JAST 2015 M.U.C. Women s College, Burdwan ISSN a peer reviewed multidisciplinary research journal Vol.-01, Issue- 01

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1 JAST 05 M.U.C. Women s College, Burdwan ISSN a peer reviewed multidisiplinary researh journal Vol.-0, Issue- 0 On Type II Fuzzy Parameterized Soft Sets Pinaki Majumdar Department of Mathematis, M.U.C Women s College, Burdwan West-Bengal, India Mail: pmajumdar@rediffmail.om Abstrat: In this paper the notion of Type II fuzzy parameterized soft sets has been introdued. Several operations on these sets have been defined and basi set theoreti properties have been investigated. An appliation of Type II Fuzzy Parameterized Soft sets in deision making has been shown. Notion of entropy of these sets are also disussed. Keywords: Fuzzy set, Soft sets, Fuzzy Parameterized soft set, Type II fuzzy parameterized soft set, Deision making, entropy. 00 AMS Classifiation: 037, 0375, 6C86 0. Introdution There are several tehniques to represent and solve various types of unertainties prevailing in this physial world. There are theories namely theory of probability, theory of fuzzy sets, theory of multisets, theory of rough sets, theory of vague sets et. whih an handle unertainties of various types. In 999, Molodtsov [] initiated the theory of soft sets as a new mathematial tool for dealing with physial problems involving unertainties, whih traditional mathematial tools annot handle. He has shown several appliations of this theory in solving many pratial problems in eonomis, engineering, soial siene, medial siene, et. Later other authors like Maji et al [8-0] have further studied the theory of soft sets and used this theory to solve some deision making problems. They have also introdued the onept of fuzzy soft set, a more generalised onept, whih is a ombination of fuzzy set and soft set and studied its properties. Several authors [-5] has defined many hybrid versions of soft sets and applied them in deision making. Reently, in 00, P. Majumdar & S. K. Samanta [] have defined the notion of generalised fuzzy Soft Sets, whih further generalises the onept of fuzzy soft sets as introdued by Maji et. al. [8]. The notion of fuzzy parameterised soft sets was first introdued by Cagman et. al. [] whih is a further generalization of soft sets. In this generalization of soft sets, a degree has been attahed with the parameterization proess, while assigning a risp set to eah parameter. This definition is more realisti as it involves unertainty in the seletion of a risp set, i.e. elements of the universe, orresponding to eah value of the parameter. In this paper we have introdued a new sophistiated version of fuzzy parameterized soft sets alled Type II fuzzy parameterized soft sets. Here orresponding to eah parameter we get a fuzzy subset of the [Artile History: Reeived on , Aepted on ] [63]

2 On Type II Fuzzy Parameterized Soft Sets Author: P. Majumdar power set of the universal set. This set has greater powers of handling unertain data and the fuzzy parameterized soft sets an be dedued as a speial ase of these Type II fuzzy parameterized soft sets. This paper is organised as follows: In setion one, some basi definitions and results regarding soft sets and fuzzy parameterized soft sets are disussed. The notion of type II fuzzy parameterized soft sets are introdued in setion two. Some algebrai operations and basi properties of Type II fuzzy parameterized soft sets are also given there. In setion three, an appliation of type II fuzzy parameterized soft sets in solving a deision making problem has been shown. ntropy of these sets is disussed in setion four. Setion five onludes this paper.. Preliminaries In this setion some definitions, results and eamples regarding soft sets are given whih will be used in rest of this paper. The idea of soft sets was first given by Molodtsov. Later Maji & Roy have defined operations on these soft sets and studied their properties. Definition. [0] Let U be an initial universal set and let be a set of parameters. Let PU ) denote the power set ofu. Let A be a subset of. A pair F, A ) is alled a soft set over U iff F is a mapping given by F : A P U). ample. As an illustration, onsider the following eample. Suppose a soft set F, ) desribes attrativeness of the shirts whih the authors are going to wear. U = the set of all shirts under onsideration = {,,, }, 3 5 = {olorful, bright, heap, warm} = { e, e, e 3, e }are the parameters of attration. Let e ) {, }, Here F F e ) {,, }, F e 3 ) { }, F e ) {, }. F ei )' 3 s are alled e-approimations. 5 So, the soft set F, ) is a family { F e i ), i,,3, } of subsets of PU ). Definition.3 [0] For two soft sets F, and G, B) over a ommon universe U, we say t hat F, is a soft subset of G, B) if i) A B, ii) A, F ) G ). Definition. [0] quality of two soft sets) Two soft sets F, and G, B) over a ommon universe U are said to be equal if F, is a soft subset of G, B) and G, B) soft subset of F,. is a JAST-05, Vol.-0, Issue-0 [6]

