research article hesitant fuzzy soft set and its...

Research ArticleHesitant Fuzzy Soft Set and Its Applications inMulticriteria Decision Making

Fuqiang Wang Xihua Li and Xiaohong Chen

School of Business Central South University Changsha Hunan 410083 China

Correspondence should be addressed to Xihua Li 251560913qqcom

Received 16 April 2014 Accepted 16 June 2014 Published 3 July 2014

Academic Editor Zhenfu Cao

Copyright copy 2014 Fuqiang Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Molodtsovrsquos soft set theory is a newly emerging mathematical tool to handle uncertainty However the classical soft sets are notappropriate to deal with imprecise and fuzzy parameters This paper aims to extend the classical soft sets to hesitant fuzzy soft setswhich are combined by the soft sets and hesitant fuzzy setsThen the complement ldquoANDrdquo ldquoORrdquo union and intersection operationsare defined on hesitant fuzzy soft sets The basic properties such as DeMorganrsquos laws and the relevant laws of hesitant fuzzy softsets are proved Finally with the help of level soft set the hesitant fuzzy soft sets are applied to a decision making problem and theeffectiveness is proved by a numerical example

1 Introduction

In the real world there are many complicated problems ineconomics engineering environment social science andmanagement science They are characterized with uncer-tainty imprecision and vagueness We cannot successfullyutilize the classical methods to deal with these problems be-cause there are various types of uncertainties involved inthese problems Moreover though there are many theoriessuch as theory of probability theory of fuzzy sets theory ofinterval mathematics and theory of rough sets to be con-sidered as mathematical tools to deal with uncertaintiesMolodtsov [1] pointed out all these theories had their ownlimitations Moreover in order to overcome these difficultiesMolodtsov [1] firstly proposed a new mathematical toolnamed soft set theory to deal with uncertainty and impreci-sion This theory has been demonstrated to be a useful toolin many applications such as decision making measurementtheory and game theory

The soft set model can be combined with other mathe-matical models Maji et al [2] firstly presented the concept offuzzy soft set by combining the theories of fuzzy set and softset together Furthermore Maji et al [3ndash5] established thenotion of intuitionistic fuzzy soft set which was based on

a combination of the intuitionistic fuzzy set [6 7] and soft setmodels By combining the interval-valued fuzzy set [8 9] andsoft set Yang et al [10] presented the concept of the interval-valued fuzzy soft set Jiang et al [11] initiated the concept ofinterval-valued intuitionistic fuzzy soft sets as an interval-valued fuzzy extension of the intuitionistic fuzzy soft set the-ory or an intuitionistic fuzzy extension of the interval-valuedfuzzy soft set theory Recently Xiao et al [12] presented thetrapezoidal fuzzy soft set and Yang et al [13] introduced themultifuzzy soft set respectively and applied them in decisionmaking problems Roy and Maji [14] Kong et al [15] Fenget al [16] and Jiang et al [17] also applied the fuzzy soft set inthe decision making problems Jun [18] initiated the applica-tion of soft sets in BCKBCI-algebras and introduced the con-cept of soft BCKBCI-algebras Furthermore Jun and Park[19] and Jun et al [20 21] applied the soft sets in ideal theoryof BCKBCI-algebras and d-algebras Feng et al [22 23] ini-tiated the concept of rough soft sets soft rough sets and softrough fuzzy sets

Recently in order to tackle the difficulty in establishingthe degree of membership of an element in a set Torra andNarukawa [24] and Torra [25] proposed the concept of a hesi-tant fuzzy set This new extension of fuzzy set can handle thecases that the difficulty in establishing the membership

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 643785 10 pageshttpdxdoiorg1011552014643785

2 Journal of Applied Mathematics

degree does not arise from a margin of error (as in intuition-istic or interval-valued fuzzy sets) or a specified possibilitydistribution of the possible values (as in type-2 fuzzy sets)but arises from our hesitation among a few different values[26] Thus the hesitant fuzzy set can more accurately reflectthe peoplersquos hesitancy in stating their preferences over objectscompared to the fuzzy set and its many classical extensionsThe purpose of this paper is to extend the soft set model tothe hesitant fuzzy set and thus we establish a new soft setmodel named hesitant fuzzy soft set

The rest of this paper is organized as follows We first re-view some background on soft set fuzzy soft set and hesitantfuzzy set in Section 2 In Section 3 the concepts and oper-ations of hesitant fuzzy soft set are proposed and their prop-erties are discussed in detail In Section 4 we apply theproposed hesitant fuzzy soft set to a decisionmaking problemand give an explicit algorithm Finally we conclude inSection 5

2 Preliminaries

21 Soft Sets Suppose that 119880 is an initial universe set 119864 is aset of parameters 119875(119880) is the power set of 119880 and 119860 sub 119864

Definition 1 (see [1]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by 119865 119860 rarr 119875(119880)

In other words a soft set over 119880 is a parameterizedfamily of subsets of the universe 119880 For 119890 isin 119860 119865(119890) may beconsidered as the set of 119890-approximate elements of the soft set(119865 119860)

Example 2 Suppose that 119880 = ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 is a set

of houses and 119860 = 1198901 1198902 1198903 1198904 1198905 is a set of parameters

which stands for the parameters ldquocheaprdquo ldquobeautifulrdquo ldquosizerdquoldquolocationrdquo and ldquosurrounding environmentrdquo respectively Inthis case a soft set (119865 119860) can be defined as a mapping fromparameter set 119860 to the set of all subsets of 119880 In this waythe set of the houses with specific characteristics can bedescribed

