Some operators on interval-valued Hesitant fuzzy soft sets
aa r X i v : . [ m a t h . G M ] A p r Some operators on interval-valued Hesitant fuzzy soft sets
Manash Jyoti Borah and Bipan Hazarika ∗ Department of Mathematics, Bahona College, Jorhat-785 101, Assam, IndiaEmail:[email protected] Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, IndiaEmail: bh [email protected]
Abstract.
The main aim of this paper is to introduced the operations ”Union” and”Intersection,” and four operators O , O , O , O on interval-valued hesitant fuzzy softsets and discuss some of their properties.Keywords: Fuzzy soft sets; Interval-valued Hesitant fuzzy sets; Hesitant fuzzy softsets.AMS subject classification no: 03E72. Introduction
Interval arithmetic was first suggested by Dwyer [8] in 1951. Development of intervalarithmetic as a formal system and evidence of its value as a computational device was pro-vided by Moore [17] in 1959 and Moore and Yang [18] in 1962. Further works on intervalnumbers can be found in Dwyer [9], Fischer [10]. Furthermore, Moore and Yang [19], havedeveloped applications to differential equations. Chiao in [7] introduced sequence of intervalnumbers and defined usual convergence of sequences of interval number.A set consisting of a closed interval of real numbers x such that a ≤ x ≤ b is called aninterval number. A real interval can also be considered as a set. Thus we can investigate someproperties of interval numbers, for instance arithmetic properties or analysis properties.Wedenote the set of all real valued closed intervals by I R . Any elements of I R is called closedinterval and denoted by x. That is x = { x ∈ R : a ≤ x ≤ b } . An interval number x isa closed subset of real numbers [7]. Let x l and x r be first and last points of x intervalnumber, respectively. For x , x ∈ I R , we have x = x ⇔ x l = x l , x r = x r . x + x = { x ∈ R : x l + x l ≤ x ≤ x r + x r } , and if α ≥ , then αx = { x ∈ R : αx l ≤ x ≤ αx r } and if α < , then αx = { x ∈ R : αx r ≤ x ≤ αx l } ,x .x = (cid:26) x ∈ R : min { x l .x l , x l .x r , x r .x l , x r .x r } ≤ x ≤ max { x l .x l , x l .x r , x r .x l , x r .x r } (cid:27) . The Hesitant fuzzy set, as one of the extensions of Zadeh [29] fuzzy set, allows the mem-bership degree that an element to a set presented by several possible values, and it canexpress the hesitant information more comprehensively than other extensions of fuzzy set.In 2009, Torra and Narukawa [22] introduced the concept of hesitant fuzzy set. In 2011,Xu and Xia [28] defined the concept of hesitant fuzzy element, which can be considered asthe basic unit of a hesitant fuzzy set, and is a simple and effective tool used to express thedecision makers hesitant preferences in the process of decision making. So many researchershas done lots of research work on aggregation, distance, similarity and correlation measures, ∗ The corresponding author. clustering analysis, and decision making with hesitant fuzzy information. In 2013, Babithaand John [3] defined another important soft set Hesitant fuzzy soft sets. They introducedbasic operations such as intersection, union, compliment and De Morgan’s law was proved.In2013, Chen et al. [6] extended hesitant fuzzy sets into interval-valued hesitant fuzzy envi-ronment and introduced the concept of interval-valued hesitant fuzzy sets. In 2015, Zhanget al. [30] introduced some operations such as complement, ”AND”,”OR”, ring sum andring product on interval-valued hesitant fuzzy soft sets.There are many theories like theory of probability, theory of fuzzy sets, theory of intu-itionistic fuzzy sets, theory of rough sets etc. which can be considered as mathematical toolsfor dealing with uncertain data, obtained in various fields of engineering, physics, computerscience, economics, social science, medical science, and of many other diverse fields. Butall these theories have their own difficulties. The most appropriate theory for dealing withuncertainties is the theory of fuzzy sets, introduced by L.A. Zadeh [29] in 1965. This theorybrought a paradigmatic change in mathematics. But there exists difficulty, how to set themembership function in each particular case. The theory of intuitionistic fuzzy sets (see[1, 2]) is a more generalized concept than the theory of fuzzy sets, but this theory has thesame difficulties. All