International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February ISSN

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1 International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February KEY PROPERTIES OF HESITANT FUZZY SOFT TOPOLOGICAL SPACES ASREEDEVI, DRNRAVI SHANKAR Abstract In this paper, we define hesitant fuzzy soft ( HF soft in short) topological spaces and its fundamental properties such as HF soft interior, HF soft closure, HF soft exterior, HF soft boundary of a HF soft set Moreover, we investigate some roperties on HF soft compact space, especially finite intersection property on HF soft topological spaces Key words Hesitant fuzzy soft topological space, HF soft sub space, HF soft interior, HF soft exterior, HF soft boundary, HF soft closure, HF Soft open cover, HF soft compact space 1 INTRODUCTION mbiguity imperfection and uncertainty and are main making problem Afactors in the real life The significance and uses of ambiguity In this paper, we present hesitant fuzzy soft sets (Shortly HF is identified by researchers for the last 5 decades It is pointed out as a complicated one to be dealt within some specifications To excel in dealing uncertainty, different theories Soft) and HF soft topological spaces and prove related properties We also define HF Soft point, HF Soft neighborhood system and verify corresponding results Later we define HF Soft such as vague, fuzzy and rough theories are initiated and developed drastically In 1965, a new mathematical tool to cope Soft exterior points and examine few of their properties We interior point, HF Soft closure, HF Soft boundary point, HF up with uncertainty, [2] Zadeh has introduced fuzzy sets and also define HF Soft compactness, for this we introduce HF Soft this is used in various fields In that process, [16] Molodtsov s open cover and HF Soft finite sub cover Next we prove some has given one more useful approach, ie Soft set theory in the theorems related to them year 1999 It is an advanced frame work to handle ambiguity Let us suppose that is any universal set and E is a set of parameters corresponding to the elements of throughout the Parameterization is the main concept in it Without any specific condition on parameters, the soft theory has established paper recently in presenting uncertain concepts in a systematic manner Torra [8] has generalized the fuzzy sets and proposed hesitant 2 PRELIMINARIES fuzzy sets which permit the membership degree to avail a set of possible values and the membership is contained in the closed interval [0, 1] This new concept has grabbed the attention of number of authors In 2013 K V Babithaa and Definition 21 [16] A soft set is defined as a pair (s, A), denoted by s A over, where s: A (), A E, ρ() is the set of all subsets of The class of all soft sets defined on with respect to the parameters in A is denoted by s A Sunil Jacob John [6] have combined the hesitant fuzzy sets and Definition 22 If s soft sets ie They have initiated the notion of hesitant fuzzy and q B over a common universe are soft sets then s soft sets A vigorous study on topological structures on soft OR q B denoted by s A q B is defined as s sets is continued by number of researchers S Atmaca and I q B = (f, A B) where f(a, b) = s(a) q(b) Zorlutuna have explored the topological structures of fuzzy Definition 23 [16] A fuzzy soft set is defined as a pair (, parameterized soft sets and mappings on fuzzy parameterized A) denoted by A over, where : A ρ(), A E, ρ() soft sets They have given the notion of quasi-coincidence for fuzzy parameterized soft sets and have examined some basic concepts in fuzzy parameterized soft topological spaces In is a set of all fuzzy sets in The class of all fuzzy soft sets over with respect