A new approach to soft set through applications of cubic set
aa r X i v : . [ m a t h . L O ] O c t A new approach to soft set through applications of cubic set
Saleem Abdullah, Imran Khan, and Muhammad Aslam
Abstract.
The notions of (internal, external) cubic soft sets, P-(R-)order,P-(R-)union, P-(R-)intersection and P-OR, R-OR, P-AND and R-AND are in-troduced, and related properties are investigated. We show that the P-unionand the P-intersection of internal cubic soft sets are also internal cubic softsets. We provide conditions for the P-union (resp. P-intersection) of two ex-ternal cubic soft sets to be an internal cubic soft set. We give conditions forthe P-union (resp. R-union and R-intersection) of two external cubic soft setsto be an external cubic soft set. We consider conditions for the R-intersection(resp.P-intersection) of two cubic sof sets to be both an external cubic soft setand an internal cubic soft set.
1. INTRODUCTION
In order to deal with many complicated problems in the fields of engineering,social science, economics, medical science etc involving uncertainties, classical meth-ods are found to be inadequate in recent times. Molodstov [ ] pointed out that theimportant existing theories viz. probability theory, fuzzy set theory, intuitionisticfuzzy set theory, rough set theory etc, which can be considered as mathematicaltools for dealing with uncertainties, have their own difficulties. He further pointedout that the reason for these difficulties is, possibly, the inadequacy of the param-eterization tool of the theory. In 1999 he proposed a new mathematical tool fordealing with uncertainties which is free of the difficulties present in these theories.He introduced the novel concept of soft sets and established the fundamental resultsof the new theory. He also showed how soft set theory is free from parameterizationinadequacy syndrome of fuzzy set theory, rough set theory and probability theoryetc. Many of the established paradigms appear as special cases of soft set theory.In 2003, P. K .Maji, R. Biswas and A. R. Roy [ ] studied the theory of soft setsinitiated by Molodstov. They defined equality of two soft sets, subset and superset of a soft set, complement of a soft set, null soft set, and absolute soft set withexamples. Soft binary operations like AND, OR and also the operations of union,intersection were also defined. Pei and Miao [ ] and Chen et al. [ ] improved thework of Maji et al. [ ?, 6 ]. In 2009, M. Irfan Ali et al., [ ] gave some new notionssuch as the restricted intersection, the restricted union, the restricted difference andthe extended intersection of two soft sets along with a new notion of complementof a soft set. Sezgin and Atag¨un [ ] studied on soft set operations. Babitha and Key words and phrases.
Fuzzy set, Cubic set, Soft set, Cubic soft, Internal cubic soft set,External cubic soft set.
Sunil introduced soft set relations and functions [ ]. Majumdar and Samanta,worked on soft mappings [ ] were proposed and many related concepts were dis-cussed too. Moreover, the theory of soft sets has gone through remarkably rapidstrides with a wide-ranging applications especially in soft decision making as in thefollowing studies: [
27, 28 ] and some other fields such as [
29, 30, 31, 32 ]. Sinceits inception, it has received much attention in the mean of algebraic structures.In [ ], Aktas. and C. a˘gman applied the concept of soft set to groups theory andintroduced soft group of a group. Feng et.al, studied soft semirings by using softset [ ]. Recently, Acar studied soft rings [ ]. Jun et. al, applied the concept ofsoft set to BCK/BCI-algebras [
12, 13, 14 ]. Sezgin and Atag¨un initiated th con-cept of normalistic soft groups [ ]. Zhan et.al, worked on soft ideal of BL-algebras[ ]. In [ ], Kazancı et. al, used the concept of soft set to BCH-algebras. Sezginet. al, studied soft nearrings [ ]. Atag¨un and Sezgin [ ] defined the concepts ofsoft subrings and ideals of a ring, soft subfields of a field and soft submodules ofa module and studied their related properties with respect to soft set operationsalso union soft substructues of nearrings and nearring modules are studied in [ ].C. a˘gman et al. defined two new types of group actions on a soft set, called group SI -action and group SU -action [ ], which are based on the inclusion relation andthe intersection of sets and union of sets, respectively.Recently, Sezgin describeda new view of ring through soft intersection properties and discussed some funda-mental results [ ]. The concept of soft equality and some related properties arederived by Qin and HongIn recent times, researches have contributed a lot towards fuzzification of softset theory. Maji et al. [ ] introduced some properties of fuzzy soft set. Theseresults were further revised and improved by Ahmad and Kharal [ ]. This theoryhas proven useful in many different fields such as decision making [
33, 34, 35, 36,37, 38, 39 ]. In [ ],Y. B. Jun et al., introduced a new notion, called a (internal,external) cubic set by using a fuzzy set and an interval-valued fuzzy set, and inves-tigate several properties. They also defined P-union, P-intersection, R-union andR-intersection of cubic sets, and investigate several related properties.In this paper, using a fuzzy set and an interval-valued fuzzy set, we introducenew notions, is called (internal, external) cubic soft sets, P-(R-)order,P-(R-)union,P-(R-)intersection and P-OR, R-OR, P-AND and R-AND are introduced, and re-lated properties are investigated. We show that the P-union and the P-intersectionof internal cubic soft sets are also internal cubic soft sets. We provide conditionsfor the P-union (resp. P-intersection) of two external cubic soft sets to be aninternal cubic soft set. We give conditions for the P-union (resp. R-union andR-intersection) of two external cubic soft sets to be an external cubic soft set. Weconsider conditions for the R-intersection (resp.P-intersection) of two cubic sof setsto be both an external cubic soft set and an internal cubic soft set.
2. Preliminaries
We introduce below necessary notions and present a few auxiliary results thatwill be used throughout the paper. Recall first the basic terms and definitions fromthe cubic set theory.A map λ : X → [0 ,
1] is called a fuzzy subset of X. For any two fuzzy subsets λ and µ of X , λ ≤ µ means that, for all x ∈ X , λ ( x ) ≤ µ ( x ). The symbols λ ∧ µ ,and λ ∨ µ will mean the following fuzzy subsets of X NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 3 ( λ ∧ µ )( x ) = λ ( x ) ∧ µ ( x )( λ ∨ µ )( x ) = λ ( x ) ∨ µ ( x )for all x ∈ X. Let X be a non-empty set. A function A : X → [ I ] is called an interval-valuedfuzzy set (shortly, an IVF set) in X . Let [ I ] X stands for the set of all IVF setsin X . For every A ∈ [ I ] X and x ∈ X , A ( x ) = [ A − ( x ) , A + ( x )] is called the degreeof membership of an element x to A , where A − : X → I and A + : X → I arefuzzy sets in X which are called a lower fuzzy set and an upper fuzzy set in X ,respectively. For simplicity, we denote A = [ A − , A + ]. For every A, B ∈ [ I ] X , wedefine A ⊆ B if and only if A ( x ) (cid:22) B ( x ) for all x ∈ X . [ ] Definition . [ ] Let A = [ A − , A + ] , and B = [ B − , B + ] be two interval val-ued fuzzy sets in X (briefly, IVF sets). Then, we define r min { A ( x ) , B ( x ) } =[min { A − ( x ) , B − ( x ) } , min { A + ( x ) , B + ( x ) } ] ,r max { A ( x ) , B ( x ) } = [max { A − ( x ) , B − ( x ) } , max { A + ( x ) , B + ( x ) } ] . Definition . [ ] Let A = [ A − , A + ] , and B = [ B − , B + ] be two interval valuedfuzzy sets in X (briefly, IVF sets). Then, we define “ (cid:22) ”, and “ (cid:23) ”, as A ( x ) (cid:22) B ( x ) if and only if A − ( x ) ≤ B − ( x ) and A + ( x ) ≤ B + ( x ) . Similarly, A ( x ) (cid:23) B ( x ) if andonly if A − ( x ) ≥ B − ( x ) and A + ( x ) ≥ B + ( x ) for all x ∈ X. Definition . [ ] Let X be a non-empty set. By a cubic set in X, we meana structure A = { < x, A ( x ) , λ ( x ) > : x ∈ X } in which A is an interval valuedfuzzy sets in X (briefly, IVF set) and λ is a fuzzy set in X . A cubic set A = {
2, 4, 7, 9, 23, 40, 41 ].
3. CUBIC SOFT SETS
In this section we apply cubic set to soft set. We define a new extension offuzzy soft by using cubic set. We defined two types of cubic soft set and relatedproperties.