Assume that 119865(1198901) = ℎ

2 ℎ4 119865(119890

2) = ℎ

1 ℎ3 119865(119890

3) =

ℎ3 ℎ4 ℎ5119865(1198904) = ℎ

1 ℎ3 ℎ5 and119865(119890

5) = ℎ

2 whichmean

ldquohouses (cheap)rdquo whose function-value is the set ℎ2 ℎ4

ldquohouses (beautiful)rdquo whose function-value is the set ℎ1 ℎ3

ldquohouses (big size)rdquo whose function-value is the set ℎ3 ℎ4 ℎ5

ldquohouses (in good location)rdquo whose function-value is the setℎ1 ℎ3 ℎ5 ldquohouses (great surrounding environment)rdquowhose

function-value is the set ℎ2 respectively Then we can view

the soft set (119865 119860) as consisting of the following collection ofapproximations

(119865 119860) = (1198901 ℎ2 ℎ4) (1198902 ℎ1 ℎ3) (1198903 ℎ3 ℎ4 ℎ5)

(1198904 ℎ1 ℎ3 ℎ5) (1198905 ℎ2)

(1)

For the purpose of storing a soft set in a computer we canrepresent the soft set defined in Example 2 in tabular form asTable 1 shows

Table 1 The tabular representation of the soft set (119865 119860)

119880 ldquoCheaprdquo ldquoBeautifulrdquo ldquoSizerdquo ldquoLocationrdquoldquoGreat

surroundingenvironmentrdquo

ℎ1

0 1 0 1 0ℎ2

1 0 0 0 1ℎ3

0 1 1 1 0ℎ4

1 0 1 0 0ℎ5

0 0 1 1 0ℎ6

0 0 0 0 0

Table 2 The tabular representation of the fuzzy soft set (119865 119860)



ℎ1

02 06 04 03 06ℎ2

05 05 06 02 03ℎ3

03 06 08 05 05ℎ4

03 07 03 07 04ℎ5

04 04 04 05 07ℎ6

06 03 07 08 03

22 Fuzzy Soft Sets

Definition 3 (see [2]) Let (119880) be the set of all fuzzy subsetsof 119880 a pair (119865 119860) is called a fuzzy soft set over 119880 where 119865 isa mapping given by 119865 119860 rarr (119880)

Example 4 Reconsider Example 2 In real life the perceptionof the people is characterized by a certain degree of vaguenessand imprecision thus when people consider if a house ℎ

1

is cheap the information cannot be expressed with only twocrisp numbers 0 and 1 Instead it should be characterizedby a membership function 120583

119860(119909) which associates with each

element a real number in the interval [0 1] Then fuzzy softset (119865 119860) can describe the characteristics of the house underthe fuzzy information

119865 (1198901) =

ℎ1

02ℎ2

05ℎ3

03ℎ4

03ℎ5

04ℎ6

06

119865 (1198902) =

ℎ1

06ℎ2

05ℎ3

06ℎ4

07ℎ5

04ℎ6

03

119865 (1198903) =

ℎ1

04ℎ2

06ℎ3

08ℎ4

03ℎ5

04ℎ6

07

119865 (1198904) =

ℎ1

03ℎ2

02ℎ3

05ℎ4

07ℎ5

05ℎ6

08

119865 (1198905) =

ℎ1

06ℎ2

03ℎ3

05ℎ4

04ℎ5

07ℎ6

03

(2)

Similarly for the purpose of storing a fuzzy soft set in acomputer we could also represent the fuzzy soft set definedin Example 4 in Table 2

Journal of Applied Mathematics 3

23 Hesitant Fuzzy Sets

Definition 5 (see [25]) A hesitant fuzzy set (HFS) on 119880 isin terms of a function that when applied to 119880 returns asubset of [0 1] which can be represented as the followingmathematical symbol

119860 = ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880 (3)

where ℎ119860(119906) is a set of values in [0 1] denoting the possible

membership degrees of the element 119906 isin 119880 to the set 119860 Forconvenience we call ℎ

119860(119906) a hesitant fuzzy element (HFE)

and119867 the set of all HFEs

Furthermore Torra [25] defined the empty hesitant setand the full hesitant set

Definition 6 (see [25]) Given hesitant fuzzy set 119860 if ℎ(119906) =

0 for all 119906 in 119880 then 119860 is called the null hesitant fuzzy setdenoted by 120601 If ℎ(119906) = 1 for all 119906 in 119880 then 119860 is called thefull hesitant fuzzy set denoted by 1

Definition 7 (see [27]) For a HFE ℎ 119904(ℎ) = (1119897(ℎ))sum120574isinℎ

120574

is called the score function of ℎ where 119897(ℎ) is the number ofthe values in ℎ For two HFEs ℎ

1and ℎ2 if 119904(ℎ

1) gt 119904(ℎ

2) then

ℎ1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2

Let ℎ1and ℎ

2be two HFEs It is noted that the number

of values in different HFEs ℎ1and ℎ2are commonly different

that is 119897(ℎ1) = 119897(ℎ

2) For convenience let 119897 = max119897(ℎ

1) 119897(ℎ2)

To operate correctly we should extend the shorter one untilboth of them have the same length when we compare themTo extend the shorter one the best way is to add the samevalue several times in it [28] In fact we can extend the shorterone by adding any value in it The selection of this valuemainly depends on the decision makersrsquo risk preferencesOptimists anticipate desirable outcomes and may add themaximum value while pessimists expect unfavorable out-comes and may add the minimum value For example letℎ1= 01 02 03 let ℎ

2= 04 05 and let 119897(ℎ

1) gt 119897(ℎ

2) To

operate correctly we should extend ℎ2to ℎ1015840

2= 04 04 05

until it has the same length of ℎ1 the optimist may extend

ℎ2as ℎ1015840

2= 04 05 05 and the pessimist may extend it as

ℎ1015840

2= 04 04 05 Although the results may be different if

we extend the shorter one by adding different values thisis reasonable because the decision makersrsquo risk preferencescan directly influence the final decision The same situationcan also be found in many existing references [29ndash31] In thispaper we assume that the decision makers are all pessimistic(other situations can be studied similarly)

We arrange the elements in ℎ119860(119906) in decreasing order and

let ℎ120590(119895)119860

(119906) be the 119895th largest value in ℎ119860(119906)