the above mentioned theories are successful to some extent in dealingwith problems arising due to vagueness present in the real world. But there are also caseswhere these theories failed to give satisfactory results, possibly due to inadequacy of theparameterization tool in them. As a necessary supplement to the existing mathematicaltools for handling uncertainty, Molodtsov [15] introduced the theory of soft sets as a newmathematical tool to deal with uncertainties while modelling the problems in engineering,physics, computer science, economics, social sciences, and medical sciences. Molodtsov et al[16] successfully applied soft sets in directions such as smoothness of functions, game the-ory, operations research, Riemann integration, Perron integration, probability, and theoryof measurement. Maji et al [12] gave the first practical application of soft sets in decision-making problems. Maji et al [13] defined and studied several basic notions of the soft settheory. Also C¸ aˇgman et al [5] studied several basic notions of the soft set theory. V. Torra[21, 22] and Verma and Sharma [23] discussed the relationship between hesitant fuzzy setand showed that the envelope of hesitant fuzzy set is an intuitionistic fuzzy set. Zhang etal [30] introduced weighted interval-valued hesitant fuzzy soft sets and finally applied it indecision making problem. Thakur et al [20] proposed four new operators O , O , O , O onhesitant fuzzy sets.In this paper, in section 3, we study operations union and intersetion on hesitant interval-valued fuzzy soft sets and some interesting properties of this noton. In section 4, we intro-duce four operators O , O , O , O in interval-valued hesitant fuzzy soft sets. Also variousproposition are proved by using them.2. 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. [14]
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 softset over U where F : A → ˜ P ( U ) is a mapping from A into ˜ P ( U ) . Definition 2.2. [15] F E is called a soft set over U if and only if F is a mapping of E intothe set of all subsets of the set U. In other words, the soft set is a parameterized family of subsets of the set U. Every set F ( ǫ ) , ǫ ˜ ∈ E, from this family may be considered as the set of ǫ -element of the soft set F E oras the set of ǫ -approximate elements of the soft set. ome operators on interval-valued... 3 Definition 2.3. [2, 26]
Let intuitionistic fuzzy value IFV(X) denote the family of all IFVsdefined on the universe X, and let α, β ∈ IF V ( X ) be given as: α = ( µ α , ν α ) , β = ( µ β , ν β ) , (i) α ∩ β = (min( µ α , µ β ) , max( ν α , ν β )(ii) α ∪ β = (max( µ α , µ β ) , min( ν α , ν β )(iii) α ∗ β = ( µ α + µ β µ α .µ β +1) , ν α + ν β ν α .ν β +1) ) . Definition 2.4. [21]
Given a fixed set X, then a hesitant fuzzy set (shortly HFS) in X isin terms of a function that when applied to X return a subset of [0 , . We express the HFSby a mathematical symbol: F = { < h, µ F ( x ) > : h ∈ X } , where µ F ( x ) is a set of some values in [0 , , denoting thepossible membership degrees of the element h ∈ X to the set F. µ F ( x ) is called a hesitantfuzzy element (HFE) and H is the set of all HFEs. Definition 2.5. [21]
Given an hesitant fuzzy set F, define below it lower and upper boundaslower bound F − ( x ) = min F ( x ) . upper bound F + ( x ) = max F ( x ) . Definition 2.6. [21]
Let µ , µ ∈ H and three operations are defined as follows: (1) µ C = ∪ γ ∈ µ { − γ } ;(2) µ ∪ µ = ∪ γ ∈ µ ,γ ∈ µ max { γ , γ } ;(3) µ ∩ µ = ∩ γ ∈ µ ,γ ∈ µ min { γ , γ } . Definition 2.7. [6]
Let X be a reference set, and D [0 , be the set of all closed subintervalsof [0 , . An IVHFS on X is F = { < h i , µ F ( h i ) > : h i ∈ X, i = 1 , , ...n } , where µ F ( h i ) : X → D [0 , denotes all possible interval-valued membership degrees of the element h i ∈ X to the set F. For convenience, we call µ F ( h i ) an interval -valued hesitant fuzzy element(IVHFE), which reads µ F ( h i ) = { γ : γ ∈ µ F ( h i ) } . Here γ = [ γ L , γ U ] is an interval number. γ L = inf γ and γ U = sup γ represent the lowerand upper limits of γ, respectively. An IVHFE is the basic unit of an IVHFS and it can beconsidered as a special case of the IVHFS. The relationship between IVHFE and IVHFS issimilar to that between interval-valued fuzzy number and interval-valued fuzzy set. Example 2.8.
