function that when applied to X return a subset of [0, 1]. We express the HFS by a mathematical symbol: is a set of some values in [0,1], denoting the possible membership degrees of the element h ∈ X to the set F. F (x) is called a hesitant fuzzy element (HFE) and H is the set of all HFEs.
Definition 2.4 Torra (2010) Let 1 , 2 ∈ H and three operations are defined as follows: (1) C 1 = ∪ Definition 2. 5 Wang, Li, and Chen (2014) Let U be an initial universe and E be a set of parameters. Let F (U) be the set of all hesitant fuzzy subsets of U. Then, F E is called a hesitant fuzzy soft set (HFSS) over U, where F : E →F(U).
A HFSS is a parameterized family of hesitant fuzzy subsets of U, i.e. F (U). For all ∈E,F( ) is referred to as the set of − approximate elements of the HFSS F E . It can be written as F Since HFE can represent the situation, in which different membership function are considered possible (see Torra, 2010), Example 2.6 Suppose U = {a, b} be an initial universe and E = {e 1 , e 2 , e 3 , e 4 } be a set of parameters. Let A = {e 1 , e 2 }. Then, the hesitant fuzzy soft set F A is given as Definition 2.7 Wang, Li, and Chen (2014)

Proof
From (E) and (F), we get the result.
From (I) and (J), we get the result. (ii) From (K) and (L), we get the result.