Definition 8 Given two hesitant fuzzy sets = ⟨119906 ℎ(119906)⟩ |

119906 isin 119880 and = ⟨119906 ℎ(119906)⟩ | 119906 isin 119880 is called the fuzzy

subset of if and only if ℎ120590(119895)119872

(119906) le ℎ120590(119895)

119873(119906) for forall119906 isin 119880 and

119895 = 1 2 119897 which can be denoted by sube

Definition 9 and are two hesitant fuzzy sets we call and is hesitant fuzzy equal if and only if

(1) sube

(2) supe

which can be denoted by = Given three HFEs ℎ ℎ

1 and ℎ

2 Torra [25] and Torra and

Narukawa [24] defined the following HFE operations

(1) ℎ119888 = cup120574isinℎ

1 minus 120574

(2) ℎ1cup ℎ2= ℎ isin (ℎ

1cup ℎ2) | ℎ ge max(ℎminus

1 ℎminus

2)

(3) ℎ1cap ℎ2= ℎ isin (ℎ

1cup ℎ2) | ℎ le min(ℎ+

1 ℎ+

2)

where ℎminus(119906) = min ℎ(119906) and ℎ

+(119906) = max ℎ(119906) are the lower

bound and upper bound of the given hesitant fuzzy elementsrespectively

Therefore giving two hesitant fuzzy soft sets 119860 and 119861 wecan define the following operations

(1) 119860119888 = ⟨119906 ℎ119888

119860(119906)⟩ | 119906 isin 119880

(2) 119860 cup 119861 = ⟨119906 ℎ119860(119906) cup ℎ

119861(119906)⟩ | 119906 isin 119880

(3) 119860 cap 119861 = ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880

Moreover for the aggregation of hesitant fuzzy informa-tion Xia and Xu [27] defined the following new operationson HFEs ℎ ℎ

1 and ℎ

2

(1) ℎ120582 = cup120574isinℎ

120574120582

(2) 120582ℎ = cup120574isinℎ

1 minus (1 minus 120574)120582

(3) ℎ1oplus ℎ2= cup1205741isinℎ11205742isinℎ2

1205741+ 1205742minus 12057411205742

(4) ℎ1otimes ℎ2= cup1205741isinℎ11205742isinℎ2

12057411205742

3 Hesitant Fuzzy Soft Sets

31 The Concept of Hesitant Fuzzy Soft Sets

Definition 10 Let (119880) be the set of all hesitant fuzzy sets in119880 a pair (119865 119860) is called a hesitant fuzzy soft set over119880 where119865 is a mapping given by

119865 119860 997888rarr (119880) (4)

A hesitant fuzzy soft set is a mapping from parameters to(119880) It is a parameterized family of hesitant fuzzy subsetsof 119880 For 119890 isin 119860 119865(119890) may be considered as the set of 119890-approximate elements of the hesitant fuzzy soft set (119865 119860)

Example 11 Continue to consider Example 2Mr X evaluatesthe optional six houses under various attributes with hesitantfuzzy element then hesitant fuzzy soft set (119865 119860) can describethe characteristics of the house under the hesitant fuzzyinformation


Table 3 The tabular representation of the hesitant fuzzy soft set (119865 119860)

119880 ldquoCheaprdquo ldquoBeautifulrdquo ldquoSizerdquo ldquoLocationrdquo ldquoGreat surrounding environmentrdquoℎ1

02 03 04 06 07 02 04 03 05 06 06ℎ2

05 06 05 07 08 06 07 02 02 03 05ℎ3

03 06 08 08 09 05 05 07ℎ4

03 05 07 09 03 05 06 07 02 04ℎ5

04 05 03 04 05 04 06 05 06 05 07ℎ6

06 07 03 07 08 03 05

Consider

119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119865 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

119865 (1198904) =

ℎ1

03 05 06

ℎ2

02

ℎ3

05

ℎ4

06 07

ℎ5

05 06

ℎ6

08

119865 (1198905) =

ℎ1

06

ℎ2

02 03 05

ℎ3

05 07

ℎ4

02 04

ℎ5

05 07

ℎ6

03 05

(5)

Similarly we can also represent the hesitant fuzzy soft setin the form of Table 3 for the purpose of storing the hesitantfuzzy soft set in a computer

Definition 12 Let119860 119861 isin 119864 (119865 119860) and (119866 119861) are two hesitantfuzzy soft sets over119880 (119865 119860) is said to be a hesitant fuzzy softsubset of (119866 119861) if

(1) 119860 sube 119861

(2) For all 119890 isin 119860 119865(119890) sube 119866(119890)

In this case we write (119865 119860) sube (119866B)

Example 13 Given two hesitant fuzzy soft sets (119865 119860) and(119866 119861) 119880 = ℎ

1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 is the set of the optional

houses 119860 = 1198901 1198902 = cheap beautiful 119861 = 119890

1 1198902 1198903 =

cheap beautiful size and

119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119866 (1198901) =

ℎ1

03

ℎ2

06 08

ℎ3

04 05

ℎ4

03 04 05

ℎ5

05

ℎ6

07

119866 (1198902) =

ℎ1

04 05 06 07

ℎ2

07 08

ℎ3

09

ℎ4

08 09

ℎ5

05

ℎ6

05

119866 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

(6)

Then we have (119865 119860) sube (119866 119861)

Definition 14 Two hesitant fuzzy soft sets (119865 119860) and (119866 119861)

are said to be hesitant fuzzy soft equal if (119865 119860) is a hesitantfuzzy soft subset of (119866 119861) and (119866 119861) is a hesitant fuzzy softsubset of (119865 119860)

In this case we write (119865 119860) cong (119866 119861)

Definition 15 A hesitant fuzzy soft set (119865 119860) is said to beempty hesitant fuzzy soft set denoted by Φ

119860 if 119865(119890) = 120601 for

all 119890 isin 119860

Definition 16 A hesitant fuzzy soft set (119865 119860) is said to be fullhesitant fuzzy soft set denoted by