Let U = { h , h } be a reference set and let µ F ( h ) = { [0 . , . , [0 . , . } , µ F ( h ) = { [0 . , . } be the IVHFEs of h i ( i = 1 , to a set F respectively. Then IVHFS F can be writ-ten as F = { < h , { [0 . , . , [0 . , . } >, < h , { [0 . , . } > } . Definition 2.9. [27]
Let ˜ a = [˜ a L , ˜ a U ] and ˜ b = [˜ b L , ˜ b U ] be two interval numbers and λ ≥ , then (i) ˜ a = ˜ b ⇔ ˜ a L = ˜ b L and ˜ a U = ˜ b U ;(ii) ˜ a + ˜ b = [˜ a L + ˜ b L , ˜ a U + ˜ b U ];(iii) λ ˜ a = [ λ ˜ a L , λ ˜ a U ] , especially λ ˜ a = 0 , if λ = 0 . Definition 2.10. [27]
Let ˜ a = [˜ a L , ˜ a U ] and ˜ b = [˜ b L , ˜ b U ] , and let l a = ˜ a U − ˜ a L and l b = ˜ b U − ˜ b L ; then the degree of possibility of ˜ a ≥ ˜ b is formulated by p (˜ a ≥ ˜ b ) = max { − max ( ˜ b U − ˜ a L l ˜ a + l ˜ b , , } . Above equation is proposed in order to compare two interval numbers, and to rank all theinput arguments.
Definition 2.11. [6]
For an IVHFE ˜ µ, s (˜ µ ) = l ˜ µ P ˜ γ ∈ ˜ µ ˜ γ is called the score function of ˜ µ with l ˜ µ being the number of the interval values in ˜ µ, and s (˜ µ ) is an interval value belongingto [0 , . For two IVHFEs ˜ µ and ˜ µ , if s ( ˜ µ ) ≥ s ( ˜ µ ) , then ˜ µ ≥ ˜ µ . We can judge the magnitude of two IVHFEs using above equation.
Borah, Hazarika
Definition 2.12. [6]
Let ˜ µ, ˜ µ and ˜ µ be three IVHFEs, then (i) ˜ µ C = { [1 − ˜ γ U , − ˜ γ L ] : ˜ γ ∈ ˜ µ } ;(ii) ˜ µ ∪ ˜ µ = { [ max ( ˜ γ L , ˜ γ L ) , max ( ˜ γ U , ˜ γ U )] : ˜ γ ∈ ˜ µ , ˜ γ ∈ ˜ µ } ;(iii ˜ µ ∩ ˜ µ = { [ min ( ˜ γ L , ˜ γ L ) , min ( ˜ γ U , ˜ γ U )] : ˜ γ ∈ ˜ µ , ˜ γ ∈ ˜ µ } ;(iv) ˜ µ ⊕ ˜ µ = { [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ] : ˜ γ ∈ ˜ µ , ˜ γ ∈ ˜ µ } ;(v) ˜ µ ⊗ ˜ µ = { [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ] : ˜ γ ∈ ˜ µ , ˜ γ ∈ ˜ µ } . Proposition 2.13. [6]
For three IVHFEs ˜ µ, ˜ µ and ˜ µ , we have (i) ˜ µ C ∪ ˜ µ C = ( ˜ µ ∩ ˜ µ ) C ;(ii) ˜ µ C ∩ ˜ µ C = ( ˜ µ ∪ ˜ µ ) C ; Definition 2.14. [24]
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, that is, ˜ F ( U ) . For all ǫ ˜ ∈ E,F ( ǫ ) is referred to as the set of ǫ − approximate elements of the HFSS F E . It can be writtenas ˜ F ( ǫ ) = { < h, µ ˜ F ( ǫ )( x ) > : h ∈ U } . Since HFE can represent the situation, in which different membership function are consideredpossible (see [21] ), µ ˜ F ( ǫ )( x ) is a set of several possible values, which is the hesitant fuzzymembership degree. In particular, if ˜ F ( ǫ ) has only one element, ˜ F ( ǫ ) can be called a hesitantfuzzy soft number. For convenience, a hesitant fuzzy soft number (HFSN) is denoted by { < h, µ ˜ F ( ǫ )( x ) > } . Example 2.15.
Suppose U = { h , h } be an initial universe and E = { e , e , e , e } be aset of parameters. Let A = { e , e } . Then the hesitant fuzzy soft set F A is given as F A = { F ( e ) = { < h , { . , . } >, < h , { . , . , . } > } , F ( e ) = { < h , { . , . , . } >, < h , { . } > } . Definition 2.16. [30]
Let ( U, E ) be a soft universe and A ⊆ E. Then F A is called an intervalvalued hesitant fuzzy soft set over U, where F is a mapping given by F : A → IV HF ( U ) . An interval-valued hesitant fuzzy soft set is a parameterized family of interval-valued hesitantfuzzy sub set of U. That is to say, F ( e ) is an interval-valued hesitant fuzzy subset in U, ∀ e ∈ A. Following the standard notations, F ( e ) can be written as ˜ F ( e ) = { < h, µ ˜ F ( e )( x ) > : h ∈ U } . Example 2.17.