to the parameters in A is denoted by ( A ) 2014, [17] S Broumi, F Smarandach have worked on new operations over interval valued intuitionistic hesitant fuzzy sets Definition 24 [8] A hesitant fuzzy set is defined as a function Hesitant fuzzy topological space and some related properties that when applied on returns a subset of [0, 1] are introduced by Serkan Karata [13] Later [9] Verma and Sharma have given the fundamental operations on hesitant ie : [0, 1] A hesitant fuzzy set over is denoted by fuzzy sets Few more operations are established by the author and the class of all hesitant fuzzy sets defined over is denoted J Wang et al on Hesitant fuzzy soft sets Their purpose is to use the hesitant fuzzy soft sets in multi criteria group decision by HF 2016

2 International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February (x) = min { (x): for all }= {01}, Definition 25 [8] The lower and upper bound of a hesitant fuzzy set over are defined as below Lower bound of is defined and denoted by (x) = min { (x): for all } Upper bound of is defined and denoted by +(x) = max { (x): for all } Example 26 Let = {h1, h2, h3} and, ( ) Such that = {(h1,[02, 045]), ( h2, [06, 08]), ( h3,[01, 07])}, 085])} Definition 215 [6] We define the hesitant fuzzy null set as = {(h1,[005, 05]), ( h2, [07, 09]), ( h3,[02, 07])}, a HF set such that (x) = {0} for all x It is denoted by Then, we have that (x) = min { (x): for all }= {01}, Definition 216 [6] We define the hesitant fuzzy full set as a HF set such that (x) = {1} for all x It is denoted by +(x) = max { (x): for all } ={08}, (x) = min { (x): for all }= {005}, Definition 217 [6] A pair (, A) is called a hesitant + (x) = max { (x): for all }= {09} fuzzy soft set (HF soft set in short cut) if is a mapping such Definition 27 [8] The complement of any hesitant fuzzy set that : A ( ) ie (e) ( ), for all e A where A (x) is defined as c (x) = E The set of all hesitant fuzzy soft sets is denoted by HFS Example 28 (x) = = {(h1,[08, 055]), Example 218 Let be set of team members played in a game ={h1, h2, h3, h4} Let A = {cooperation, energetic, (h2, [04, 02]), ( h3,[09, 03])}, Complement of communication, commitment}, a set of parameters And : B Definition 29 [8] Let, ( ) Then is said ( ) Then hesitant fuzzy soft set A given below is the to be a subset of if (x) (x) for all x and is evaluation of the performance of 4 players p1, p2, p3, p4 by denoted by four judges Definition 210 [8] Two hesitant fuzzy sets, are (cooperation) = {h1 = (075,1,08, 08), h2 = (06,065,07,075), said to be equal if if and only if and h3 = (08,075,1,09), h4 = (06,075,09,085)}; (energetic) = Example 211 It can be seen from example 26 clearly that Definition 212 [8] The intersection of two hesitant fuzzy sets and is a hesitant fuzzy set such that ( )(x) = {h ( ) / h min( +, +)} Definition 213 [8] The union of two hesitant fuzzy sets and is a hesitant fuzzy set such that ( )(x) = {h ( (x) (x)) / h max(, )} Example 214 Let = {h1, h2, h3} and, ( ) such that = {(h1,[04, 0 5]), ( h2, [06, 09]), ( h3,[03, 085])}, = {(h1,[0 5, 02]), ( h2, [08, 07]), ( h3,[01, 02])}, Then, we have that (x) = min { (x): for all }= {03}, + (x) = max { (x): for all } ={09}, + (x) = max { (x): for all }= {08} max(, ) = {03} and h max(, ) h 03 also ( (x) (x)) = {(h1,[02, 04, 0 5]), ( h2, [06, 07, 08, 09]), ( h3,[01, 02, 03, 085])} ( )(x) = {h ( (x) (x)) / h max(, )}= {(h1,[04, 0 5]), ( h2, [06, 07, 08, 09]), ( h3,[03, {h1 = (075,07,08,06), h2 = (085,07,08,08), h3 = (07,07,09,08), h4 = (1,09,099,09)}; (communication) = {h1 = (075,09,08,09), h2 = (06,085,08,09), h3 = (0 8,09,075,07), h4 = (09,065,085,07)}; (commitment) = {h1 = (08,09,09,1), h2 = (07,06,07,08), h3 = (08,1,09,09), h4 = (09,07,075,08)} Definition 219 [6] For two hesitant fuzzy soft sets A and B over a common universe we say that A is hesitant fuzzy soft subset of A B Here : B ( ) B It is denoted by if (i) A B (ii) (a) (a) for every a in A Definition 220 [6] Let A be hesitant fuzzy soft set Then the complement of A is defined by A = (, A) where (a) is the complement of the hesitant fuzzy set (a) Note (i) Let A and B be two hesitant fuzzy soft sets over Then they satisfy Demorgan laws 2016