119860 if119865(119890) = 1 for all 119890 isin 119860


Table 4 The result of ldquoANDrdquo operation on (119865 119860) and (119866 119861)

119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

02 03 02 03 02 03 03 04 05 06 07 02 04ℎ2

05 06 05 06 05 06 05 06 07 08 05 07 08 05 06 07ℎ3

03 03 03 04 05 06 08 06 08ℎ4

03 05 03 05 03 05 03 04 05 07 08 09 03 05ℎ5

03 04 05 04 05 04 05 03 04 05 03 04 05 03 04 05ℎ6

03 05 06 07 03 03 03

32 Operations on Hesitant Fuzzy Soft Sets

Definition 17 The complement of a hesitant fuzzy soft set(119865 119860) is denoted by (119865 119860)

119888 and is defined by

(119865 119860)119888

= (119865119888 119860) (7)

where 119865119888 119860 rarr (119880) is a mapping given by 119865119888(119890) = (119865(119890))

119888

for all 119890 isin 119860

Clearly (119865119888)119888 is the same as 119865 and ((119865 119860)119888

)119888

= (119865 119860)It is worth noting that in the above definition of comple-

ment the parameter set of the complement (119865 119860)119888 is still the

original parameter set 119860 instead of not119860

Example 18 Reconsider Example 11 the (119865 119860)119888 can be calcu-

lated as follows

119865119888(1198901) =

ℎ1

07 08

ℎ2

04 05

ℎ3

07

ℎ4

05 07

ℎ5

05 06

ℎ6

03 04

119865119888(1198902) =

ℎ1

03 04 06

ℎ2

02 03 05

ℎ3

02 04

ℎ4

01 03

ℎ5

05 06 07

ℎ6

07

119865119888(1198903) =

ℎ1

06 08

ℎ2

03 04

ℎ3

01 02

ℎ4

05 07

ℎ5

04 06

ℎ6

03

119865119888(1198904) =

ℎ1

04 05 07

ℎ2

08

ℎ3

05

ℎ4

03 04

ℎ5

04 05

ℎ6

02

119865119888(1198905) =

ℎ1

04

ℎ2

05 07 08

ℎ3

03 05

ℎ4

06 08

ℎ5

03 05

ℎ6

05 07

(8)

Definition 19 The AND operation on two hesitant fuzzy softsets (119865 119860) and (119866 119861) which is denoted by (119865 119860) and (119866 119861) is

defined by (119865 119860)and(119866 119861) = (119869 119860times119861) where 119869(120572 120573) = 119865(120572)cap

119866(120573) for all (120572 120573) isin 119860 times 119861

Definition 20 The OR operation on the two hesitant fuzzysoft sets (119865 119860) and (119866 119861) which is denoted by (119865 119860) or (119866 119861)

is defined by (119865 119860) or (119866 119861) = (119874 119860 times 119861) where 119874(120572 120573) =

119865(120572) cup 119866(120573) for all (120572 120573) isin 119860 times 119861

Example 21 The results of ldquoANDrdquo and ldquoORrdquo operations onthe hesitant fuzzy soft sets (119865 119860) and (119866 119861) in Example 13 areshown in Tables 4 and 5 respectively

Theorem 22 (De Morganrsquos laws of hesitant fuzzy soft sets)Let (119865 119860) and (119866 119861) be two hesitant fuzzy soft sets over U wehave

(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888

Proof (1) Suppose that (119865 119860)and (119866 119861) = (119869 119860times119861) Therefore((119865 119860) and (119866 119861))

119888

= (119869 119860 times 119861)119888

= (119869119888 119860 times 119861) Similarly

(119865 119860)119888

or (119866 119861)119888

= (119865119888 119860) or (119866

119888 119861) = (119874 119860 times 119861) Now take

(120572 120573) isin 119860 times 119861 therefore

119869119888(120572 120573)

= (119869 (120572 120573))119888

= (119865 (120572) cap 119866 (120573))119888

= ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880

119888

= ⟨119906 ℎ isin (ℎ119860cup ℎ119861) | ℎ le min (ℎ

+

119860 ℎ+

119861)⟩ | 119906 isin 119880

119888

= ⟨119906 1 minus ℎ | ℎ isin (ℎ119860cup ℎ119861)

and ℎ le min (ℎ+

119860 ℎ+

119861)⟩ | 119906 isin 119880

= ⟨119906 1 minus ℎ | (1 minus ℎ) isin (1 minus ℎ119860) cup (1 minus ℎ

119861)

and (1 minus ℎ) ge max ((1 minus ℎ119860)minus

(1 minus ℎ119861)minus

)⟩ | 119906 isin 119880

= ⟨119906 (1 minus ℎ) isin (1 minus ℎ119860) cup (1 minus ℎ

119861) | (1 minus ℎ)

ge max ((1 minus ℎ119860)minus


)⟩ | 119906 isin 119880


Table 5 The result of ldquoORrdquo operation on (119865 119860) and (119866 119861)

119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

03 04 05 06 07 02 03 04 04 06 07 04 05 06 07 04 06 07ℎ2

06 08 07 08 06 07 06 07 08 07 08 06 07 08ℎ3

04 05 09 08 09 06 08 09 08 09ℎ4

03 04 05 08 09 03 05 07 09 08 09 07 09ℎ5

05 05 04 05 06 05 05 04 05 06ℎ6

07 06 07 07 07 05 07

119874 (120572 120573)

= 119865119888(120572) cup 119866

119888(120573)

= ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880

119888

cup ⟨119906 ℎ119861(119906)⟩ | 119906 isin 119880

119888

= ⟨119906 1 minus ℎ119860(119906)⟩ | 119906 isin 119880 cup ⟨119906 1 minus ℎ

119861(119906)⟩ | 119906 isin 119880

= ⟨119906 (1 minus ℎ119860) cup (1 minus ℎ

119861)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880

(9)