Suppose U = { h , h } be an initial universe and E = { e , e , e , e } be aset of parameters. Let A = { e , e } . Then the interval valued hesitant fuzzy soft set F A isgiven as F A = { e = { < h , [0 . , . >, < h , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Definition 2.18. [30] U be an initial universe and let E be a set of parameters. Supposingthat A, B ˜ ⊆ E, F A and F B are two interval-valued hesitant fuzzy soft sets, one says that F A isan interval-valued hesitant fuzzy soft subset of G B if and only if (i) A ˜ ⊆ B, (ii) γ σ ( k )1 ˜ ≤ γ σ ( k )2 , where for all e ˜ ∈ A, x ˜ ∈ U, γ σ ( k )1 and γ σ ( k )2 stand for the kth largest interval number in theIVHFEs µ F ( e )( x ) and µ G ( e )( x ) , respectively. In this case, we write F A ˜ ⊆ G A . Definition 2.19. [30]
The complement of F A , denoted by F CA , is defined by F CA ( e ) = {
An interval-valued hesitant fuzzy soft set is said to be an emptyinterval-valued hesitant fuzzy soft set, denoted by ˜ φ, if F : E → IV HF ( U ) such that ˜ F ( e ) = { < h, µ ˜ F ( e )( x ) > : h ∈ U } = { < h, { [0 , } > : h ∈ U } , ∀ e ˜ ∈ E. Definition 2.21. [30]
An interval-valued hesitant fuzzy soft set is said to be an full interval-valued hesitant fuzzy soft set, denoted by ˜ E, if F : E → IV HF ( U ) such that ˜ F ( e ) = { < h, µ ˜ F ( e )( x ) > : h ∈ U } = { < h, { [1 , } > : h ∈ U } , ∀ e ˜ ∈ E. Definition 2.22. [30]
The ring sum operation on the two interval-valued hesitant fuzzy softsets F A , G B over ( U, E ) , denoted by F A ⊕ G A = H, is a mapping given by H : E → IV HF ( U ) such that ∀ e ˜ ∈ E ˜ H ( e ) = { < h, µ ˜ H ( e )( x ) > : h ∈ U } = { < h, µ ˜ H ( e )( x ) ⊕ µ ˜ G ( e )( x ) > : h ∈ U } , ∀ e ˜ ∈ E. Definition 2.23. [30]
The ring product operation on the two interval-valued hesitant fuzzysoft sets F A , G B over ( U, E ) , denoted by F A ⊗ G A = H, is a mapping given by H : E → IV HF ( U ) such that ∀ e ˜ ∈ E ˜ H ( e ) = { < h, µ ˜ H ( e )( x ) > : h ∈ U } = { < h, µ ˜ H ( e )( x ) ⊗ µ ˜ G ( e )( x ) > : h ∈ U } , ∀ e ˜ ∈ E. Main Results
Definition 3.1.
The union of two interval-valued hesitant fuzzy soft sets F A and G B over ( U, E ) , is the interval-valued hesitant fuzzy soft set H C , where C = A ∪ B and ∀ e ˜ ∈ C,µ H ( e ) = µ F ( e ) , if e ˜ ∈ A − B ; µ G ( e ) , if e ˜ ∈ A − B ; µ F ( e ) ∪ µ G ( e ) , if e ˜ ∈ A ∩ B. We write F A ˜ ∪ G B = H C . Example 3.2.
Let F A = { e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } .G B = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Now rearrange the membership value of F A and G B with the help of Definitions 2.9 , 2.10and assumptions given by [6] , we have F A = { e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } .G B = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Therefore F A ˜ ∪ G B = H A ˜ ∪ B = H C = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Definition 3.3.
The intersection of two interval-valued hesitant fuzzy soft sets F A and G B with A ∩ B = φ over ( U, E ) , is the interval-valued hesitant fuzzy soft set H C , where C = A ∩ B, and ∀ e ˜ ∈ C, µ H ( e ) = µ F ( e ) ∩ µ G ( e ) . We write F A ˜ ∩ G B = H C . Borah, Hazarika
Example 3.4.
From Example 3.2, we have F A ˜ ∩ G B = H A ˜ ∩ B = H C = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . . , . >, < h , [0 . , . , [0 . , . > } . Proposition 3.5.