3 International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February (ii) [ A B ] = A B B = B ; A B = B (iii) [ A B ] = ( A) ( B ) Definition 221 The hesitant fuzzy soft null set is a HF soft set A over is said to be HF soft null set denoted by if e A, (e) is the hesitant fuzzy null set of where (x) = {0}, x Definition 222 A HF soft set A over is said to be HF soft full set denoted by if e A, (e) is the hesitant fuzzy full set of where (x) = {1}, x In this section, we will introduce hesitant fuzzy soft topological spaces and related properties 3 Hesitant fuzzy soft topological space Definition 31[13] A hesitant fuzzy topology HFτ is a class of hesitant fuzzy sets over which satisfies the following conditions: (i), HFτ, (ii) If A, B HFτ, then A B HFτ, (iii) If { Ai }i I HFτ, then i I Ai HFτ HFSτ* for all Proof (i) Since (HFS, HFSτ) is a HF Soft topological space clearly, HFSτ The pair (HF, HFτ) is called a hesitant fuzzy topological space Every element of HFτ is said to be hesitant fuzzy open, set The complement of a hesitant fuzzy open set is said to be a are HF Soft open sets in (HFS, hesitant fuzzy closed set Definition 32 A hesitant fuzzy soft topology HFSτ is a, are HF Soft closed sets in (HFS, class of hesitant fuzzy soft sets over which satisfies the following conditions:, HFSτ* (i), HFSτ, If A, B HFSτ, then A B HFSτ, If { Hi }i I HFSτ, then i I Hi HFSτ The pair (HFS, HFSτ) is called a hesitant fuzzy soft topological space Every element of HFSτ is said to be hesitant fuzzy soft open set Definition 33 If the complement of a HF soft set hesitant fuzzy soft open set, then A is a A is called HF soft closed set Example 35 Consider a collection HFSτ of HF Soft sets over as HFSτ = {, A, B } (i), HFSτ (ii) = ; A = ; B = ; A = A ; (iii) = ; A = A ; Thus (HFS B = B, A = ; B = ; A B = A ; A B = and also A B =, HFSτ) is a HFS Topological space The HFS open sets are,, A, B and their complements are HF Soft closed sets Theorem 36 Let (HFS, HFSτ) be a HF soft topological space and HFSτ* denote the collection of all HF soft closed sets in (HFS, Then (i), HFSτ* (ii) If A, B HFSτ* then A B HFSτ* and (iii) If HFSτ* for all then, HFSτ* because = and = (ii) Suppose A, B HFSτ* A, B are HF Soft closed sets in (HFS, A, B are HF Soft open sets in (HFS, A, B HFSτ A B HFSτ, By definition 532 A B is a HF Soft open set in (HFS, ( A B ) is a HF Soft closed set in (HFS, ie A B is a HF Soft open set, by Demorgan laws and definition of HF Soft closed set 2016

4 (iii) Suppose HFSτ* for all For each, (HFS, For each, (HFS, is a HF Soft closed set in is a HF Soft open set in HFSτ By definition 532 the following properties: (i) If B N HFSτ ( A e), then A e B, is HF Soft open set in (HFS, (ii) If B N HFSτ ( A e) and B D then D ) is HF Soft closed set in (HFS, N HFSτ ( A e), ie (by Demorgan s law) is HF Soft closed set (iii) If B, D N HFSτ ( A e), then B D in (HFS, N HFSτ ( A e), HFSτ* for all (iv) If B N HFSτ ( A e), then there is a M Hence proved Definition 37 Let HFSτ and HSHτ* be two HF Soft topologies N HFSτ ( A e A e) such that M B N HFSτ ( A e ) for each on (HFS, HFSτ is said to be coarser than HFSτ* Proof (i) If B NHFSτ( A e), then there is a D or that is HFSτ* finer than HFSτ iff HFSτ HFSτ* If neither HFSτ HFSτ HFSτ* nor HFSτ* HFSτ then these topologies are not such that A e D B