Hence 119869119888(120572 120573) = 119874(120572 120573) is proved(2) Similar to the above process it is easy to prove that

((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888

Theorem 23 Let (119865 119860) (119866 119861) and (119869 119862) be three hesitantfuzzy soft sets over119880Then the associative law of hesitant fuzzysoft sets holds as follows

(119865 119860) and ((119866 119861) and (119869 119862)) = ((119865 119860) and (119866 119861)) and (119869 119862)

(119865 119860) or ((119866 119861) or (119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)

(10)

Proof For all 120572 isin 119860 120573 isin 119861 and 120575 isin 119862 because 119865(120572) cap

(119866(120573) cap 119869(120575)) = (119865(120572) cap 119866(120573)) cap 119869(120575) we can conclude that(119865 119860) and ((119866 119861) and (119869 119862)) = ((119865 119860) and (119866 119861)) and (119869 119862) holds

Similarly we can also conclude that (119865 119860) or ((119866 119861) or

(119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)

Remark 24 The distribution law of hesitant fuzzy soft setsdoes not hold That is

(119865 119860) and ((119866 119861) or (119869 119862))

= ((119865 119860) and (119869 119862)) or ((119866 119861) and (119869 119862))

(119865 119860) or ((119866 119861) and (119869 119862))

= ((119865 119860) or (119869 119862)) and ((119866 119861) or (119869 119862))

(11)

For all 120572 isin 119860 120573 isin 119861 and 120575 isin 119862 because 119865(120572) cap (119866(120573) cup

119869(120575)) = (119865(120572) cap 119866(120573)) cup (119865(120572) cap 119869(120575)) for the hesitant fuzzysets we can conclude that (119865 119860) and ((119866 119861) or (119869 119862)) = ((119865 119860)

Table 6 The Union of hesitant fuzzy soft sets (119865 119860) and (119866 119861)

119880 1198901

1198902

1198903

ℎ1

03 04 05 06 07 02 04ℎ2

06 08 07 08 06 07ℎ3

04 05 09 08 09ℎ4

03 04 05 08 09 03 05ℎ5

05 05 04 06ℎ6

07 05 07

and(119869 119862))or((119866 119861)and(119869 119862)) Similarly we can also conclude that(119865 119860) or ((119866 119861) and (119869 119862)) = ((119865 119860) or (119869 119862)) and ((119866 119861) or (119869 119862))

Definition 25 Union of two hesitant fuzzy soft sets (119865 119860) and(119866 119861) over 119880 is the hesitant fuzzy soft set (119869 119862) where 119862 =

119860 cup 119861 and for all 119890 isin 119862

119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(12)

We write (119865 119860) cup (119866 119861) = (119869 119862)

Example 26 Reconsider Example 13 the Union of hesitantfuzzy soft sets (119865 119860) and (119866 119861) is shown in Table 6

Definition 27 Intersection of two hesitant fuzzy soft sets(119865 119860) and (119866 119861) with 119860 cap 119861 = 120601 over 119880 is the hesitant fuzzysoft set (119869 119862) where 119862 = 119860 cap 119861 and for all 119890 isin 119862 119869(119890) =

119865(119890) cap 119866(119890)

We write (119865 119860) cap (119866 119861) = (119869 119862)

Example 28 Reconsider Example 13 the Intersection of hes-itant fuzzy soft sets (119865 119860) and (119866 119861) is shown in Table 7

Theorem 29 Let (119865 119860) and (119866 119861) be two hesitant fuzzy softsets over 119880 Then

(1) (119865 119860) cup (119865 119860) = (119865 119860)(2) (119865 119860) cap (119865 119860) = (119865 119860)(3) (119865 119860) cup Φ

119860= (119865 119860)

(4) (119865 119860) cap Φ119860= Φ119860

(5) (119865 119860) cup 119860= 119860


Table 7The Intersection of hesitant fuzzy soft sets (119865 119860) and (119866 119861)

119880 1198901

1198902

ℎ1

02 03 04 05 06 07ℎ2

05 06 05 07 08ℎ3

03 06 08ℎ4

03 05 07 08 09ℎ5

03 04 05 03 04 05ℎ6

03 03

(6) (119865 119860) cap 119860= (119865 119860)

(7) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)(8) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)


(1) ((119865 119860) cup (119866 119861))119888

sub (119865 119860)119888

cup (119866 119861)119888

(2) (119865 119860)119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888

Proof (1) Suppose that (119865 119860) cup (119866 119861) = (119869 119862) where 119862 =


119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(13)

Thus ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(14)

Similarly suppose that (119865 119860)119888

cup (119866 119861)119888

= (119865119888 119860) cup (119866

119888 119861) =

(119874 119862) where 119862 = 119860 cup 119861 and for all 119890 isin 119862

119874 (119890) = 119865119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cup 119866

119888(119890) if 119890 isin 119860 cap 119861

(15)

Obviously for all 119890 isin 119862 119869119888(119890) sube 119874(119890) Part (1) of Theorem 30is proved

(2) Similar to the above process it is easy to prove that(119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888


(1) (119865 119860)119888

cap (119866 119861)119888

sub((119865 119860) cup (119866 119861))119888

(2) ((119865 119860) cap (119866 119861))119888

sub(119865 119860)119888

cup (119866 119861)119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(16)

Then ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(17)

Similarly suppose (119865 119860)119888

cap (119866 119861)119888

= (119865119888 119860) cap (119866

119888 119861) = (119874

119869) where 119869 = 119860 cap 119861Then for all 119890 isin 119869119874(119890) = 119865119888(119890) cap 119866

119888(119890)

Hence it is obvious that 119869 sube 119862 and for all 119890 isin 119869 119874(119890) =

119869119888(119890) thus (119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cup (119866 119861))119888 is proved

(2) Similar to the above process it is easy to prove that((119865 119860) cap (119866 119861))

119888

sub (119865 119860)119888

cup (119866 119861)119888

Theorem 32 Let (119865 119860) and (119866 119860) be two hesitant fuzzy softsets over 119880 then