Let F A be a interval-valued hesitant fuzzy soft set. Then the followingare true: (i) F A ˜ ∪ F A = F A (ii) F A ˜ ∩ F A = F A (iii) F A ˜ ∪ ˜ φ A = F A (iv) F A ˜ ∩ ˜ φ A = ˜ φ A (v) F A ˜ ∪ ˜ E A = ˜ E A (vi) F A ˜ ∩ ˜ E A = F A . Proof.
Obvious. (cid:3)
Proposition 3.6.
Let F A and G A are two interval-valued hesitant fuzzy soft sets. Then (i) ( F A ˜ ∪ G A ) C = F CA ˜ ∩ G CA (ii) ( F A ˜ ∩ G A ) C = F CA ˜ ∪ G CA . Proof. (i) Let F CA ˜ ∩ G CA = H A . We have ∀ e ˜ ∈ A, µ H ( e ) = µ F C ( e ) ∩ µ G C ( e ) . ..................(A1)Suppose that F A ˜ ∪ G A = L A Therefore, ( F A ˜ ∪ G A ) C = L CA . We have ∀ e ˜ ∈ A, µ L C ( e ) = ( µ F ( e ) ∪ µ G ( e ) ) C = µ F C ( e ) ∩ µ G C ( e ) . ..................(A2)From (A1) and (A2), ( F A ˜ ∪ G A ) C = F CA ˜ ∩ G CA . (ii) Let F CA ˜ ∪ G CA = P A . We have ∀ e ˜ ∈ A, µ P ( e ) = µ F C ( e ) ∪ µ G C ( e ) . ..................(B1)Suppose that F A ˜ ∩ G A = Q A Therefore ( F A ˜ ∩ G A ) C = Q CA . We have ∀ e ˜ ∈ A, µ Q C ( e ) = ( µ F ( e ) ∩ µ G ( e ) ) C = µ F C ( e ) ∪ µ G C ( e ) . ..................(B2)From (B1) and (B2), ( F A ˜ ∩ G A ) C = F CA ˜ ∪ G CA . (cid:3) Proposition 3.7.
Let F A and G B are two interval-valued hesitant fuzzy soft sets. Then thefollowing are satisfied: (i) F CA ˜ ∩ G CB ˜ ⊆ ( F A ˜ ∪ G B ) C (ii) ( F A ˜ ∩ G B ) C ˜ ⊆ F CA ˜ ∪ G CB (iii) F CA ˜ ∩ G CB ˜ ⊆ ( F A ˜ ∩ G B ) C (iv) ( F A ˜ ∪ G B ) C ˜ ⊆ F CA ˜ ∪ G CB . Proof.
From Example 3.2(i) ( F A ˜ ∪ G B ) C = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } .F CA = { e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } .G CB = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . > } ome operators on interval-valued... 7 e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } .F CA ˜ ∩ G CB = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . . , . >, < h , [0 . , . , [0 . , . > } . Hence F CA ˜ ∩ G CB ˜ ⊆ ( F A ˜ ∪ G B ) C . (ii) ( F A ˜ ∩ G B ) C = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . . , . >, < h , [0 . , . , [0 . , . > } .F CA ˜ ∪ G CB = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Hence ( F A ˜ ∩ G B ) C ˜ ⊆ F CA ˜ ∪ G CB . (iii) From (i) and (ii) we get the result.(iv) From (i) and (ii) we get the result. (cid:3) Proposition 3.8.
Let F A , G B and H C are three interval-valued hesitant fuzzy soft sets.Then the following are satisfied: (i) F A ˜ ∪ G B = G B ˜ ∪ F A (ii) F A ˜ ∩ G B = G B ˜ ∩ F A (iii) F A ˜ ∪ ( G B ˜ ∪ H C ) = ( F A ˜ ∪ G B )˜ ∪ H C (iv) F A ˜ ∩ ( G B ˜ ∩ H C ) = ( F A ˜ ∩ G B )˜ ∩ H C . Proof.
The proof can be obtained from definition 3.1 and definition 3.3. (cid:3)
Proposition 3.9.
Let F A , G A and H A are three interval-valued hesitant fuzzy soft sets.Then the following propositiones are satiesfied: (i) F A ˜ ∪ G A = G A ˜ ∪ F A (ii) F A ˜ ∩ G A = G A ˜ ∩ F A (iii) F A ˜ ∪ ( G A ˜ ∪ H A ) = ( F A ˜ ∪ G A )˜ ∪ H A (iv) F A ˜ ∩ ( G A ˜ ∩ H A ) = ( F A ˜ ∩ G A )˜ ∩ H A . Proof.