comparable Therefore, we have A e B Definition 38 Let (HFS, HFSτ) be a HF Soft topological (ii) Let B NHFSτ( A e) and B D space and for Y, B E, : B ρ() Then the HF Soft Since B NHFSτ( A e), then there is a M HFSτ topology, HFSτ*= { Y B A : A HFSτ} equipped such that A e M B with the class of HF soft subsets HFQ is called HF Soft Therefore, we have A e M B D and subspace of (HFS, HFSτ) and is denoted by (HFQ, so D NHFSτ( A e) HFSτ*) Definition 39 A HF Soft set A is said to be a HF Soft (iii) If B, D NHFSτ( A e), then there exist M, point in (HFS, HFSτ), denoted by A e, if for the element K HFSτ e A, (e) and (e ) =, e A -{e} such that A e M B Definition 310 A HF Soft point A e is said to be in a and A e K D HF Soft topological space (HFS, HFSτ) denoted by A e Hence A e [ M K ] [ B D ] B, if for the element e A, A (e) B (e) Since [ M K ] HFSτ, we have [ B D ] Definition 311 Let (HFS, HFSτ) be a HF Soft topological NHFSτ( A e) space over ; (iv) If B NHFSτ( A e), then there is a K HFSτ B be a HF soft subset of A and A x be a HF soft such that A e K B point in (HFS, HFSτ), then B is called a HF Soft neighbourhood Put M = K Then for each A e M, of A x if there exists a HF Soft open set C A e M K B such that A x C B The neighbourhood system of a HF Soft point A e, denoted by NHFSτ( A e), is the collection of all its neighbourhoods Definition 312 A HF Soft set International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February A B HFSτ* by definition of HFSτ* N in a HF Soft topological space (HFS (shortly nbd) of the HF Soft set, HFSτ) is called a HF Soft neighbourhood A if there exists a HF Soft open set HFSB such that A B N Theorem 313 The neighbourhood system N HFSτ ( A e in a HF Soft topological space (HFS A e) at, HFSτ) possesses Hence B NHFSτ( A e ) Definition 314 The HF Soft interior of a HF Soft set denoted by I( A A ) is defined by the union of all HF Soft 2016

5 International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February open sub sets of A we have B from (1), being intersection of all HF ie = i { : is HF Soft open sub set of A } Soft closed super sets Similarly, as A is HF Soft closed and also we know that Theorem 315 By the above definition, the following statements can be proved easily A A, we get that A (3) (i) I( A ) is a HF Soft open set, because arbitrary union of Hence from (2) and (3), A = HF Soft open sets is HF Soft open Definition 318 A HF Soft point A e is said to be a HF (ii) I( A ) A, because union of subsets of a set is Soft boundary point of a HF Soft set A if A e again subset of it, and (iii) If A is HF Soft open then A = I( A ) (iv) I( A ) is the largest HF Soft open set contained in Definition 319 The set of all HF Soft boundary points over a HF Soft set A is called HF Soft boundary of the set A Definition 316 A and is denoted by B( A ) The HF Soft closure of a HF Soft set A Definition 320 Let (HFS, HFSτ) be a HF Soft topological denoted by is defined by the intersection of all HF Soft space Let A be a closed super sets of A HF Soft set over Then the HF Soft exterior of A, denoted by Ao, ie = { : is HF Soft closed super set of i A } is defined as Ao = I( Ac )ie the exterior of a HF Soft set Theorem 317 For any HF Soft set A, we have the following Theorem 321 Let (HFS, HFSτ) be a HF Soft topological A is interior of the complement of the HF Soft set A (i) is a HF Soft closed set (Since is the arbitrary space Let A and intersection of HF Soft closed sets) B are fuzzy soft sets over Then (ii) A (Since is the intersection of HF Soft (1) ( A ) o = ( Ac ) o super sets of A ) (2) (( A ) ( B )) o = ( A ) o ( B o ) and (iii) If A is HF Soft closed then A = (3) ( A ) o ( B ) o (( A ) ( B )) o Proof: Let (HFS, HFSτ) be a HF Soft topological space Theorem 322 Let (HFS, HFSτ) be a HF Soft topological over space Let Let A be a HF Soft set over such that A = A be HF Soft set over Then (1) (B( To prove that A is HF Soft closed A )) = ( A ) o (( A ) o = ( A ) o I( A ) (2) = ( We have = { : is HF Soft closed super i A ) o B( A ) (3) ( A ) o = ( A ) B( A ) set of A } (1) (4) B( A ) = = ( A ) o Being an arbitrary intersection of HF Soft closed sets, is Proof (1) ( A ) o (( A ) o = ((( A ) o ) ) ([( A ) o ] ) HF Soft closed set, = [(( A ) o ) ((( A ) o ) ] = [ ] = (B( A )) and since A =, we get A is also HF Soft closed (2) ( Conversely, suppose that A is HF Soft closed in (HFS, A ) o B( A ) = ( A ) o = [( A ) o ] [( A ) o ] To prove that A = = [( A ) o ] It is clear from definition (1), being a superset, (2) And for any HF Soft closed super set B of A, A = = ( A ) (3) ( A ) B( A ) = ( A ) (B( A )) = ( A ) ( ( A ) o (( A ) ) o ) (by (a)) 2016

6 International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February = [( A ) ( ( A ) o ] [( A ) (( A ) ) o ] So A It shows that A is a HF Soft closed = ( A ) o = ( A ) o (4) B( A ) = ( A ) o = (( A ) o ) = Theorem 323 Let (HFS space Let A be fuzzy soft set over Then (1) B( A ) ( A ) o = (2) B( A ) ( A ) o = Proof (1) B( A ) ( A ) o = ( A ) o B( A ) = ( A ) o (, HFSτ) be a HF Soft topological = ( A ) Soft closed set over o (iii) ( = A ) o = ( A ) o ) c = [ ( ] (2) B( A ) I( A ) (By definition 219 and 32, ( A ), HFSτ = (( A ) o ( ) )) = (( A ) o = ( [ ] ) = = B( A ) = (Since HFSτ, is HF Soft open set for each, = is HF Soft closed for each and arbitrary intersection Theorem 324 The following properties can be easily established with the help of the above definition of HF Soft closed sets is HF Soft closed set and we know that [ ] = ) For any HF Soft topological space (HFS, HFSτ) and C Theorem 325 Let (HFS, HFSτ) be a HF Soft topological HFSτ then space Let (i) B( C ) A and ie The HF Soft boundary of a HF Soft set is subset of the HF B are HF Soft sets over Then Soft boundary of the set (1) B( A B ) B[ A B ] [B( B ) (ii) A is HF soft open set if and only if A B( A ) ] [B( A B( B )] = (iii) A is HF Soft closed if and only if and only if B( A ) A (iv) B( A ) = ( A ) o Proof (i) By definition it is clear (ii) Let A be a HF Soft open set over Then A = ( A ) o A B( A ) = ( A ) o B( A ) = (by theorem 323) Conversely, let A B( A ) = Then A = That is, A = set Hence 2016 A is HF Soft open (iii) Let A be HF Soft closed set over Then A = Now B( A ) = = A That is, B( A ) A Conversely, let B( A ) A Then B( A ) A = Since B( A ) =B( A ) =, we have B( A ) A = Then by (1) A is HF Soft open and hence A is a HF (2) B( A B ) [B( A ) [B( B ) ] [B( A B(HFQB] Proof (1) B ( A B ) = = [ ] [ ] = ( )] [ ( )] = ( )] [ ] = [B( A ) ] [B( B ) ]

7 International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February [B( A B( B ] is HF Soft open subspace of HFS (2) B ( A B ) = [ ] = [ ] [ ] = [ ] = ] [ = [B( A ) [B( B ) ] [B( A ) B( B )] 4 Hesitant fuzzy soft compact spaces In this section we define HF Soft open cover, HF Soft sub cover, HF Soft compact space, HF Soft compact subspace Definition 41 Let (HFS, HFSτ) be a HF Soft topological space A collection { } of HF Soft subsets of HFS is said to be a HF Soft open cover of HFS if each HF Soft point HFS = in HFS is in at least one ie HFS = (4) Definition 42 A sub class {,, Since HFS we get = HFS } of a HF Soft open cover { } which itself forms an = [ open cover for HFS is called a HF Soft sub cover of HFS ], Using (4) ie When HFS = =[( ( ) ( )] ( ) Definition 43 A HF Soft compact space is a HF Soft topological space in which every HF Soft open cover has a finite = = HF Soft sub cover Definition 44 A HF Soft compact subspace of a HF Soft,,,, } is a finite topological space is a HF Soft subspace which