3 JAST-a peer reviewed multidisiplinary researh journal Vol.-0, Issue-0 In 008, Majumdar & Samanta[] have given a new definition of omplement of soft sets as follows: Definition.5 [] Complement of a soft set) The omplement of a soft set F, is denoted by C C F, and is defined by F, F,, given by F C ) U F ), A. C where C F : A P U ) is a mapping Definition.6[0] Null soft set) A soft set F, over U is said to be null soft set denoted by, if A, F ). Definition.7[0] Absolute soft set) A soft set F, over U is said to be absolute soft set denoted by A ~, if A, F ) U. Definition.8[0] Union of two soft sets F, and G, B) over a ommon universe U is the soft set H, C), where C A B, and e C, H e) F e), e A B, G e), e B A, F e) G e), e A B. We write H, C) F, ~ G, B). Definition.9[0] Intersetion of two soft sets F, and G, B) over a ommon universe U is the soft set H, C), where C A B, and e C, H e) F e) G e). We write H, C) F, ~ G, B). Proposition.0 [0] The following results hold here. ~ i) F, F, F,, ~ ii) F, F, F,, ~ iii) F, F,, ~ iv) F,, ~ ~ ~ F, A A v), [65]

4 ~ ~ F, A F A vi), ). On Type II Fuzzy Parameterized Soft Sets Author: P. Majumdar Proposition. The following results hold here. i) F, G, B)) C F, C G, B) C, ii) F, G, B)) C F, C G, B) C. First notion of Fuzzy parameterized soft sets are given in [] by Cagman at. al.. We here state the definition and some important properties of Fuzzy parameterized soft sets in a modified version. Definition. LetU be an initial universal set and let be a set of parameters. Then the pair U, ) will be alled a soft universe. Let PU ) denote the power set ofu and be a fuzzy set over. A triplet U,, ) is alled a fuzzy parameterized soft set FPS set in F short) over U, ) iff F is a mapping given by F : P U) I and is defined as follows: For e, F e) F e), e)), where F e) U. Note.3 If we take, then we atually get the ordinary soft set. Thus FPS sets are more general than its predeessors. ample.3 LetU {,, 3} be the universal set and e e {, } be the parameter set. e e Further be a fuzzy set over, whih is defined as follows: {, }. Let the mapping F : P U) I be defined as follows: F e ) {, },0.8)& F e ) {, },0.63). Then U,, is a fuzzy 3 3 parameterized soft set over U, ). F Definition. Let U,, and U,, be two FPS sets over U, ). Then the F FPS set U,, is said to be a soft subset of U,, F U,, F U,, G, if i) & ii) F e) G e), e. G G, whih will be denoted by JAST-05, Vol.-0, Issue-0 [66]

5 JAST-a peer reviewed multidisiplinary researh journal Vol.-0, Issue-0 Consequently, they will be equal if U,, F U,, G and U,, G U,, F. ample.5 U {,, 3} be the universal set and e e {, } be our parameter set. e e Further be a fuzzy set over, whih is defined as follows: {, }. Let the mapping G : P U) I be defined as follows: G e ) { },0.6)& G e ) { },0.3). Then U,, is a fuzzy parameterized 3 G F soft set over U, ) whih is a subset of the FPS set U,, defined in ample.3. Definition.6 Let U,, be a FPS set over U, ) suh that F e) U,.0), e. F Then this FPS set is alled an absolute fuzzy parameterized soft set AFPS set in short), denoted by A. Definition.7 Let U,, be a FPS over U, ) suh that F e),0.0), e. F Then this FPS set is alled a null fuzzy parameterized soft set NFPS in short), denoted by. Definition.8 Let U,, and U,, be two FPS over U, ). Then their F union is denoted as U,, F U,, G U,, H, is a FPS defined as follows: H e) F e) G e), e) e)), e. G Definition.9 Let U,, and U,, be two FPS set over U, ). Then their F intersetion U,, F U,, G U,, H, is a FPS set defined as follows: G H e) F e) G e), e) e)), e. Definition.0 Let U,, be a FPS set over U, ). Then its omplement, denoted by U,, C F F, is a FPS set defined as follows: C F e) F e)), e)) e, where F e) F e), e)). [67]