The proof can be obtained from definition 3.1 and definition 3.3. (cid:3)
Proposition 3.10.
Let F A , G A and H A are three interval-valued hesitant fuzzy soft sets.Then the following are satisfied: (i) F A ˜ ∪ ( G A ˜ ∩ H A ) = ( F A ˜ ∪ G A )˜ ∩ ( F A ˜ ∪ H A )(ii) F A ˜ ∩ ( G A ˜ ∪ H A ) = ( F A ˜ ∩ G A )˜ ∪ ( F A ˜ ∩ H A ) . Proof.
Obvious. (cid:3)
Proposition 3.11.
Let F A , G B and H C are three interval-valued hesitant fuzzy soft sets.Then the following are not satisfied: (i) F A ˜ ∪ ( G B ˜ ∩ H C ) = ( F A ˜ ∪ G B )˜ ∩ ( F A ˜ ∪ H C )(ii) F A ˜ ∩ ( G B ˜ ∪ H C ) = ( F A ˜ ∩ G B )˜ ∪ ( F A ˜ ∩ H C ) . Proof.
We consider a example.Let H C = { e = { < h , [0 . , . , [ o. , . , [0 . , . >, < h , [0 . , . , > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . Now by Definitions 2.9 and 2.10 and assumptions given by [6] H C = { e = { < h , [0 . , . , [ o. , . , [0 . , . >, < h , [0 . , . , > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . Borah, Hazarika (i) From example 3.2, we have F A ˜ ∪ H C = { e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . ( F A ˜ ∪ G B )˜ ∩ ( F A ˜ ∪ H C )= { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Again G B ˜ ∩ H C = { e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Therefore F A ˜ ∪ ( G B ˜ ∩ H C )= { e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Hence F A ˜ ∪ ( G B ˜ ∩ H C ) = ( F A ˜ ∪ G B )˜ ∩ ( F A ˜ ∪ H C ) . (ii) From example 3.2 and 3.4. G B ˜ ∪ H C = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Therefore F A ˜ ∩ ( G B ˜ ∪ H C )= { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . . , . > } e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . Again F A ˜ ∩ H C = { e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . Therefore( F A ˜ ∩ G B )˜ ∪ ( F A ˜ ∩ H C )= { e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . Hence F A ˜ ∩ ( G B ˜ ∪ H C ) = ( F A ˜ ∩ G B )˜ ∪ ( F A ˜ ∩ H C ) . (cid:3) Definition 3.12.
Let ℜ = { ( F i ) A i : i ˜ ∈ I } be a family of hesitant fuzzy soft sets over ( U, E ) . Then the union of hesitant fuzzy soft sets in ℜ is a hesitant fuzzy soft set H K , K = ∪ i A i and ∀ e ˜ ∈ E, K ( e ) = ∪ i ( △ i ) A i ( e ) , where ( △ i ) A i ( e ) = ( F i ( e ) , if e ˜ ∈ A i φ, if e ˜ / ∈ A i . Example 3.13.
Let ( F ) A = { e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . ( F ) A = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . ( F ) A = { e = { < h , [0 . , . , [ o. , . , [0 . , . >, < h , [0 . , . , > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . Therefore ( F ) A ˜ ∪ ( F ) A ˜ ∪ ( F ) A = { e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } ome operators on interval-valued... 9 e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } e = { < h , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . , [0 . , . > } . Definition 3.14.
Let ℜ = { ( F i ) A i : i ˜ ∈ I } be a family of hesitant fuzzy soft sets with ∩ i A i = φ over ( U, E ) . Then the intersection of hesitant fuzzy soft sets in ℜ is a hesitant fuzzy soft set H K , K = ∩ i A i and ∀ e ˜ ∈ E, K ( e ) = ∩ i A i ( e ) . Example 3.15.
From Example 3.13, we have ( F ) A ˜ ∩ ( F ) A ˜ ∩ ( F ) A = { e = { < h , [0 . , . , [0 . , . , [0 . , . >, < h , [0 . , . , [0 . , . > } . Proposition 3.16.
Let ℜ = { ( F i ) A i : i ˜ ∈ I } be a family of hesitant fuzzy soft sets over ( U, E ) . Then (i) ˜ T i ( F i ) A i C ˜ ⊆ ( ˜ S i ( F i ) A i ) C (ii) ( ˜ T i ( F i ) A i ) C ˜ ⊆ ˜ S i ( F i ) A i C . Proof.
Obvious. (cid:3)
Proposition 3.17.