itself is a HF HF Soft sub cover of Soft compact as a HF Soft topological space in its own right Theorem 45 Any HF Soft closed subspace of a HF Soft Hence every HF Soft open cover of has a finite HF Soft compact space is HF Soft compact Proof Let (HFS, HFSτ) be a HF Soft compact space Let A be a HF Soft closed subspace of HFS Now we have to prove that is HF Soft compact space For this we show that every HF Soft open cover of contains a finite HF Soft sub cover Suppose { } is a HF Soft open cover of By definition 32, = (1) Since each is a HF Soft open set in, = HFS (2), where HFS for each i by definition of relative HF Soft topology Since is HF Soft closed subspace of HFS, 2016 And it implies that HFS = (3) = (Using (1) = ) = ) ], Using (2) = ( ) ( )] By distributive property = [( ) HFSE], Using (3) = [ ], Since each and are HF Soft subsets of HFS { { }, } is a HF Soft cover of HFS Since HFS is HF Soft compact, this HF Soft open cover has a finite HF Soft sub cover Let it be {,,,, } sub cover And hence is a HF Soft compact space Thus any HF Soft closed subspace of a HF Soft compact space is HF Soft compact Hence the theorem is proved 5 CONCLUSION In this chapter, we have presented mappings on hesitant fuzzy soft sets,hf soft topological spaces and proved related properties We have also defined HF Soft point, HF Soft neighborhood system and verified corresponding results Later we have defined HF Soft interior point, HF Soft closure, HF Soft boundary point, and HF Soft exterior points and their properties are verified We have also defined HF Soft compactness, for this we have introduced HF Soft open cover and HF Soft finite sub cover Next we have proved some theorems related

8 International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February to them REFERENCES [1] Maji, Pabitra Kumar, R Biswas, and A R Roy "Fuzzy soft sets" Journal of Fuzzy Mathematics 93 (2001): [2] LA Zadeh, Fuzzy sets, Information and Control 8 (1965) [3] Y Zou, Z iao, Data analysis approaches of soft sets under incomplete information, Knowledge-Based Systems 21 (8) (2008) [4] Maji PK, Biswas R and Roy AR, Soft set theory Computers and mathematics with application 45 (2003) [5] D Dubois, H Prade, Fuzzy sets and systems: theory and applications Academic Press, New York, 1980 [6] Babithaa, K V, and Sunil Jacob John "Hesitant fuzzy soft sets" Journal of New Results in Science 3 (2013): [7] Qiu-Mei Sun, Zi-Long Zhang, Jing Liu, Soft Sets and Soft Modules Rough Sets and Knowledge Technology,Vol 5009 (2008) [8] V Torra, Hesitant fuzzy sets International Journal of Intelligent Systems 25 (2010) [9] R Verma, B D Sharma, New operations over hesitant fuzzy sets, Fuzzy Inf Eng 2 (2013) [10] M Shabir, M Naz, On soft toplogical spaces, Computers and mathematics with application 61, 7 (2011) [11] N Çağman, S Karataş and S Enginoğlu, Soft Topology, Computers and mathematics with application (2011), 62 (2011) [12] B Tanay, M B Kandemir, Toplogical structure of fuzzy soft sets, Computers and Mathematics with application,61 (2011) [13] Karataş, Serkan "Hesitant fuzzy topological spaces and some properties"contemporary Analysis and Applied Mathematics 31 (2015) [14] M ia, Z u, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52 (2011) [15] C L Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications 24 (1968) [16] DA Molodtsov, Soft set theory-first results, Computers and Mathematics with Applications 37 (1999) [17] S Broumi, F Smarandache, New operations over interval valued intuitionistic hesitant fuzzy set, Mathematics and Statistics 2 (2) (2014) Author Profile A Sreedevi Asst Prof, CMR Engineering College, Hyderabad, India DrNRavi Shankar Assoc Prof, HOD, Dept of Applied Mathematics, GIS, GITAM University, Visakhapatnam 2016