(1) ((119865 119860) cup (119866 119860))119888

= (119865 119860)119888

cap (119866 119860)119888

(2) ((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888

Proof (1) Suppose that (119865 119860) cup (119866 119860) = (119869 119860) and for all119890 isin 119860

119869 (119890) = 119865 (119890) cup 119866 (119890) (18)

Then ((119865 119860) cup (119866 119860))119888

= (119869 119860)119888

= (119869119888 119860) and for all 119890 isin 119860

119869119888(119890) = 119865

119888(119890) cap 119866

119888(119890) (19)


cap (119866 119860)119888

= (119865119888 119860) cap (119866

119888 119860) =

(119874 119860) and for all 119890 isin 119860

119874 (119890) = 119865119888(119890) cap 119866

119888(119890) (20)

Hence119874(119890) = 119869119888(119890) the ((119865 119860) cup (119866 119860))

119888

= (119865 119860)119888

cap (119866 119860)119888

is proved(2) Similar to the above process it is easy to prove that

((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888

4 Application of Hesitant Fuzzy Soft Sets

Roy and Maji [14] presented an algorithm to solve the deci-sion making problems according to a comparison table fromthe fuzzy soft set Later Kong et al [15] pointed out that thealgorithm in Roy and Maji [14] was incorrect by giving acounterexample Kong et al [15] also presented a modifiedalgorithm which was based on the comparison of choice val-ues of different objects [13] However Feng et al [16] furtherexplored the fuzzy soft set based decision making problems


Table 8 The tabular representation of the hesitant fuzzy soft set (119865 119860) in Example 35

119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08

more deeply and pointed out that the concept of choice val-ues was not suitable to solve the decision making problemsinvolving fuzzy soft sets though it was an efficient methodto solve the crisp soft set based decision making problemsThey proposed a novel approach by using the level soft setsto solve the fuzzy soft set based decision making problemsThus some decision making problems that cannot be solvedby the methods in Roy and Maji [14] and Kong et al [15] canbe successfully solved by this approach In the following wewill apply this level soft sets method to hesitant fuzzy soft setbased decision making problems

Let 119880 = 1199061 1199062 119906

119899 and (119865 119860) be a hesitant fuzzy

soft set over 119880 For every 119890 isin 119860 119865(119890) = 1199061ℎ119860(1199061)

1199062ℎ119860(1199062) 119906

119899ℎ119860(119906119899) we can calculate the score of each

hesitant fuzzy element by Definition 7Thus we define an in-duced fuzzy set with respect to 119890 in 119880 as 120583

119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this

method we can change a hesitant fuzzy set 119865(119890) intoan induced fuzzy set 120583

119865(119890) Furthermore once the in-

duced fuzzy soft set of a hesitant fuzzy soft set hasbeen obtained we can determine the optimal alternativeaccording to Fengrsquos [16] algorithm The explicit algorithm ofthe decision making based on hesitant fuzzy soft set is givenas follows

Algorithm 33 Consider the following

(1) Input the hesitant fuzzy soft set (119865 119860)(2) Compute the induced fuzzy soft set Δ

119865= (Γ 119860)

(3) Input a threshold fuzzy set 120582 119860 rarr [0 1] (or givea threshold value 119905 isin [0 1] or choose the midleveldecision rule or choose the top-level decision rule)for decision making

(4) Compute the level soft set 119871(Δ119865 120582) ofΔ

119865with respect

to the threshold fuzzy set 120582 (or the 119905-level soft set119871(Δ119865 119905) or the mid-level soft set 119871(Δ

119865mid) or the

top-level soft set 119871(Δ119865max))

(5) Present the level soft set 119871(Δ119865 120582) (or 119871(Δ

119865 119905) or

119871(Δ119865mid) or 119871(Δ

119865max)) in tabular form and

compute the choice value 119888119894of 119906119894 for all 119894

(6) The optimal decision is to select 119906119895= argmax

119894119888119894

(7) If there are more than one 119906119895s then any one of 119906

119895may

be chosen

Remark 34 In order to get a unique optimal choice accordingto the above algorithm the decision makers can go back to

Table 9 The tabular representation of the induced fuzzy soft setΔ119865= (Γ 119860) in Example 35

119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075

the third step and change the threshold (or decision criteria)in case that there is more than one optimal choice that canbe obtained in the last step Moreover the final optimaldecision can be adjusted according to the decision makersrsquopreferences

We adopt the following example to illustrate the idea ofalgorithm given above

Example 35 Consider a retailer planning to open a new storein the city There are five sites 119909

119894(119894 = 1 2 5) to be

selected Five attributes are considered market (1198901) traffic

(1198902) rent price (119890

3) competition (119890

4) and brand improve-

ment (1198905) Suppose that the retailer evaluates the optional

five sites under various attributes with hesitant fuzzy elementthen hesitant fuzzy soft set (119865 119860) can describe the character-istics of the sites under the hesitant fuzzy information whichis shown in Table 8

In order to select the optimal sites to open the storebased on the above hesitant fuzzy soft set according to Algo-rithm 33 we can calculate the score of each hesitant fuzzyelement and obtain the induced fuzzy soft set Δ

119865= (Γ 119860)

which is shown as in Table 9As an adjustable approach one can use different rules (or

the thresholds) to get different decision results accordingto his or her preference For example we use the midleveldecision rule in our paper and the midlevel thresholdfuzzy set of Δ

119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890

5 055) Furthering we can get the

midlevel soft set 119871(Δ119865mid) of Δ

119865 whose tabular form is as

Table 10 shows The choice values of each site are also shownin the last column of Table 10

According to Table 10 themaximum choice value is 3 andso the optimal decision is to select 119909

3 Therefore the retailer

should select site 3 as the best site to open a store


Table 10 The tabular representation of the midlevel soft set119871(Δ119865mid) with choice values in Example 35