SALEEM ABDULLAH, IMRAN KHAN, AND MUHAMMAD ASLAM
Definition . A pair (cid:16) e
F, I (cid:17) is called cubic soft set over X if and only if e F is a mapping of I ( ⊆ E ) into the set of all cubic sets in X, i.e., e F : I −→ CP ( X ) where I is any subset of parameter’s set E , X is an initial universe set and CP ( X ) is the collection of all cubic sets in X . Here we denote and define cubic soft setas (cid:16) e F, I (cid:17) = n e F ( e i ) = A i = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o in this setcorresponding to each e i ∈ I , A i = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } is a cubic setin X in which A e i ( x ) is an interval valued fuzzy set (briefly, an IVF set) and λ e i ( x ) is a fuzzy set. Example . Let X = { p , p , p , p } be the set of cricket players underconsideration and E = { e , e , e , e } be the set of parameters, where e , e ,e , e represent fitness, good current form, good domestic cricket record and goodmoral character, respectively. Let I = { e , e , e } ⊆ E. Then, the cubic soft set (cid:16) e
F, I (cid:17) = n e F ( e i ) = A i = (cid:8) < p, A ei ( p ) , λ e i ( p ) > : p ∈ X (cid:9) e i ∈ I, i = 1 , , . o in X is p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . , , . p [0 . , . , . . , . , . . , , . Internal cubic soft set (ICSS).
Definition . A cubic soft set (cid:16) e
F, I (cid:17) is said to be an internal cubic softset (ICSS), if for all e i ∈ I ⊆ E ( E is set of parameters) F ( e i ) = A i is so that A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X . Example . Let X = { p , p , p , p } be the set of cricket players underconsideration and E = { e , e , e , e } be the set of parameters, where e , e ,e , e represent fitness, good current form, good domestic cricket record and goodmoral character, respectively. Let I = { e , e , e } ⊆ E. Then, the cubic soft set (cid:16) e
F, I (cid:17) = n e F ( e i ) = A i = (cid:8) < p, A ei ( p ) , λ e i ( p ) > : p ∈ X (cid:9) e i ∈ I, i = 1 , , . o in X is p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , .
35 [0 . , . , . . , . , . p [0 . , . , .
65 [0 . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , , .
85 [0 . , . , . an internal cubic soft set (ICSS) in X . External Cubic Soft sets.
Definition . A cubic soft set (cid:16) e
F, I (cid:17) is said to be an external cubic softset if for all e i ∈ I ⊆ E ( E is set of parameters), F ( e i ) = A i is so λ e i ( x ) / ∈ ( A − e i ( x ) , A + e i ( x )) for all e i ∈ I and for all x ∈ X . NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 5
Example . Let X = { p , p , p , p } be the set of cricket players underconsideration and E = { e , e , e , e } be the set of parameters, where e , e ,e , e represent fitness, good current form, good domestic cricket record and goodmoral character, respectively. Let I = { e , e , e } ⊆ E. Then, the cubic soft set (cid:16) e
F, I (cid:17) = n e F ( e i ) = A i = (cid:8) < p, A ei ( p ) , λ e i ( p ) > : p ∈ X (cid:9) e i ∈ I, i = 1 , , . o in X is p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , .
75 [0 . , , p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , .
65 [0 . , . , . an external cubic soft set (ECSS) in X . Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o be a cubic soft set in X which is not an ECSS. Then, there exists at least one e i ∈ I for which there exists some x ∈ X such that λ e i ( x ) ∈ ( A − e i ( x ) , A + e i ( x )) . Proof.
By definition of an external cubic soft set (ECSS) we know that λ e i ( x ) / ∈ ( A − e i ( x ) , A + e i ( x )) for all x ∈ X corresponding to each e i ∈ I . But giventhat (cid:16) e F, I (cid:17) is not an ECSS so for at least one e i ∈ I there exists some x ∈ X suchthat λ e i ( x ) ∈ ( A − e i ( x ) , A + e i ( x )) . Hence the result. (cid:3)
Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o be a cubic soft set in X. If (cid:16) e F, I (cid:17) is both an ICSS and ECSS in X , then forall x ∈ X corresponding to each e i ∈ I, λ e i ( x ) ∈ ( U ( A e i ) ∪ L ( A e i )) , where U ( A e i ) = (cid:8) A + e i ( x ) : x ∈ X (cid:9) and L ( A e i ) = (cid:8) A − e i ( x ) : x ∈ X (cid:9) . Proof.
Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = A i = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o is both an ICSS and ECSS, then by definition of ICSS corresponding to each e i ∈ I and for all x ∈ X we have λ e i ( x ) ∈ ( A − e i ( x ) , A + e i ( x )) . By definition of ECSS corre-sponding to each e i ∈ I and for all x ∈ X we have λ e i ( x ) / ∈ ( A − e i ( x ) , A + e i ( x )) . Since (cid:16) e
F, I (cid:17) is both an ICSS and ECSS, so the only possibility is that λ e i ( x ) = A − e i ( x )or λ e i ( x ) = A + e i ( x ) for all x ∈ X corresponding to each e i ∈ I . Hence λ e i ( x ) ∈ ( U ( A e i ) ∪ L ( A e i )) for all e i ∈ I and for all x ∈ X . (cid:3) Definition . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = A i = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = B i = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o be two cu-bic soft sets in X, I and J are any subsets of E (set of parameters). Then we havethe following (cid:16) e F, I (cid:17) = (cid:16)f G, J (cid:17) if and only if the following conditions are satisfied(a) I = J and(b) e F ( e i ) = e G ( e i ) for all e i ∈ I if and only if A e i ( x ) = B e i ( x ) and λ e i ( x ) = µ e i ( x ) for all e i ∈ I. SALEEM ABDULLAH, IMRAN KHAN, AND MUHAMMAD ASLAM If (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are two cubic soft sets then we denote and defineP-order as (cid:16) e
F, I (cid:17) ⊆ p (cid:16)f G, J (cid:17) if and only if the following conditions aresatisfied(a) I ⊆ J and(b) e F ( e i ) ≤ P e G ( e i ) for all e i ∈ I if and only if A e i ( x ) B e i ( x ) and λ e i ( x ) ≤ µ e i ( x ) for all x ∈ X corresponding to each e i ∈ I. If (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are two cubic soft sets then we denote and defineR-order as (cid:16) e
F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) if and only if the following conditions aresatisfied:(a) I ⊆ J and(b) e F ( e i ) ≤ R e G ( e i ) for all e i ∈ I if and only if A e i ( x ) B e i ( x ) and λ e i ( x ) ≥ µ e i ( x ) for all x ∈ X corresponding to each e i ∈ I . Example . (P-order)Let (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) be the cubic soft sets in X defined as follows, (cid:16) e F, I (cid:17) = n e F ( e i ) = A i = { < x, A i ( p ) , λ i ( p ) > : p ∈ X } e i ∈ J, i = 1 , , o p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , , . . , . , . p [0 . , . , . . , . , . . , . , . and (cid:16)f G, J (cid:17) = n e G ( e i ) = B i = (cid:8) < x, B e i ( p ) , µ e i ( p ) > : p ∈ X (cid:9) e i ∈ J, i = 1 , , o is p e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , , . . , . , . p [0 . , . , . . , . , . . , . , . Thus clearly we have that (cid:16) e
F, I (cid:17) ⊆ p (cid:16)f G, J (cid:17) . Example . (R-order)Let (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) be the cubic soft sets in X defined as follows, (cid:16) e F, I (cid:17) = p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 7 and (cid:16)f
G, J (cid:17) = n e G ( e i ) = B i = (cid:8) < p, B e i ( p ) , µ e i ( p ) > : p ∈ X (cid:9) e i ∈ J, i = 1 , , o is p e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . Thus clearly we have that (cid:16) e
F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) . P-union, P-intersection, R-union and R-intersection of any two Cubicsoft sets in X . Definition . Let (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) be two cubic soft sets in X, where I and J are any subsets of parameter’s set E. Then, we define
P-union as (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∪ J and e H ( e i ) = e F ( e i ) If e i ∈ I − J, e G ( e i ) If e i ∈ J − I, e F ( e i ) ∨ P e G ( e i ) If e i ∈ I ∩ J ,where e F ( e i ) ∨ P e G ( e i ) is defined as e F ( e i ) ∨ P e G ( e i ) = n < x, r max { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ ei )( x ) > : x ∈ X o , e i ∈ I ∩ J P-intersection as (cid:16) e F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∩ J , H ( e i ) = e F ( e i ) ∧ P e G ( e i ) and e i ∈ I ∩ J. Here e F ( e i ) ∧ P e G ( e i ) is defined as e F ( e i ) ∧ P e G ( e i ) = e H ( e i ) = (cid:8) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X , e i ∈ I ∩ J (cid:9) . R-union as (cid:16) e F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∪ J and H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ R e G ( e i ) If e i ∈ I ∩ J Example . Here e F ( e i ) ∨ R e G ( e i ) is defined as e F ( e i ) ∨ R e G ( e i ) = (cid:8) < x, r max { A e i ( x ) , B e i ( p ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X (cid:9) , e i ∈ I ∩ J R-intersection as (cid:16) e F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∩ J, H ( e i ) = e F ( e i ) ∧ R e G ( e i ) and e i ∈ I ∩ J . Here e F ( e i ) ∧ R e G ( e i ) is defined as e F ( e i ) ∧ R e G ( e i ) = e H ( e i ) = (cid:8) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X e i ∈ I ∩ J (cid:9) (1) SALEEM ABDULLAH, IMRAN KHAN, AND MUHAMMAD ASLAM
Example . (P-union) Let X = { p , p , p , p } be initial universe, I = { e , e } and J = { e , e , e } are any subsets of parameter’s set E = { e , e , e } . Let (cid:16) e
F, I (cid:17) be cubic soft set define as: p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . p [0 . , . , . . , , . p [0 . , . , . . , . , . p [0 . , . , . . , . , . Let (cid:16)f
G, J (cid:17) = e G ( e i ) = B i = (cid:8) < p, B e i ( p ) , µ e i ( p ) > : p ∈ X , e i ∈ J, i = 1 , , (cid:9) be a cubic soft set over Xp e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . Then, (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is a cubic soft set over X and defined as: p e F ( e ) ∨ P e G ( e ) = < r max { A e ( p ) ,B e ( p ) } , ( λ e ∨ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∨ P e G ( e ) = < r max { A e ( p ) ,B e ( p ) } , ( λ e ∨ µ e )( p ) >p [0 . , . , . p [0 . , , . p [0 . , . , . p [0 . , . , . p e G ( e ) = B = < B e ( p ) , λ e ( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 9 (P-intersection) Let (cid:16) e
F, I (cid:17) be a cubic soft set over U and defined as: p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , .
8] 0 . . , . , . Let (cid:16)f
G, J (cid:17) = e G ( e i ) = B i = (cid:8) < p, B e i ( p ) , µ e i ( p ) > : p ∈ X , e i ∈ J, i = 1 , , (cid:9) be a cubic soft set over Xp e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , , . p [0 . , . , . . , , . . , . , . p [0 . , . , . . , . , . . , . , . Then, P-intersection is denoted by (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) and defined as: p e F ( e ) ∧ P e G ( e ) = < r min { A e ( p ) , B e ( p ) } , ( λ e ∧ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∧ P e G ( e ) = < r min { A e ( p ) , B e ( p ) } , ( λ e ∧ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∧ P e G ( e ) = < r min { A e ( p ) , B e ( p ) } , ( λ e ∧ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . (R-union) Let (cid:16) e F, I (cid:17) be a cubic soft set over X and defined as: p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . Let (cid:16)f
G, J (cid:17) = e G ( e i ) = B i = { < p, B e i ( p ) , µ i ( p ) > : p ∈ X, e i ∈ J, i = 1 , , } be a cubic soft and defined as: p e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , , . p [0 . , . , . . , , . . , . , . p [0 . , . , . . , . , . . , . , . . Then, their R-union is denoted by (cid:16) e
F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) and defined as p e F ( e ) ∨ R e G ( e ) = < r max { A e ( p ) , B e ( p ) } , ( λ e ∧ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∨ R e G ( e ) = < r max { A e ( p ) , B e ( p ) } , ( λ e ∧ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , , . p [0 . , . , . p e F ( e ) ∨ R e G ( e ) = < r max { A e ( p ) , B e ( p ) } , ( λ e ∧ µ e )( p ) >p [0 . , . , . p [0 . , , . p [0 . , . , . p [0 . , . , . (R-intersection) Let (cid:16) e F, I (cid:17) be a cubic soft set over X and defined as: p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 11
Let (cid:16)f
G, J (cid:17) = n e G ( e i ) = B i = { < p, B i ( p ) , µ i ( p ) > : p ∈ X } e i ∈ J, i = 1 , , o be a cubic sot set defined as: p e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . Then, the R-intersection is is denoted by (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) and definedas below: p e F ( e ) ∧ R e G ( e ) = < r min { A e ( p ) , B e ( p ) } , ( λ e ∨ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∧ R e G ( e ) = < r min { A e ( p ) , B e ( p ) } , ( λ e ∨ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∧ R e G ( e ) = < r min A e ( p ) , B e ( p ) , ( λ e ∨ µ e )( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . P-OR, R-OR, P-AND and R-AND of cubic soft sets.
Definition . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = A i = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = B i = (cid:8) < x, B i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o be cubicsoft sets in X. Then, (1)
P-OR is denoted by (cid:16) e
F, I (cid:17) ∨ P (cid:16)f G, J (cid:17) and defined as (cid:16) e
F, I (cid:17) ∨ P (cid:16)f G, J (cid:17) = (cid:16)f H, I × J (cid:17) where e H ( α i , β i ) = e F ( α i ) ∪ P e G ( β i ) for all ( α i , β i ) ∈ I × J. (2) R-OR is denoted by (cid:16) e
F, I (cid:17) ∨ R (cid:16)f G, J (cid:17) and defined as (cid:16) e
F, I (cid:17) ∨ R (cid:16)f G, J (cid:17) = (cid:16)f H, I × J (cid:17) where e H ( α i , β i ) = e F ( α i ) ∪ R e G ( β i ) for all ( α i , β i ) ∈ I × J. (3) P-AND is denoted by (cid:16) e
F, I (cid:17) ∧ P (cid:16)f G, J (cid:17) and defined as (cid:16) e
F, I (cid:17) ∧ P (cid:16)f G, J (cid:17) = (cid:16)f H, I × J (cid:17) where e H ( α i , β i ) = e F ( α i ) ∩ P e G ( β i ) for all ( α i , β i ) ∈ I × J. (4) R-AND is denoted by (cid:16) e
F, I (cid:17) ∧ R (cid:16)f G, J (cid:17) and defined as (cid:16) e
F, I (cid:17) ∧ R (cid:16)f G, J (cid:17) = (cid:16)f H, I × J (cid:17) where e H ( α i , β i ) = e F ( α i ) ∩ R e G ( β i ) for all ( α i , β i ) ∈ I × J. Example . Let X = { p , p , p , p } be initial universe, I = { e , e } and J = { e , e , e } are any subsets of parameter’s set E = { e , e , e } . Let (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) be two cubic soft over X and defined as below, respectively. p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , , . p [0 . , . , . . , . , . p [0 . , . , . . , . , . p [0 . , . , . . , . , . and p e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . p [0 . , . , . . , . , . p [0 . , . , . . , . , . p [0 . , . , . . , . , . .Then P-OR is denoted as (cid:16)f H, I × J (cid:17) = (cid:16) e F, I (cid:17) ∨ P (cid:16)f G, J (cid:17) , where I × J = { ( e , e ) , ( e , e ) , ( e , e ) , ( e , e ) } , is defined p e H ( e , e ) = e F ( e ) ∪ P e G ( e ) e H ( e , e ) = e F ( e ) ∪ P e G ( e ) e H ( e , e ) = e F ( e ) ∪ P e G ( e ) e H ( e , e ) = e F ( e ) ∪ P e G ( e ) p [0 . , . , . . , . , . . , , . . , , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . R-OR is denoted by (cid:16)f
H, I × J (cid:17) = (cid:16) e F, I (cid:17) ∨ R (cid:16)f G, J (cid:17) , where I × J = { ( e , e ) , ( e , e ) , ( e , e ) , ( e , e ) } , is defined p e H ( e , e ) = e F ( e ) ∪ R e G ( e ) e H ( e , e ) = e F ( e ) ∪ R e G ( e ) e H ( e , e ) = e F ( e ) ∪ R e G ( e ) e H ( e , e ) = e F ( e ) ∪ R e G ( e ) p [0 . , . , . . , . , . . , , . . , , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 13
P-AND is denoted by (cid:16)f
H, I × J (cid:17) = (cid:16) e F, I (cid:17) ∧ P (cid:16)f G, J (cid:17) , where I × J = { ( e , e ) , ( e , e ) , ( e , e ) , ( e , e ) } , is defined p e H ( e , e ) = e F ( e ) ∩ P e G ( e ) e H ( e , e ) = e F ( e ) ∩ P e G ( e ) e H ( e , e ) = e F ( e ) ∩ P e G ( e ) e H ( e , e ) = e F ( e ) ∩ P e G ( e ) p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . R-AND is denoted by (cid:16)f
H, I × J (cid:17) = (cid:16) e F, I (cid:17) ∧ P (cid:16)f G, J (cid:17) , where I × J = { ( e , e ) , ( e , e ) , ( e , e ) , ( e , e ) } , is defiend p e H ( e , e ) = e F ( e ) ∩ R e G ( e ) e H ( e , e ) = e F ( e ) ∩ R e G ( e ) e H ( e , e ) = e F ( e ) ∩ R e G ( e ) e H ( e , e ) = e F ( e ) ∩ R e G ( e ) p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . , . , . Definition . The complement of a cubic soft set (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o is denoted by (cid:16) e F, I (cid:17) c and defined as (cid:16) e F, I (cid:17) c = ( e F c , ¬ I ) , where e F c : ¬ I −→ CP ( X ) and e F c ( e i ) = ( e F ( ¬ e i )) c for all e i ∈ ¬ I = ( e F ( e i )) c (as ¬ ( ¬ e i ) = e i ) (cid:16) e F, I (cid:17) c = { (( e F ( e i )) c = { < x, A ce i ( x ) , λ ce i ( x ) > : x ∈ X } e i ∈ I } . Example . Let X = { p , p , p , p } be initial universe and E = { e , e , e } parameter’s set. Let (cid:16) e F, I (cid:17) be a cubic soft set over X and defined as: p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . . Then, (cid:16) e