6 On Type II Fuzzy Parameterized Soft Sets Author: P. Majumdar ample. Let U {,, 3} be the universal set and e e Let be a fuzzy set over, whih is defined as follows: mapping F : P U) I be defined as follows: 3 3 {, } be the parameter set. e e {, }. Let the F e ) {, },0.8)& F e ) {, },0.63). Then U,, is a fuzzy parameterized soft set over U, ). Further be a fuzzy set over, whih is defined as e e follows: {, }. Let the mapping G : P U) I be defined as follows: G e ) {, },0.6)& G e ) {, },0.9). Then their union is the FPS set 3 U,, F U,, G U,, H, is defined as follows: and H : P U) I be suh that H e) U,0.8)& H e) U,0.9). F e e {, } And their intersetion U,, F U,, G U,, K, is a FPS set defined as follows: e e {, } and : ) 3 K P U I be suh that K e ) { },0.6)& K e ) { },0.63). Also the omplement of U,, is denoted by U,, follows: F C F, is a FPS set defined as F e ) { },0.)& F e ) { },0.37). Definition. Let U,, and U,, be two FPS sets over U, ). Then the F AND operation is denoted as U,, F U,, G U,, H and is defined as follows: H e, f ) F e) G f ), e) f )), e, f ). The resultant will be a FPS set over U, ). Definition.3 Let U,, and U,, be two FPS sets over U, ). Then the F OR operation is denoted as U,, F U,, G U,, H and is defined as follows: H e, f ) F e) G f ), e) f )), e, f ). The resultant is again a FPS set over U, ). G G JAST-05, Vol.-0, Issue-0 [68]

7 JAST-a peer reviewed multidisiplinary researh journal Vol.-0, Issue-0 ample. Consider the two FPS sets U,, and U,, F G defined in ample..then the OR operation is denoted as U,, F U,, G U,, H and is given by: H e, e ) U,0.8), H e, e ) U,0.63), H e, e ) {, },0.9), H e, e ) U,0.63). 3 Again the AND operation is denoted as: U,, F U,, G U,, K and is given by, K e, e ) { },0.6), K e, e ) { },0.63), K e, e ) {, },0.8), K e, e ) { },0.63). 3 3 From now on we will denote a FPS set U,, over U, ) by F for simpliity. Proposition.5 Let F, G & H be any three FPS sets over U, ), ) i F F F ii) F F F iii) F A A iv) F A F ) v F vi) F F vii) F G G F viii) F G G F i) F G ) H F G H ) ) F G ) H F G H ) i) F G ) H F H ) G H ) ii) F G ) H F H ) G H ) Proof. The proofs are straightforward. F then the following holds: Proposition.6 Let F & G i) F G ) F G C C C ii) F G ) F G C C C be any two FPS sets over U, ), then the following holds: [69]

8 On Type II Fuzzy Parameterized Soft Sets Author: P. Majumdar Proof. The proofs are straightforward and follow from definition. Proposition.7 Let F, G & H be any three FPS sets over U, ), then the following holds: i) F G ) H F G H ) ii) F G ) H F G H ) iii) F G ) H F H ) G H ) iv) F G ) H F H ) G H ) Proof. The proofs of these properties are straightforward. Proposition.8 Let F i) F G ) F G ii) F G ) F G & G C C C C C C be any two FPS over U, ), then the following holds: Proposition.9 Let F be any FPS over U, ), then the following does not holds: i) F F ) A ii) F F ) The level sets of fuzzy parameterized soft sets are defined as follows. ' Definition.30 Let U,, F ) be a FPS set over U, ). Let L : I [0,) P U) be a mapping defined as follows: L ) { U; F e) & e), e} U. Then F ' I are alled level sets of U,, F ) L ), F. ' Note that L : I [0,) P U) is a soft set overu with parameter set.. Type II Fuzzy Parameterized Soft Sets In this setion we introdue the notion of Type II Fuzzy Parameterized Soft Sets TFPS in short). We also define several set theoreti operations on them and study the basi algebrai properties of these sets. Definition. Let U, ) be the soft universe and let I [0,]. A fuzzy parameterized soft PU ) set of Type II is the pair F, A ) where A and F : A I. JAST-05, Vol.-0, Issue-0 [70]