119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion

In this paper we consider the notion of hesitant fuzzy soft setswhich combine the hesitant fuzzy sets and soft sets Then wedefine the complement ldquoANDrdquo ldquoORrdquo union and intersectionoperations on hesitant fuzzy soft sets The basic propertiessuch as De Morganrsquos laws and the relevant laws of hesitantfuzzy soft sets are proved Finally we apply it to a decisionmaking problem with the help of level soft set

Our research can be explored deeply in two directions inthe future Firstly we can combine other membership func-tions and soft sets tomake novel soft sets which have differentforms Secondly we can not only explore the application ofhesitant fuzzy soft set in decision making more deeply butalso apply the hesitant fuzzy soft set in many other areas suchas forecasting and data analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by the Innovation Group ScienceFoundation Project of the National Natural Science Founda-tion of China (no 71221006) theMajor International Collab-oration Project of theNational Natural Science Foundation ofChina (no 71210003) and theHumanities and Social SciencesMajor Project of the Ministry of Education of China (no13JZD016)

References

[1] D Molodtsov ldquoSoft set theorymdashfirst resultsrdquo Computers ampMathematics with Applications vol 37 no 4-5 pp 19ndash31 1999

[2] P K Maji R Biswas and A R Roy ldquoFuzzy soft setsrdquo Journal ofFuzzy Mathematics vol 9 no 3 pp 589ndash602 2001

[3] P K Maji R Biswas and A R Roy ldquoIntuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 9 no 3 pp 677ndash6922001

[4] P KMaji A R Roy and R Biswas ldquoOn intuitionistic fuzzy softsetsrdquo Journal of Fuzzy Mathematics vol 12 no 3 pp 669ndash6832004

[5] P KMaji ldquoMore on intuitionistic fuzzy soft setsrdquo in Proceedingsof the 12th International Conference on Rough Sets Fuzzy SetsDataMining andGranular Computing (RSFDGrC rsquo09) H SakaiM K Chakraborty A E Hassanien D Slezak and W Zhu

Eds vol 5908 of Lecture Notes in Computer Science pp 231ndash240 2009

[6] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[7] K T Atanassov Intuitionistic Fuzzy Sets Physica-Verlag NewYork NY USA 1999

[8] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[9] M B Gorzałczany ldquoA method of inference in approximatereasoning based on interval-valued fuzzy setsrdquo Fuzzy Sets andSystems vol 21 no 1 pp 1ndash17 1987

[10] X Yang T Y Lin J Yang Y Li and D Yu ldquoCombination ofinterval-valued fuzzy set and soft setrdquo Computers ampMathemat-ics with Applications vol 58 no 3 pp 521ndash527 2009

[11] Y Jiang Y Tang Q Chen H Liu and J Tang ldquoInterval-valuedintuitionistic fuzzy soft sets and their propertiesrdquo Computers ampMathematics with Applications vol 60 no 3 pp 906ndash918 2010

[12] Z Xiao S Xia K Gong and D Li ldquoThe trapezoidal fuzzysoft set and its application in MCDMrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 36 no 12 pp 5844ndash5855 2012

[13] Y Yang X Tan and C C Meng ldquoThe multi-fuzzy soft setand its application in decision makingrdquo Applied MathematicalModelling Simulation and Computation for Engineering andEnvironmental Systems vol 37 no 7 pp 4915ndash4923 2013

[14] A R Roy and P K Maji ldquoA fuzzy soft set theoretic approachto decision making problemsrdquo Journal of Computational andApplied Mathematics vol 203 no 2 pp 412ndash418 2007

[15] Z Kong L Gao and L Wang ldquoComment on ldquoA fuzzy softset theoretic approach to decision making problemsrdquordquo Journalof Computational and Applied Mathematics vol 223 no 2 pp540ndash542 2009

[16] F Feng Y B Jun X Liu and L Li ldquoAn adjustable approach tofuzzy soft set based decision makingrdquo Journal of Computationaland Applied Mathematics vol 234 no 1 pp 10ndash20 2010

[17] Y Jiang Y Tang and Q Chen ldquoAn adjustable approach tointuitionistic fuzzy soft sets based decision makingrdquo AppliedMathematical Modelling Simulation and Computation for Engi-neering and Environmental Systems vol 35 no 2 pp 824ndash8362011

[18] Y B Jun ldquoSoft BCKBCI-algebrasrdquo Computers amp Mathematicswith Applications vol 56 no 5 pp 1408ndash1413 2008

[19] Y B Jun andCH Park ldquoApplications of soft sets in ideal theoryof BCKBCI-algebrasrdquo Information Sciences vol 178 no 11 pp2466ndash2475 2008

[20] Y B Jun K J Lee and C H Park ldquoSoft set theory applied toideals in 119889-algebrasrdquo Computers amp Mathematics with Applica-tions vol 57 no 3 pp 367ndash378 2009

[21] Y B Jun K J Lee and J Zhan ldquoSoft 119901-ideals of soft BCI-algebrasrdquo Computers amp Mathematics with Applications vol 58no 10 pp 2060ndash2068 2009

[22] F Feng X Liu V L Leoreanu-Fotea and YB Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences An International Journalvol 181 no 6 pp 1125ndash1137 2011

[23] F Feng C Li B Davvaz andM I Ali ldquoSoft sets combined withfuzzy sets and rough sets a tentative approachrdquo Soft Computingvol 14 no 9 pp 899ndash911 2010

[24] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju-do Republic of Korea August 2009


[25] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[26] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013

[27] M Xia and Z Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[28] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011

[29] H Liu and G Wang ldquoMulti-criteria decision-making methodsbased on intuitionistic fuzzy setsrdquo European Journal of Opera-tional Research vol 179 no 1 pp 220ndash233 2007

[30] J M Mendel ldquoComputing with words when words can meandifferent things to different peoplerdquo in Proceedings of the 3rdInternational ICSC Symposium on Fuzzy Logic and Applicationspp 158ndash164 Rochester NY USA 1999