F, I (cid:17) c == { (( e F ( e i )) c = { < x, A ce i ( x ) , λ ce i ( x ) > : x ∈ X } e i ∈ I } is definedas: p (( e F ( e )) c = A c = < A ce ( p ) , λ c ( p ) > = < (cid:2) − A + e ( p ) , − A − e ( p ) (cid:3) , − λ e ( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p (( e F ( e )) c = A c = < A ce ( p ) , λ c ( p ) > = < (cid:2) − A + e ( p ) , − A − e ( p ) (cid:3) , − λ e ( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . p (( e F ( e )) c = A c = < A ce ( p ) , λ ce ( p ) > = < (cid:2) − A + e ( p ) , − A − e ( p ) (cid:3) , − λ e ( p ) >p [0 . , . , . p [0 . , . , . p [0 . , . , . p [0 . , . , . Proposition . Let X be initial universe and I , J , L and S subset of E .Then, for any cubic soft sets (cid:16) e F, I (cid:17) , (cid:16)f G, J (cid:17) , (cid:16)f E, L (cid:17) and (cid:16) e
T, S (cid:17) the followingproperties hold. (1) If (cid:16) e F, I (cid:17) ⊆ P (cid:16)f G, J (cid:17) and (cid:16)f
G, J (cid:17) ⊆ P (cid:16)f E, L (cid:17) , then (cid:16) e F, I (cid:17) ⊆ P ( f E, L ) . (2) If (cid:16) e F, I (cid:17) ⊆ P (cid:16)f G, J (cid:17) , then (cid:16)f G, J (cid:17) c ⊆ P (cid:16) e F, I (cid:17) c if I = J. (3) If (cid:16) e F, I (cid:17) ⊆ P (cid:16)f G, J (cid:17) and (cid:16) e
F, I (cid:17) ⊆ P (cid:16)f E, L (cid:17) , then (cid:16) e F, I (cid:17) ⊆ P (cid:16)f G, J (cid:17) ∩ P (cid:16)f E, L (cid:17) . (4) If (cid:16) e F, I (cid:17) ⊆ P (cid:16)f G, J (cid:17) and (cid:16)f
E, L (cid:17) ⊆ P (cid:16)f G, J (cid:17) , then (cid:16) e F, I (cid:17) ∪ P (cid:16)f E, L (cid:17) ⊆ P (cid:16)f G, J (cid:17) . (5) If (cid:16) e F, I (cid:17) ⊆ P (cid:16)f G, J (cid:17) and (cid:16)f
E, L (cid:17) ⊆ P (cid:16) e T, S (cid:17) , then (a) (cid:16) e F, I (cid:17) ∪ P (cid:16)f E, L (cid:17) ⊆ P (cid:16)f G, J (cid:17) ∪ P (cid:16) e T, S (cid:17) and (b) (cid:16) e
F, I (cid:17) ∩ P (cid:16)f E, L (cid:17) ⊆ P (cid:16)f G, J (cid:17) ∩ P (cid:16) e T, S (cid:17) . (6) If (cid:16) e F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) and (cid:16)f
G, J (cid:17) ⊆ R (cid:16)f E, L (cid:17) , then (cid:16) e F, I (cid:17) ⊆ R (cid:16)f E, L (cid:17) . (7) If (cid:16) e F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) , then (cid:16)f G, J (cid:17) c ⊆ R (cid:16) e F, I (cid:17) c if I = J. NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 15 (8) If (cid:16) e F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) and (cid:16) e
F, I (cid:17) ⊆ R (cid:16)f E, L (cid:17) , then (cid:16) e F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) ∩ R (cid:16)f E, L (cid:17) . (9) If (cid:16) e F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) and (cid:16)f
E, L (cid:17) ⊆ R (cid:16)f G, J (cid:17) , then (cid:16) e F, I (cid:17) ∪ R (cid:16)f E, L (cid:17) ⊆ R (cid:16)f G, J (cid:17) . (10) If (cid:16) e F, I (cid:17) ⊆ R (cid:16)f G, J (cid:17) and (cid:16)f
E, L (cid:17) ⊆ R (cid:16) e T, S (cid:17) , then (a) (cid:16) e F, I (cid:17) ∪ R (cid:16)f E, L (cid:17) ⊆ R (cid:16)f G, J (cid:17) ∪ R (cid:16) e T, S (cid:17) and(b) (cid:16) e
F, I (cid:17) ∩ R (cid:16)f E, L (cid:17) ⊆ R (cid:16)f G, J (cid:17) ∩ R (cid:16) e T, S (cid:17) . Proof. Proof.
Proof is straightforward. (cid:3)(cid:3)
Theorem . Let (cid:16) e
F, I (cid:17) be a cubic soft set over X. (1) If (cid:16) e F, I (cid:17) is an internal cubic soft set, then (cid:16) e
F, I (cid:17) c is also an internalcubic soft set (ICSS).(2) If (cid:16) e F, I (cid:17) is an external cubic soft set, then (cid:16) e
F, I (cid:17) c is also an externalcubic soft set (ECSS). Proof. (1) Given (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o is an ICSS this implies A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X, this implies 1 − A + e i ( x ) ≤ − λ e i ( x ) ≤ − A − e i ( x ) for all e i ∈ I and for all x ∈ X. Also we haveHence (cid:16) e
F, I (cid:17) c is an ICSS.(2 ) Given (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o is anECSS this implies λ e i ( x ) / ∈ ( A − e i ( x ) , A + e i ( x )) for all e i ∈ I and for all x ∈ X. Since λ e i ( x ) / ∈ (( A − e i ( x ) , A + e i ( x )) and 0 ≤ A − e i ( x ) ≤ A + e i ( x ) ≤ . So we have λ e i ( x ) ≤ A − e i ( x ) or A + e i ( x ) ≤ λ e i ( x ) this implies 1 − λ e i ( x ) ≥ − A − e i ( x ) or 1 − A + e i ( x ) ≥ − λ e i ( x ) . Thus 1 − λ e i ( x ) / ∈ (1 − A − e i ( x ) , − A + e i ( x )) for all e i ∈ I and for all x ∈ X. Hence (cid:16) e
F, I (cid:17) c is an ECSS. (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o be internal cu-bic soft sets. Then, (1) (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is an ICSS. (2) (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) is an ICSS.
Proof. (1) Since (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) are internal cubic soft sets. So for (cid:16) e
F, I (cid:17) we have A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X. Also for (cid:16)f
G, J (cid:17) we have B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X. Then we havemax { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∨ µ e i )( x ) ≤ max { A + e i ( x ) , B + e i ( x ) } for all e i ∈ I ∪ J and for all x ∈ X . Now by definition of P-union of (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) , we have (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) where C = I ∪ J and H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ P e G ( e i ) If e i ∈ I ∩ J If e i ∈ I ∩ J, then e F ( e i ) ∨ P e G ( e i ) is defined as e F ( e i ) ∨ P e G ( e i ) = e H ( e i ) = (cid:26) < x, r max { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27) Thus (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is an ICSS if e i ∈ I ∩ J. If e i ∈ I − J or e i ∈ J − I , thenthe result is trivial. Hence, (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is an ICSS in all cases.(2) Since (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∩ J and e H ( e i ) = e F ( e i ) ∧ P e G ( e i ) . If e i ∈ I ∩ J, then e F ( e i ) ∧ P e G ( e i ) is defined as e F ( e i ) ∧ P e G ( e i ) = e H ( e i ) = (cid:26) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27) Also given that (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) are internal cubic soft sets. So for (cid:16) e
F, I (cid:17) wehave A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X. And for (cid:16)f
G, J (cid:17) we have B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X. This impliesmin { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∧ µ e i )( x ) ≤ min { A + e i ( x ) , B + e i ( x ) } for all e i ∈ I ∩ J. Hence (cid:16)f
H, C (cid:17) = (cid:16) e F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) is an internal cubic soft set (ICSS). (cid:3)
Theorem . Let (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) be ICSSs over X such that max { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∧ µ e i )( x ) for all e i ∈ I ∩ J and for all x ∈ X. Then, the R-union of (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) is also an ICSS.