Let ℜ = { ( F i ) A : i ˜ ∈ I } be a family of hesitant fuzzy soft sets over ( U, E ) . Then (i) ˜ T i ( F i ) AC = ( ˜ S i ( F i ) A ) C (ii) ( ˜ T i ( F i ) A ) C = ˜ S i ( F i ) AC . Proof.
Obvious. (cid:3) New operators on interval-valued hesitant fuzzy soft elements
Definition 4.1.
Let ˜ µ , ˜ µ be two interval-valued hesitant fuzzy soft elements (IVHFSEs)of same set of parameters, then (i) ˜ µ O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ](ii) ˜ µ O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ](iii) ˜ µ O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | , | γ U − γ U | ](iv) ˜ µ O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L ∗ γ L | , | γ U ∗ γ U | ] . Proposition 4.2. If ˜ µ , ˜ µ and ˜ µ be two interval-valued hesitant fuzzy soft elements. Thenthe following identites are true: (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊕ ˜ µ , (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊗ ˜ µ , (v) ( ˜ µ ˜ ∪ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) , (vi) ( ˜ µ ˜ ∩ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) , Proof. (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , min { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ]= ˜ µ O ˜ µ . (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , max { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ] = S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ]= ˜ µ ⊕ ˜ µ . (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , min { ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ]= ˜ µ O ˜ µ . (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ h ,γ ˜ ∈ h [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , max { ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ]= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])= ˜ µ ⊗ ˜ µ . (v) ( ˜ µ ˜ ∪ ˜ µ ) O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L , ˜ γ L } , max { ˜ γ U , ˜ γ U } ] O S γ ˜ ∈ µ [ ˜ γ L , γ U ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [ | max { γ L .γ L }− γ L | | max { γ L .γ L }− γ L | , | max { γ U .γ U }− γ U | | max { γ U .γ U }− γ U | ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [max { | γ L − γ L | | γ L − γ L | , | γ L − γ L | | γ L − γ L | } , max { | γ U − γ U | | γ U − γ U | , | γ U − γ U | | γ U − γ U | } ]= ( ˜ µ O ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) . (vi) ( ˜ µ ˜ ∩ ˜ µ ) O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L , ˜ γ L } , min { ˜ γ U , ˜ γ U } ] O S γ ˜ ∈ h [ ˜ γ L , γ U ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [ | min { γ L .γ L }− γ L | | min { γ L .γ L }− γ L | , | min { γ U .γ U }− γ U | | min { γ U .γ U }− γ U | ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { | γ L − γ L | | γ L − γ L | , | γ L − γ L | | γ L − γ L | } , min { | γ U − γ U | | γ U − γ U | , | γ U − γ U | | γ U − γ U | } ]= ( ˜ µ O ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) . (cid:3) Proposition 4.3. If ˜ µ , ˜ µ and ˜ µ be two interval-valued hesitant fuzzy soft elements. Thenthe following identites are true: (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊕ ˜ µ , (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊗ ˜ µ , (v) ( ˜ µ ˜ ∪ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) , (vi) ( ˜ µ ˜ ∩ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) . Proof. (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , min { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ]= ˜ µ O ˜ µ . (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , max { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ] ome operators on interval-valued... 11 = ˜ µ ⊕ ˜ µ . (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , min { ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ]= ˜ µ O ˜ µ . (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | | γ L − γ L | , | γ U − γ U | | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L . ˜ γ L , | γ L − γ L | | γ L − γ L | } , max { ˜ γ U . ˜ γ U , | γ U − γ U | | γ U − γ U | } ]= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])= ˜ µ ⊗ ˜ µ . (v) ( ˜ µ ˜ ∪ ˜ µ ) O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L , ˜ γ L } , max { ˜ γ U , ˜ γ U } ] O S γ ˜ ∈ µ [ ˜ γ L , γ U ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [ | max { γ L .γ L }− γ L | | max { γ L .γ L }− γ L | , | max { γ U .γ U }− γ U | | max { γ U .γ U }− γ U | ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { | γ L − γ L | | γ L − γ L | , | γ L − γ L | | γ L − γ L | } , max { | γ U − γ U | | γ U − γ U | , | γ U − γ U | | γ U − γ U | } ]= ( ˜ µ O ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) . (vi) ( ˜ µ ˜ ∩ ˜ µ ) O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L , ˜ γ L } , min { ˜ γ U , ˜ γ U } ] O S γ ˜ ∈ µ [ ˜ γ L , γ U ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [ | min { γ L .