9 ample. Let JAST-a peer reviewed multidisiplinary researh journal Vol.-0, Issue-0 U {, y, z}, { e, e }. Let A and F e ) f, F e ) f, where f, f : P U ) I aredefined as follows : f { } { y, z} {, y} { y} {, z} f {, y} {, z} { z} {, y, z} { y, z} { } and Now F, A ) is a TFPS. It an be interpreted as follows: Corresponding to the parameter we shall have the set { y, }, has the truth value 0.85 and so on. Here other subsets ofu have grades 0 and hene not shown in the tables of f& f. Note.3 If, in partiular, for eah e A, Se U andt e I suh that F e) S ) t and F e) P) 0, P S e e e Then F, A ) will redue to a FPS set. Definition. Let F, and G, B ) be two TFPS. Now F, A ) is said to be a subset of GBif, ) the followings are satisfied: i) A B & ii) e A, F e) G e). e, Definition.5 Let F, and G, B ) be two TFPS. Then the union of F, and G, B ) is again a TFPS H, C), where C A B, and is defined as follows: H e) F e); for e A B, H e) G e) and for e B A, H e) F e) G e) e A B. Definition.6 Let F, and G, B ) be two TFPS. Then the intersetion of F, and G, B ) is again a TFPS K, C), where C A B, and is defined as follows: e A B, H e) F e) G e). ample.7 Consider two TFPS F, and G, B ) defined over the soft universe, ) follows: Let U {,,,, }& { e, e, e, e, e }; A { e, e, e, e }, B { e, e, e, e } Then A B & A B { e, e, e }. U as [7]

10 On Type II Fuzzy Parameterized Soft Sets Author: P. Majumdar ) { ',, } ) { ',, } 5 3 3) { ', } { } {,, } {, } {, } {, } {, } {, } { } { } {, } {,, } {,, } {, } { } ) {,, } and Ge {, } {,, } {, } {,, } { } {, } {, } { } { } {, } {,, } {, } { } {, } ) { ',, } ) { ',, } Ge ) { ', } Ge ) {,, } Ge Then their union H, C), where C A B, is given as follows: { } {, } {,, } {, } {,, } He { } {, 5} {, 5} { 3} {, } He ) { ',,, } { } {, 5} {, 3, } He 3) { ', } { } {, 5} { 3} {, He ) { ',, } {, 3, 5}, } { 3, 5} { 3} { 3, } He 5) {,, } ) {, ',, } The intersetion K, C), where C A B, is defined as follows: Ke {, } {,, } {, } {, } {, } { } {,, } ) { ', } ) {,, } Ke 3 5 ) { }. Ke JAST-05, Vol.-0, Issue-0 [7]

11 JAST-a peer reviewed multidisiplinary researh journal Vol.-0, Issue-0 Definition.8 Let F, A ) be a TFPS. Then the omplement of F, A ) is again a TFPS GA, ), and is defined as follows: ) G e F e), where F F. ample.9 Consider the TFPS F, A ) defined over the soft universe U, ) as follows: U {, }& A { e, e }. Let ) { ' } { } {, } {, } { } ) { ' } Then the omplement F, A ) of F, will be as follows: { } { } {, } {, } { } { } ) { ' ' } ) { ', } Definition.0 Let F, be a TFPS. Then it is said to be an absolute TFPS, denoted by, if followings hold: e A, e). Definition. Let F, be a TFPS. Then it is said to be a null TFPS denoted by, e A, e) 0. followings hold: Some Important Algebrai Properties of TFPS under the above operations: Theorem. Let F,, G, B) and H, C ) be TFPS over the soft universe U, ). Then the followings hold: if [73]

12 i) F, F, F, ii) F, F, F, iii) F, iv) F, F, v) F, F, vi) F, On Type II Fuzzy Parameterized Soft Sets Author: P. Majumdar vii) F, G, B)) H, C) F, G, B) H, C)) viii) F, G, B)) H, C) F, G, B) H, C)) i) F, G, B)) H, C) F, H, C)) G, B) H, C)) ) F, G, B)) H, C) F, H, C)) G, B) H, C)) Proof. The theorem trivially follows from definition. Theorem.3 Let F,, G, B ) be two TFPS over the soft universe U, ). Then the DeMorgan s laws hold: i) F, G, B)) F, G, B) ii) F, G, B)) F, G, B) Proof. Follows from definition. 3. Deision Making using TFPS { e, e, e, e } and X {,,,, } are respetively the set of The TFPS disussed above an be used to model several unertain situations suessfully. Here we have shown a job reruitment problem where TFPS sets an be used. Let us suppose that a famous ompany want to reruit a team leader for marketing their new produt. Their goal is to selet a person with ertain speifi qualities like leadership quality e ), hard working e ), managerial skill e 3), eperiene e ) et. These will at as the parameters of the problem. The universe is the set of appliants for the said post. Let us assume that the ompany has shortlisted 5 persons for interview after onsidering their biodata. So here parameters and the universe of elements. Now the andidates are judged. Several tehniques like skill test, written eam, viva-voe, presentations et are applied to test the suitability of the andidates for the post. The result of the interview an be epressed as a TFPS F, ) as follows: JAST-05, Vol.-0, Issue-0 [7]