Proof.
Since (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) are internal cubic soft sets in X . So for (cid:16) e F, I (cid:17) we have A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X. Also for (cid:16)f
G, J (cid:17) we have B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X. So wehave ( λ e i ∧ µ e i )( x ) ≤ max { A + e i ( x ) , B + e i ( x ) } . Also given that max { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∧ µ e i )( x ) for all e i ∈ I ∩ J and for all x ∈ X. Now (cid:16) e
F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) ,where C = I ∪ J and H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ R e G ( e i ) If e i ∈ I ∩ J, where e F ( e i ) ∨ R e G ( e i ) is defined as e F ( e i ) ∨ R e G ( e i ) = e H ( e i ) = (cid:26) < x, r max { A e i ( x ) , B e i ( p ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X , e i ∈ I ∩ J. (cid:27) NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 17
Since (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) are ICSSs so from above given condition and definition ofan ICSS we can write max { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∧ µ e i )( x ) ≤ max { A + e i ( x ) , B + e i ( x ) } for all e i ∈ I ∩ J, and for all x ∈ X. If e i ∈ I − J or e i ∈ J − I then the resultis trivial. Thus (cid:16) e F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) is an ICSS if max { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∧ µ e i )( x ) for all e i ∈ I ∪ J and for all x ∈ X. (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o be ICSSs in X satisfying the following inequality min { A + e i ( x ) , B + e i ( x ) } ≥ ( λ e i ∨ µ e i )( x ) for all e i ∈ I ∩ J and for all x ∈ X. Then (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is an ICSS.
Proof.
Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o . Then, by defini-tion of an ICSS. A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X and B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X, this implis thatmin { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∨ µ e i )( x ) . Also since (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) where C = I ∩ J and H ( e i ) = e F ( e i ) ∧ R e G ( e i ) If e i ∈ I ∩ J then e F ( e i ) ∧ R e G ( e i ) is definedas e F ( e i ) ∧ R e G ( e i ) = (cid:26) e H ( e i ) = (cid:26) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J (cid:27)(cid:27) Given condition min { A + e i ( x ) , B + e i ( x ) } ≥ ( λ e i ∨ µ e i )( x ) for all e i ∈ I ∩ J and forall x ∈ X. Thus from given condition and definition of ICSSs { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∨ µ e i )( x ) ≤ min { A + e i ( x ) , B + e i ( x ) } . Hence (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is an ICSS. (cid:3)
Definition . Given two cubic soft sets (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o , if we inter-change λ and µ , then the new cubic soft sets are denoted and defined as (cid:16) e F, I (cid:17) ∗ = n e F ( e i ) = (cid:8) < x, A e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ I, o and (cid:16)f G, J (cid:17) ∗ = n e G ( e i ) = { < x, B e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ J o respectively. Theorem . For two ECSSs (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o in X, if (cid:16) e F, I (cid:17) ∗ and (cid:16)f G, J (cid:17) ∗ are ICSSs in X then (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is an ICSS in X . Proof.
Since (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are ECSSs. Thenfor (cid:16) e F, I (cid:17) , we have λ e i ( x ) / ∈ ( A − e i ( x ) , A + e i ( x )) for all e i ∈ I and for all x ∈ X. And for (cid:16)f
G, J (cid:17) , we have µ e i ( x ) / ∈ ( B − e i ( x ) , B + e i ( x )) for all e i ∈ J and for all x ∈ X. Alsogiven that (cid:16) e
F, I (cid:17) ∗ = n e F ( e i ) = (cid:8) < x, A e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ I, o and (cid:16)f G, J (cid:17) ∗ = n e G ( e i ) = { < x, B e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ J o are ICSSs sothis implies A − e i ( x ) ≤ µ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X and B − e i ( x ) ≤ λ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X. Since (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) areECSSs, and (cid:16) e
F, I (cid:17) ∗ and (cid:16)f G, J (cid:17) ∗ are ICSSs. Thus by definition of ECSSs andICSSs, all the possibilities are as under(i)(a) λ e i ( x ) ≤ A − e i ( x ) ≤ µ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X .(i)(b) µ e i ( x ) ≤ B − e i ( x ) ≤ λ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X. (ii)(a) A − e i ( x ) ≤ µ e i ( x ) ≤ A + e i ( x ) ≤ λ e i ( x ) for all e i ∈ I and for all x ∈ X .(ii)(b) B − e i ( x ) ≤ λ e i ( x ) ≤ B + e i ( x ) ≤ µ e i ( x ) for all e i ∈ J and for all x ∈ X. (iii)(a) λ e i ( x ) ≤ A − e i ( x ) ≤ µ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X .(iii)(b) B − e i ( x ) ≤ λ e i ( x ) ≤ B + e i ( x ) ≤ µ e i ( x ) for all e i ∈ J and for all x ∈ X. (iv)(a) A − e i ( x ) ≤ µ e i ( x ) ≤ A + e i ( x ) ≤ λ e i ( x ) for all e i ∈ I and for all x ∈ X .(iv)(b) µ e i ( x ) ≤ B − e i ( x ) ≤ λ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X. Since P-union of (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) is denoted and defined as (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) where C = I ∪ J and e H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ P e G ( e i ) If e i ∈ I ∩ J, where e F ( e i ) ∨ P e G ( e i ) is defined as e F ( e i ) ∨ P e G ( e i ) = e H ( e i ) = (cid:26) < x, r max { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27) Case 1: If e H ( e i ) = e F ( e i ) that is if e i ∈ I − J, then from (i)(a) and (ii)(a), wehave λ e i ( x ) = A − e i ( x ) and λ e i ( x ) = A + e i ( x ) for all e i ∈ I and for all x ∈ X. Thus A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I − J and for all x ∈ X. Case 2: If e H ( e i ) = e G ( e i ) that is if e i ∈ J − I, then from (i)(b) and (ii)(b), wehave µ e i ( x ) = B − e i ( x ) and µ e i ( x ) = B + e i ( x ) for all e i ∈ J and for all x ∈ X. Thus B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J − I and for all x ∈ X. Case 3: If e H ( e i ) = e F ( e i ) ∨ P e G ( e i ) that is if e i ∈ I ∩ J, then from (i)(a) and(i)(b), we have A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) for all e i ∈ I and for all x ∈ X and B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) for all e i ∈ J and for all x ∈ X. Hence if e i ∈ I ∩ J, then max { A − e i ( x ) , B − e i ( x ) } ≤ ( λ e i ∨ µ e i )( x ) ≤ max { A + e i ( x ) , B + e i ( x ) } . Thus, in all the three cases (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is an ICSS in X. (cid:3) NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 19
Example . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o be cubic soft set defined by p e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) > e F ( e ) = A = < A e ( p ) , λ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . and (cid:16)f G, J (cid:17) = n e G ( e i ) = B i = (cid:8) < p, B e i ( p ) , µ e i ( p ) > : p ∈ X (cid:9) e i ∈ J, i = 1 , , o cubic soft set defined by p e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) > e G ( e ) = B = < B e ( p ) , µ e ( p ) >p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . p [0 . , . , . . , . , . . , . , . Above (cid:16) e
F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs and constructed so that their corresponding (cid:16) e
F, I (cid:17) ∗ and (cid:16)f G, J (cid:17) ∗ are ICSSs. Now consider (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is defined by p e F ( e ) ∨ P e G ( e ) = < r max { A e ( p ) ,B e ( p ) } , ( λ e ∨ µ e ) p >p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∨ P e G ( e ) = < r max { A e ( p ) ,B e ( p ) } , ( λ e ∨ µ e ) p >p [0 . , . , . p [0 . , . , . p [0 . , . , . p e F ( e ) ∨ P e G ( e ) = < r max { A e ( p ) ,B e ( p ) } ( λ e ∨ µ e ) p >p [0 . , . , . p [0 . , . , . p [0 . , . , . Which is an ICSS.
Theorem . For two ECSSs (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o in X, if (cid:16) e F, I (cid:17) ∗ = n e F ( e i ) = (cid:8) < x, A i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ I, o and (cid:16)f G, J (cid:17) ∗ = n e G ( e i ) = { < x, B i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ J o are ICSSs in X, then (cid:16) e F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) is an ICSS in X . Proof.