γ L }− γ L | | min { γ L .γ L }− γ L | , | min { γ U .γ U }− γ U | | min { γ U .γ U }− γ U | ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { | γ L − γ L | | γ L − γ L | , | γ L − γ L | | γ L − γ L | } , min { | γ U − γ U | | γ U − γ U | , | γ U − γ U | | γ U − γ U | } ]= ( ˜ µ O ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) . (cid:3) Proposition 4.4. If ˜ µ , ˜ µ and ˜ µ be two interval-valued hesitant fuzzy soft elements. Thenthe following identites are true: (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊕ ˜ µ , (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊗ ˜ µ , (v) ( ˜ µ ˜ ∪ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) , (vi) ( ˜ µ ˜ ∩ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) . Proof. (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | , | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L − γ L | } , min { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | , | γ U − γ U | ]= ˜ µ O ˜ µ . (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | , | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L − γ L | } , max { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ]= ˜ µ ⊕ ˜ µ . (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | , | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L . ˜ γ L , | γ L − γ L | } , min { ˜ γ U . ˜ γ U , | γ U − γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | , | γ U − γ U | ]= ˜ µ O ˜ µ . (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L − γ L | , | γ U − γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L . ˜ γ L , | γ L − γ L | } , max { ˜ γ U . ˜ γ U , | γ U − γ U | } ]= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])= ˜ µ ⊗ ˜ µ . (v) ( ˜ µ ˜ ∪ ˜ µ ) O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L , ˜ γ L } , max { ˜ γ U , ˜ γ U } ] O S γ ˜ ∈ µ [ ˜ γ L , γ U ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [ | max { γ L .γ L }− γ L | , | max { γ U .γ U }− γ U | ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [max { | γ L − γ L | , | γ L − γ L | } , max { | γ U − γ U | , | γ U − γ U | } ]= ( ˜ µ O ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) . (vi) ( ˜ µ ˜ ∩ ˜ µ ) O ˜ µ = S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L , ˜ γ L } , min { ˜ γ U , ˜ γ U } ] O S γ ˜ ∈ µ [ ˜ γ L , γ U ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [ | min { γ L .γ L }− γ L | , | min { γ U .γ U }− γ U | ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ ,γ ˜ ∈ µ [min { | γ L − γ L | , | γ L − γ L | } , min { | γ U − γ U | , | γ U − γ U | } ]= ( ˜ µ O ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) . (cid:3) Proposition 4.5. If ˜ µ , ˜ µ and ˜ µ be two interval-valued hesitant fuzzy soft elements. Thenthe following identites are true: (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊕ ˜ µ , (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) = ˜ µ O ˜ µ , (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) = ˜ µ ⊗ ˜ µ , (v) ( ˜ µ ˜ ∪ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∪ ( ˜ µ O ˜ µ ) , (vi) ( ˜ µ ˜ ∩ ˜ µ ) O ˜ µ = ( ˜ µ O ˜ µ )˜ ∩ ( ˜ µ O ˜ µ ) . Proof. (i) ( ˜ µ ⊕ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ h ,γ ˜ ∈ h [ | γ L .γ L | , | γ U .γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L .γ L | } , min { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U .γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L .γ L | , | γ U .γ U | ]= ˜ µ O ˜ µ . (ii) ( ˜ µ ⊕ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L .γ L | , | γ U .γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , | γ L .γ L | } , max { ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U , | γ U .γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L + ˜ γ L − ˜ γ L . ˜ γ L , ˜ γ U + ˜ γ U − ˜ γ U . ˜ γ U ]= ˜ µ ⊕ ˜ µ . (iii) ( ˜ µ ⊗ ˜ µ )˜ ∩ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∩ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L .γ L | , | γ U .γ U | ]) ome operators on interval-valued... 13 = S γ ˜ ∈ µ ,γ ˜ ∈ µ [min { ˜ γ L . ˜ γ L , | γ L .γ L | } , min { ˜ γ U . ˜ γ U , | γ U .γ U | } ]= S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L .γ L | , | γ U .γ U | ]= ˜ µ O ˜ µ . (iv) ( ˜ µ ⊗ ˜ µ )˜ ∪ ( ˜ µ O ˜ µ )= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])˜ ∪ ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ | γ L .γ L | , | γ U .γ U | ])= S γ ˜ ∈ µ ,γ ˜ ∈ µ [max { ˜ γ L . ˜ γ L , | γ L .γ L | } , max { ˜ γ U . ˜ γ U , | γ U .γ U | } ]= ( S γ ˜ ∈ µ ,γ ˜ ∈ µ [ ˜ γ L . ˜ γ L , ˜ γ U . ˜ γ U ])= ˜ µ ⊗ ˜ µ . 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