13 { } {,, } {, } {, } {, } {, } {, } { } { } {, } {,, } {,, } {, } { } ) { ',, } ) { ',, } 5 3 3) { ', ) {,, } JAST-a peer reviewed multidisiplinary researh journal Vol.-0, Issue-0 We further onsider weights to eah parameter as follows: Parameters: e e e 3 e Weights: Net we list the number of ourrenes of eah grade against eah ). i in every ) i and then find their average Net we multiply this average grades with the orresponding weights of eah ei to get the resultant grades and find the sum of all suh resultant grades to find the total grade of the element, i.e the total grade of a andidate. The ompany will selet the andidate with maimum total grades. The alulations are shown below in two tables: i ) ) Table : Average weights of the andidates ) ) Weights of e i Table : Resultant grades of the andidates 3 5 we i ) we ) we ) we ) we ) Total Grades: [75]

14 On Type II Fuzzy Parameterized Soft Sets Author: P. Majumdar Here we see that has the highest total grade ompared to the other andidates. Therefore the ompany should offer the job to. In this tehnique eah desired quality of a andidate is tested with several methods, whih is possible using TFPS sets only. For eample the leadership quality e may be tested using ) group task, interview, written test, group disussion and results are shown in whih is not possible to epress using earlier tehniques.. ntropy Measure of a TFPS set ntropy as a measure of fuzziness was first mentioned by Zadeh [7] in 965. Later De Lua- Termini [7] aiomatized the non-probabilisti entropy. Aording to them the entropy of a fuzzy set A should satisfy the following aioms: DT) 0iff A DT ) iff ) 0.5, X DT3) B) iff Aisless fuzzy than B, i. e. if ) ) 0.5 X or if ) ) 0.5, X. A DT ) A ). B A X Several other authors have investigated the notion of entropy. Kaufmann [0] proposed a distane based measure of soft entropy, Yager [6] gave another view of degree of fuzziness of any fuzzy set in terms of lak of distintion between the fuzzy set and its omplement. Kosko [6] investigated the fuzzy entropy in relation to a measure of subsethood. Szmidt & Kaprzyk [5] studied the entropy of intuitionisti fuzzy sets et. The notion of entropy of soft sets and fuzzy parameterized soft sets were given by Majumdar & Samanta [ 3]. A B Following these onepts here we propose a set of aioms that should be satisfied by any measure of entropy for TFPS. For that purpose we give two new definitions, namely onstant TFPS set and half TFPS sets. P X) Definition. Let F : I be a TFPS set over the soft universe X, ). Then F, ) is said to be a onstant TFPS set if F e), [0,], e. If, then F, ) is alled a half TFPS set and is denoted by F. Let us denote as the olletion of all TFPS sets over the universe X, ). Definition. Let FG,, and F is more unertain than G, if JAST-05, Vol.-0, Issue-0 [76]

15 G e) F e) 0.5, or G e) F e) 0.5, A P U), e. JAST-a peer reviewed multidisiplinary researh journal Vol.-0, Issue-0 Definition.3 Any funtion : [0,] whih satisfies the following onditions is a measure of entropy of TFPS set: ) P) 0 if P or, ) F ), 3) F) F ), ) F) G), if F is moreunertainthang. Let A be a fuzzy set over a finite universe and A denotes the number of elements in A. Then the following result holds: Theorem. Let F, define F) N F e)), where N is an entropy of a fuzzy set. e Then is anentropy measureof T FPS set F. Proof. The result follows from the definition of fuzzy entropy given at the beginning of setion. 5. Conlusion: It is evident from reent publiations that soft sets have more apability in handling unertain information in omparison to fuzzy sets. It has also been shown that fuzzy sets are speial types of soft sets. The notion of fuzzy parameterized soft sets is again a further generalised struture ompared to ordinary soft sets and beause of its general nature and parameterized struture, this set has greater powers of epressing and handling unertainty present in physial systems. These sets may be used in many areas suh as deision making, medial diagnosis, weather predition et. whih require handling a lot of unertain data. Here we have introdued a riher Type II version of these FPS sets. In these sets orresponding to any parameter we get fuzzy subsets of the power set of the universe instead of a pair of single risp set and its grade of belongingness in the approimation set. We have shown its potentiality for appliation in Deision making problems suh as job reruitment. This new hybrid set will be useful in many areas of theoretial study of unertainty and ould be applied in proessing vague data. Aknowledgement: The present work is partially supported by University Grants Commission RO) under minor researh projet Siene) Projet No. F.PSW-9/-3). [77]

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