By similar way to Theorem 7, we can obtain the result. (cid:3)
Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o be ECSSs in X such that (cid:16) e F, I (cid:17) ∗ = n e F ( e i ) = (cid:8) < x, A e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ I o and (cid:16)f G, J (cid:17) ∗ = n e G ( e i ) = { < x, B e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ J o be ECSSs. Then,the P-union of (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) is an ECSS in X . Proof.
Since (cid:16) e
F, I (cid:17) , (cid:16)f G, J (cid:17) , (cid:16) e F, I (cid:17) ∗ and (cid:16)f G, J (cid:17) ∗ are ECSSs so by using thedefinition of an external cubic soft set for (cid:16) e F, I (cid:17) , (cid:16)f G, J (cid:17) , (cid:16) e F, I (cid:17) ∗ and (cid:16)f G, J (cid:17) ∗ ,we have λ e i ( x ) / ∈ ( A − e i ( x ) , A + e i ( x )) for all e i ∈ I and for all x ∈ X , µ e i ( x ) / ∈ ( B − e i ( x ) , B + e i ( x )) for all e i ∈ J and for all x ∈ X, µ e i ( x ) / ∈ ( A − e i ( x ) , A + e i ( x )) for all e i ∈ I and for all x ∈ X and λ e i ( x ) / ∈ ( B − e i ( x ) , B + e i ( x )) for all e i ∈ J and for all x ∈ X re-spectively. Thus we have ( λ e i ∨ µ e i )( x ) / ∈ (max { A − e i ( x ) , B − e i ( x ) } , max { A + e i ( x ) , B + e i ( x ) } )for all e i ∈ I ∩ J and for all x ∈ X. Also since (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) where C = I ∪ J and e H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ P e G ( e i ) If e i ∈ I ∩ J, where e F ( e i ) ∨ P e G ( e i ) is defined as e F ( e i ) ∨ P e G ( e i ) = e H ( e i ) = (cid:26) < x, r max { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27) So, by definitions of an external cubic soft set (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is an ECSS in X . (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o be ECSSs in X such that ( λ e i ∧ µ e i )( x ) ∈ (cid:20) min { max { A + e i ( x ) , B − e i ( x ) } , max { A − e i ( x ) , B + e i ( x ) }} , max { min { A + e i ( x ) , B − e i ( x ) } , min { A − e i ( x ) , B + e i ( x ) }} (cid:19) for all e i ∈ I, for all e i ∈ J and for all x ∈ X. Then (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) is also anECSS.
Proof.
Consider (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , C = I ∩ J where e H ( e i ) = e F ( e i ) ∧ P e G ( e i ) is defined as e F ( e i ) ∧ P e G ( e i ) = e H ( e i ) = (cid:26) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J (cid:27) For each e i ∈ I ∩ J, take α e i = min { max { A + e i ( x ) , B − e i ( x ) } ,max { A − e i ( x ) , B + e i ( x ) }} and β e i = max { min { A + e i ( x ) , B − e i ( x ) } ,min { A − i ( x ) , B + e i ( x ) }} . Then α e i is one of A − e i ( x ), B − e i ( x ), A + e i ( x ) and B + e i ( x ) . We consider α e i = A − e i ( x ) or A + e i ( x ) only, as the remain-ing cases are similar to this one. If α e i = A − e i ( x ) , then B − e i ( x ) ≤ B + e i ( x ) ≤ A − e i ( x ) ≤ NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 21 A + e i ( x ) and so β e i = B + e i ( x ) thus B − e i ( x ) = ( A e i ∩ B e i ) − ( x ) ≤ ( A e i ∩ B e i ) + ( x ) = B + e i ( x ) = β e i < ( λ e i ∧ µ e i )( x ) . Hence ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) . If α e i = A + e i ( x ) , then B − e i ( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) and so β e i = max { A − e i ( x ) , B − e i ( x ) } . Assume that β e i = A − e i ( x ) , then B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∧ µ e i )( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) . So from this, we can write B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∧ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) or B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∧ µ e i )( x ) = A + e i ( x ) ≤ B + e i ( x ) . For the case B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∧ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) it is con-tradiction to the fact that (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs. For the case B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∧ µ e i )( x ) = A + e i ( x ) ≤ B + e i ( x ) , we have ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) because, ( λ e i ∧ µ e i )( x ) = A + e i ( x ) = ( A e i ∩ B e i ) + ( x ) . Again assumethat β e i = B − e i ( x ) , then A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∧ µ e i )( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) . From this we can write A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∧ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) or A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∧ µ e i )( x ) = A + e i ( x ) ≤ B + e i ( x ) . For the case A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∧ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) it is contra-diction to the fact that (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs. And if we take the case A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∧ µ e i )( x ) = A + e i ( x ) ≤ B + e i ( x ) we get ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ), ( A e i ∩ B e i ) + ( x )) because, ( λ e i ∧ µ e i )( x ) = A + e i ( x ) = ( A e i ∩ B e i ) + ( x ) . Hence in all the cases, (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) is an ECSS in X . (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o be cubic softsets in X such that min { max { A + e i ( x ) , B − e i ( x ) } , max { A − e i ( x ) , B + e i ( x ) }} = ( λ e i ∧ µ e i )( x )= max { min { A + e i ( x ) , B − e i ( x ) } , min { A − e i ( x ) , B + e i ( x ) }} for all e i ∈ I, for all e i ∈ J and for all x ∈ X . Then, the (cid:16) e F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) is bothan ECSS and an ICSS in X . Proof.
Consider (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∩ J. e H ( e i ) = e F ( e i ) ∧ P e G ( e i ) is defined as e F ( e i ) ∧ P e G ( e i ) = n e H ( e i ) = (cid:8) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X (cid:9) e i ∈ I ∩ J o . For each e i ∈ I ∩ J take α e i = min { max { A + e i ( x ) , B − e i ( x ) } ,max { A − e i ( x ) , B + e i ( x ) }} and β e i = max { min { A + e i ( x ) , B − e i ( x ) } ,min { A − e i ( x ) , B + e i ( x ) }} . Then α e i is one of A − e i ( x ), B − e i ( x ), A + e i ( x ) and B + e i ( x ) . We consider α e i = A − e i ( x )or A + e i ( x ) only, as the remaining cases are similar to this one. If α e i = A − e i ( x ) then B − e i ( x ) ≤ B + e i ( x ) ≤ A − e i ( x ) ≤ A + e i ( x ) and so β e i = B + e i ( x ) this implies that A − e i ( x ) = α e i = ( λ e i ∧ µ e i )( x ) = β e i = B + e i ( x ) . Thus B − e i ( x ) ≤ B + e i ( x ) = ( λ e i ∧ µ e i )( x ) = A − e i ( x ) ≤ A + e i ( x ) , which implies that ( λ e i ∧ µ e i )( x ) = B + e i ( x ) = ( A e i ∩ B e i ) + ( x ) . Hence ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) and ( A i ∩ B i ) − ( x ) ≤ ( λ e i ∧ µ e i )( x ) ≤ ( A e i ∩ B e i ) + ( x ) . If α e i = A + e i ( x ) , then B − e i ( x ) ≤ A + e i ( x ) ≤ B + e i ( x )and so ( λ e i ∧ µ e i )( x ) = A + e i ( x ) = ( A e i ∩ B e i ) + ( x ) . Hence ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ), ( A e i ∩ B e i ) + ( x )) and ( A e i ∩ B e i ) − ( x ) ≤ ( λ e i ∧ µ e i )( x ) ≤ ( A e i ∩ B e i ) + ( x ) . Consequently, we note that (cid:16) e
F, I (cid:17) ∩ P (cid:16)f G, J (cid:17) is both an ECSS and an ICSS in X . (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are ECSSs in X such that ( λ e i ∨ µ e i )( x ) ∈ (cid:18) min { max { A + e i ( x ) , B − e i ( x ) } , max { A − e i ( x ) , B + e i ( x ) }} , max { min { A + e i ( x ) , B − e i ( x ) } , min { A − e i ( x ) , B + e i ( x ) }} (cid:21) for all e i ∈ I for all e i ∈ J and for all x ∈ X then the (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is anECSS in X.
Proof.
Since (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is defined as (cid:16) e
F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) where C = I ∪ J and e H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ P e G ( e i ) If e i ∈ I ∩ J, where e F ( e i ) ∨ P e G ( e i ) is defined as e F ( e i ) ∨ P e G ( e i ) = e H ( e i ) = (cid:26) < x, r max { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J (cid:27) If e i ∈ I ∩ J, α e i = min { max { A + e i ( x ) , B − e i ( x ) } ,max { A − e i ( x ) , B + e i ( x ) }} and β e i = max { min { A + e i ( x ) , B − e i ( x ) } ,min { A − i ( x ) , B + e i ( x ) }} . Then α e i is one of A − e i ( x ), B − e i ( x ), A + i ( x ) and B + e i ( x ) . We consider α e i = A − e i ( x ) or A + e i ( x ) only, as the remain-ing cases are similar to this one. If α e i = A − e i ( x ) , then B − e i ( x ) ≤ B + e i ( x ) ≤ A − e i ( x ) ≤ A + e i ( x ) and so β e i = B + e i ( x ) . Thus ( A e i ∪ B e i ) − ( x ) = A − e i ( x ) = α e i > ( λ e i ∨ µ e i )( x ) . Hence ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∪ B e i ) − ( x ),( A e i ∪ B e i ) + ( x )) . If α e i = A + e i ( x ) , then B − e i ( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) and so β e i = max { A − e i ( x ) , B − e i ( x ) } . Assume β e i = A − e i ( x ) , then B − e i ( x ) ≤ A − e i ( x ) ≤ ( λ e i ∨ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) , so from thiswe can write B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∨ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) or B − e i ( x ) ≤ A − e i ( x ) = ( λ e i ∨ µ e i )( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) . For the case B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∨ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) , it is con-tradiction to the fact that (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs. For the case B − e i ( x ) ≤ A − e i ( x ) = ( λ e i ∨ µ e i )( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) , we have ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∪ B e i ) − ( x ),( A e i ∪ B e i ) + ( x )) because, ( A e i ∪ B e i ) − ( x ) = A − e i ( x ) = ( λ e i ∨ µ e i )( x ) . Again assume that β e i = B − e i ( x ) , then A − e i ( x ) ≤ B − e i ( x ) ≤ ( λ e i ∨ µ e i )( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) , so from this we can write A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∨ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x )or A − e i ( x ) ≤ B − e i ( x ) = ( λ e i ∨ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) . For the case A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∨ µ e i )( x ) < A + e i ( x ) ≤ B + e i ( x ) , it is contradiction to the fact (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs. And if we take the case A − e i ( x ) ≤ B − e i ( x ) = ( λ e i ∨ µ e i )( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) , we get ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∪ B e i ) − ( x ), ( A e i ∪ B e i ) + ( x )) because,( A e i ∪ B e i ) − ( x ) = B − e i ( x ) = ( λ e i ∨ µ e i )( x ) . If e i ∈ I − J or e i ∈ J − I, then we havethe trivialresult. Hence (cid:16) e F, I (cid:17) ∪ P (cid:16)f G, J (cid:17) is an ECSS in X . (cid:3) NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 23
Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are ECSSs in X such that ( λ e i ∧ µ e i )( x ) ∈ (cid:18) min { max { A + e i ( x ) , B − e i ( x ) } , max { A − e i ( x ) , B + e i ( x ) }} , max { min { A + e i ( x ) , B − e i ( x ) } , min { A − e i ( x ) , B + i ( x ) }} (cid:21) for all e i ∈ I for all e i ∈ J and for all x ∈ X then (cid:16) e F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) is also anECSS in X . Proof.
Since (cid:16) e
F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) is defined as (cid:16) e
F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∪ J and e H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ R e G ( e i ) If e i ∈ I ∩ J, where e F ( e i ) ∨ R e G ( e i ) is defined as e F ( e i ) ∨ R e G ( e i ) = e H ( e i ) = (cid:26) < x, r max { A e i ( x ) , B e i ( x ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27) If e i ∈ I ∩ J, take α e i = min { max { A + e i ( x ) , B − e i ( x ) } ,max { A − e i ( x ) , B + e i ( x ) }} and β e i = max { min { A + e i ( x ) , B − e i ( x ) } ,min { A − e i ( x ) , B + e i ( x ) }} . Then α e i is one of A − e i ( x ), B − e i ( x ), A + i ( x ) and B + e i ( x ) . We consider α e i = B − e i ( x ) or B + e i ( x ) only, as the remain-ing cases are similar to this one. If α e i = B − e i ( x ) then A − e i ( x ) ≤ A + e i ( x ) ≤ B − e i ( x ) ≤ B + e i ( x ) and so β e i = A + e i ( x ) . Thus ( A e i ∪ B e i ) − ( x ) = B − e i ( x ) = α e i > ( λ e i ∧ µ e i )( x ) . Hence ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∪ B e i ) − ( x ),( A e i ∪ B e i ) + ( x )) . If α e i = B + e i ( x ) , then A − e i ( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) and so β e i = max { A − e i ( x ) , B − e i ( x ) } . Assume β e i = A − e i ( x ) , then we have B − e i ( x ) ≤ A − e i ( x ) ≤ ( λ e i ∧ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) . So from this we can write B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∧ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) or B − e i ( x ) ≤ A − e i ( x ) = ( λ e i ∧ µ e i )( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) . For the case B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∧ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) , it is contradiction to the factthat (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs. For the case B − e i ( x ) < A − e i ( x ) = ( λ e i ∧ µ e i )( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) , we have ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∪ B e i ) − ( x ),( A e i ∪ B e i ) + ( x )) because( A i ∪ B i ) − ( x ) = A − e i ( x ) = ( λ e i ∧ µ e i )( x ) . Again assume that β e i = B − e i ( x ) , thenwe have A − e i ( x ) ≤ B − e i ( x ) ≤ ( λ e i ∧ µ e i )( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) . From this we canwrite A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∧ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) or A − e i ( x ) ≤ B − e i ( x ) =( λ e i ∧ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) . For the case A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∧ µ e i )( x )
G, J (cid:17) are ECSSs.And if we take the case A − e i ( x ) ≤ B − e i ( x ) = ( λ e i ∧ µ e i )( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) , weget ( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∪ B e i ) − ( x ),( A e i ∪ B e i ) + ( x )) because, ( A e i ∪ B e i ) − ( x ) = B − e i ( x ) = ( λ e i ∧ µ e i )( x ) . If e i ∈ I − J or e i ∈ J − I , then the result is trivial. Hence (cid:16) e F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) is an ECSS in X . (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are ECSSs in X such that ( λ e i ∨ µ e i )( x ) ∈ (cid:20) min { max { A + e i ( x ) , B − e i ( x ) } , max { A − e i ( x ) , B + e i ( x ) }} , max { min { A + e i ( x ) , B − e i ( x ) } , min { A − e i ( x ) , B + i ( x ) }} (cid:19) for all e i ∈ I for all e i ∈ J and for all x ∈ X then the (cid:16) e F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is anECSS in X . Proof.
Consider (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∩ J. e H ( e i ) = e F ( e i ) ∧ R e G ( e i ) is defined as e F ( e i ) ∧ R e G ( e i ) = e H ( e i ) = (cid:26) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J (cid:27) For each e i ∈ I ∩ J, take α e i = min { max { A + e i ( x ) , B − e i ( x ) } ,max { A − e i ( x ) , B + e i ( x ) }} and β e i = max { min { A + e i ( x ) , B − e i ( x ) } ,min { A − i ( x ) , B + e i ( x ) }} . Then α e i is one of A − e i ( x ), B − e i ( x ), A + i ( x ) and B + e i ( x ) . We consider α e i = B − e i ( x ) or B + e i ( x ) only, as the re-maining cases are similar to this one. If α e i = B − e i ( x ) , then A − e i ( x ) ≤ A + e i ( x ) ≤ B − e i ( x ) ≤ B + e i ( x ) and so β e i = A + e i ( x ) . Then by given inequality we have ( A e i ∩ B e i ) + ( x ) = A + e i ( x ) = β e i < ( λ e i ∨ µ e i )( x ) . Thus we have ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) . If α e i = B + e i ( x ) , then A − e i ( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) and so β e i = max { A − e i ( x ) , B − e i ( x ) } . Assume β e i = A − e i ( x ) , then we have B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∨ µ e i )( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) . So from this we can write B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∨ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) or B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∨ µ e i )( x ) = B + e i ( x ) ≤ A + e i ( x ) . For the case B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∨ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) , it is contradictionto the fact that (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs. For the case B − e i ( x ) ≤ A − e i ( x ) < ( λ e i ∨ µ e i )( x ) = B + e i ( x ) ≤ A + e i ( x ) , we have ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) because, ( A e i ∩ B e i ) + ( x ) = B + e i ( x ) = ( λ e i ∨ µ e i )( x ) . Again assumethat β e i = B − e i ( x ) , then we have A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∨ µ e i )( x ) ≤ B + e i ( x ) ≤ A + e i ( x ) . From this we can write A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∨ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x )or A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∨ µ e i )( x ) = B + e i ( x ) ≤ A + e i ( x ) . For the case A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∨ µ e i )( x ) < B + e i ( x ) ≤ A + e i ( x ) , it is contradiction to the fact (cid:16) e F, I (cid:17) and (cid:16)f
G, J (cid:17) are ECSSs. And if we take the case A − e i ( x ) ≤ B − e i ( x ) < ( λ e i ∨ µ e i )( x ) = B + e i ( x ) ≤ A + e i ( x ) , we get ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) because,( A e i ∩ B e i ) + ( x ) = B + e i ( x ) = ( λ e i ∨ µ e i )( x ) . Thus (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is an ECSS forall e i ∈ I ∩ J. (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I, o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are cubic softsets in X such that min { max { A + e i ( x ) , B − e i ( x ) } , max { A − e i ( x ) , B + e i ( x ) }} = ( λ e i ∧ µ e i )( x ) = max { min { A + e i ( x ) , B − e i ( x ) } , min { A − e i ( x ) , B + i ( x ) }} for all e i ∈ I for all e i ∈ J and for all x ∈ X, then (cid:16) e F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is both anECSS and an ICSS in X . NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 25
Proof.
Consider (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) where C = I ∩ J. Also e H ( e i ) = e F ( e i ) ∧ R e G ( e i ) is defined as e F ( e i ) ∧ P e G ( e i ) = e H ( e i ) = (cid:26) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27) For each e i ∈ I ∩ J, take α e i = min { max { A + e i ( x ) , B − e i ( x ) } ,max { A − e i ( x ) , B + e i ( x ) }} and β e i = max { min { A + e i ( x ) , B − e i ( x ) } ,min { A − e i ( x ) , B + e i ( x ) }} . Then α e i is one of A − e i ( x ), B − e i ( x ), A + e i ( x ) and B + e i ( x ) . We consider α e i = A − e i ( x ) or A + e i ( x ) only, as remainingcases are similar to this one. If α e i = A − e i ( x ) , then B − e i ( x ) ≤ B + e i ( x ) ≤ A − e i ( x ) ≤ A + e i ( x ) and so β e i = B + e i ( x ) . This implies that A − e i ( x ) = α e i = ( λ e i ∨ µ e i )( x ) = β e i = B + e i ( x ) . Thus B − e i ( x ) ≤ B + e i ( x ) = ( λ e i ∨ µ e i )( x ) = A − e i ( x ) ≤ A + e i ( x ) which impliesthat ( λ e i ∨ µ e i )( x ) = B + e i ( x ) = ( A e i ∩ B e i ) + ( x ) . Hence ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) and ( A e i ∩ B e i ) − ( x ) ≤ ( λ e i ∨ µ e i )( x ) ≤ ( A e i ∩ B e i ) + ( x ) . If α e i = A + e i ( x ) , then B − e i ( x ) ≤ A + e i ( x ) ≤ B + e i ( x ) and so ( λ e i ∨ µ e i )( x ) = A + e i ( x ) =( A e i ∩ B e i ) + ( x ) . Hence ( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ),( A e i ∩ B e i ) + ( x )) and( A e i ∩ B e i ) − ( x ) ≤ ( λ e i ∨ µ e i )( x ) ≤ ( A e i ∩ B e i ) + ( x ) . Consequently, we note that (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is both an ECSS and an ICSS in X . (cid:3) Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are internal cu-bic soft sets in X such that ( λ e i ∧ µ e i )( x ) ≤ max { A − e i ( x ) , B − e i ( x ) } for all e i ∈ I forall e i ∈ J and for all x ∈ X, then (cid:16) e F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) is an ECSS.
Proof.
Given that (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are internal cu-bic soft sets in X .Thus for all e i ∈ I, we have A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) , and for all e i ∈ J wehave B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) . Since (cid:16) e
F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) is defined as (cid:16) e
F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) , where C = I ∪ J and e H ( e i ) = e F ( e i ) If e i ∈ I − J e G ( e i ) If e i ∈ J − I e F ( e i ) ∨ R e G ( e i ) If e i ∈ I ∩ J, where e F ( e i ) ∨ R e G ( e i ) is defined as e F ( e i ) ∨ R e G ( e i ) = e H ( e i ) = (cid:26) < x, r max { A e i ( x ) , B e i ( x ) } , ( λ e i ∧ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27) Given condition is ( λ e i ∧ µ e i )( x ) ≤ max { A − e i ( x ) , B − e i ( x ) } for all e i ∈ I for all e i ∈ J and for all x ∈ X. This implies that( λ e i ∧ µ e i )( x ) / ∈ (( A e i ∪ B e i ) − ( x ) , ( A e i ∪ B e i ) + ( x ))= (max { A − e i ( x ) , B − e i ( x ) } , max { A + e i ( x ) , B + e i ( x ) } ) . Hence (cid:16) e
F, I (cid:17) ∪ R (cid:16)f G, J (cid:17) is an ECSS. (cid:3)
Theorem . Let (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are internal cu-bic soft sets in X such that ( λ e i ∨ µ e i )( x ) ≥ min { A + e i ( x ) , B + e i ( x ) } for all e i ∈ I, forall e i ∈ J and for all x ∈ X . Then (cid:16) e F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is an ECSS in X . Proof.
Given that (cid:16) e
F, I (cid:17) = n e F ( e i ) = { < x, A e i ( x ) , λ e i ( x ) > : x ∈ X } e i ∈ I o and (cid:16)f G, J (cid:17) = n e G ( e i ) = (cid:8) < x, B e i ( x ) , µ e i ( x ) > : x ∈ X (cid:9) e i ∈ J o are internal cu-bic soft sets in X .Thus for all e i ∈ I, we have A − e i ( x ) ≤ λ e i ( x ) ≤ A + e i ( x ) and for all e i ∈ J, wehave B − e i ( x ) ≤ µ e i ( x ) ≤ B + e i ( x ) . Since (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is defined as (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) = (cid:16)f H, C (cid:17) where C = I ∩ J and e H ( e i ) = e F ( e i ) ∧ R e G ( e i ) is defined as e F ( e i ) ∧ R e G ( e i ) = (cid:26) e H ( e i ) = (cid:26) < x, r min { A e i ( x ) , B e i ( x ) } , ( λ e i ∨ µ e i )( x ) > : x ∈ X, e i ∈ I ∩ J. (cid:27)(cid:27) . Given condition is that ( λ e i ∨ µ e i )( x ) ≥ min { A + e i ( x ) , B + e i ( x ) } for all e i ∈ I, for all e i ∈ J and for all x ∈ X. This implies that( λ e i ∨ µ e i )( x ) / ∈ (( A e i ∩ B e i ) − ( x ) , ( A ∩ B ) + ( x ))= (min { A − e i ( x ) , B − e i ( x ) } , min { A + e i ( x ) , B + e i ( x ) } ) . Hence, (cid:16) e
F, I (cid:17) ∩ R (cid:16)f G, J (cid:17) is an ECSS in X . (cid:3)
4. Conclusion
In order to deal with many complicated problems in the fields of engineer-ing, social science, economics, medical science etc involving uncertainties, classicalmethods are found to be inadequate in recent times. In 1999 [ ], Molodstov pro-posed a new mathematical tool for dealing with uncertainties which is free of thedifficulties present theories. He introduced the novel concept of soft sets and es-tablished the fundamental results of the new theory. He also showed how softset theory is free from parameterization inadequacy syndrome of fuzzy set theory,rough set tTheory and probability theory etc. In this paper we discuss a new ap-proach to soft set through applications of cubic set. By combine of cubic set andsoft set, we introduce a new mathematical model which is called cubic soft set. Weintroduce two types of cubic soft set 1) Internal cubic soft set (ICSS), Exter-nal cubic soft set (ICSS).
We describe P-(R-)order,P-(R-)union, P-(R-)intersectionand P-OR, R-OR, P-AND and R-AND are introduced, and related properties areinvestigated. We show that the P-union and the P-intersection of internal cubicsoft sets are also internal cubic soft sets. We provide conditions for the P-union(resp. P-intersection) of two external cubic soft sets to be an internal cubic softset. We give conditions for the P-union (resp. R-union and R-intersection) of twoexternal cubic soft sets to be an external cubic soft set. We consider conditions for
NEW APPROACH TO SOFT SET THROUGH APPLICATIONS OF CUBIC SET 27 the R-intersection (resp.P-intersection) of two cubic sof sets to be both an externalcubic soft set and an internal cubic soft set.In future we will focuse on Applications of cubic soft in information sciencesand knowledge System. We will also study cubic soft relations. We will apply cubicsoft set to algebraic stractures. This work also extended to topological space.
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