[31] J M Merigo and A M Gil-Lafuente ldquoThe induced generalizedOWA operatorrdquo Information Sciences vol 179 no 6 pp 729ndash741 2009

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Stochastic AnalysisInternational Journal of


degree does not arise from a margin of error (as in intuition-istic or interval-valued fuzzy sets) or a specified possibilitydistribution of the possible values (as in type-2 fuzzy sets)but arises from our hesitation among a few different values[26] Thus the hesitant fuzzy set can more accurately reflectthe peoplersquos hesitancy in stating their preferences over objectscompared to the fuzzy set and its many classical extensionsThe purpose of this paper is to extend the soft set model tothe hesitant fuzzy set and thus we establish a new soft setmodel named hesitant fuzzy soft set

The rest of this paper is organized as follows We first re-view some background on soft set fuzzy soft set and hesitantfuzzy set in Section 2 In Section 3 the concepts and oper-ations of hesitant fuzzy soft set are proposed and their prop-erties are discussed in detail In Section 4 we apply theproposed hesitant fuzzy soft set to a decisionmaking problemand give an explicit algorithm Finally we conclude inSection 5

2 Preliminaries

21 Soft Sets Suppose that 119880 is an initial universe set 119864 is aset of parameters 119875(119880) is the power set of 119880 and 119860 sub 119864

Definition 1 (see [1]) A pair (119865 119860) is called a soft set over 119880where 119865 is a mapping given by 119865 119860 rarr 119875(119880)

In other words a soft set over 119880 is a parameterizedfamily of subsets of the universe 119880 For 119890 isin 119860 119865(119890) may beconsidered as the set of 119890-approximate elements of the soft set(119865 119860)

Example 2 Suppose that 119880 = ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 is a set

of houses and 119860 = 1198901 1198902 1198903 1198904 1198905 is a set of parameters

which stands for the parameters ldquocheaprdquo ldquobeautifulrdquo ldquosizerdquoldquolocationrdquo and ldquosurrounding environmentrdquo respectively Inthis case a soft set (119865 119860) can be defined as a mapping fromparameter set 119860 to the set of all subsets of 119880 In this waythe set of the houses with specific characteristics can bedescribed

Assume that 119865(1198901) = ℎ

2 ℎ4 119865(119890

2) = ℎ

1 ℎ3 119865(119890

3) =

ℎ3 ℎ4 ℎ5119865(1198904) = ℎ

1 ℎ3 ℎ5 and119865(119890

5) = ℎ

2 whichmean

ldquohouses (cheap)rdquo whose function-value is the set ℎ2 ℎ4

ldquohouses (beautiful)rdquo whose function-value is the set ℎ1 ℎ3

ldquohouses (big size)rdquo whose function-value is the set ℎ3 ℎ4 ℎ5

ldquohouses (in good location)rdquo whose function-value is the setℎ1 ℎ3 ℎ5 ldquohouses (great surrounding environment)rdquowhose

function-value is the set ℎ2 respectively Then we can view

the soft set (119865 119860) as consisting of the following collection ofapproximations

(119865 119860) = (1198901 ℎ2 ℎ4) (1198902 ℎ1 ℎ3) (1198903 ℎ3 ℎ4 ℎ5)

(1198904 ℎ1 ℎ3 ℎ5) (1198905 ℎ2)

(1)

For the purpose of storing a soft set in a computer we canrepresent the soft set defined in Example 2 in tabular form asTable 1 shows

Table 1 The tabular representation of the soft set (119865 119860)



ℎ1

0 1 0 1 0ℎ2

1 0 0 0 1ℎ3

0 1 1 1 0ℎ4

1 0 1 0 0ℎ5

0 0 1 1 0ℎ6

0 0 0 0 0

Table 2 The tabular representation of the fuzzy soft set (119865 119860)



ℎ1

02 06 04 03 06ℎ2

05 05 06 02 03ℎ3

03 06 08 05 05ℎ4

03 07 03 07 04ℎ5

04 04 04 05 07ℎ6

06 03 07 08 03

22 Fuzzy Soft Sets

Definition 3 (see [2]) Let (119880) be the set of all fuzzy subsetsof 119880 a pair (119865 119860) is called a fuzzy soft set over 119880 where 119865 isa mapping given by 119865 119860 rarr (119880)

Example 4 Reconsider Example 2 In real life the perceptionof the people is characterized by a certain degree of vaguenessand imprecision thus when people consider if a house ℎ

1

is cheap the information cannot be expressed with only twocrisp numbers 0 and 1 Instead it should be characterizedby a membership function 120583

119860(119909) which associates with each

element a real number in the interval [0 1] Then fuzzy softset (119865 119860) can describe the characteristics of the house underthe fuzzy information

119865 (1198901) =

ℎ1

02ℎ2

05ℎ3

03ℎ4

03ℎ5

04ℎ6

06

119865 (1198902) =

ℎ1

06ℎ2

05ℎ3

06ℎ4

07ℎ5

04ℎ6

03

119865 (1198903) =

ℎ1

04ℎ2

06ℎ3

08ℎ4

03ℎ5

04ℎ6

07

119865 (1198904) =

ℎ1

03ℎ2

02ℎ3

05ℎ4

07ℎ5

05ℎ6

08

119865 (1198905) =

ℎ1

06ℎ2

03ℎ3

05ℎ4

04ℎ5

07ℎ6

03

(2)

Similarly for the purpose of storing a fuzzy soft set in acomputer we could also represent the fuzzy soft set definedin Example 4 in Table 2




119860 = ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880 (3)









120574


1and ℎ2 if 119904(ℎ

1) gt 119904(ℎ

2) then

ℎ1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2

Let ℎ1and ℎ



that is 119897(ℎ1) = 119897(ℎ


1) 119897(ℎ2)


2= 04 05 and let 119897(ℎ

1) gt 119897(ℎ

2) To


2= 04 04 05


ℎ2as ℎ1015840


ℎ1015840




let ℎ120590(119895)119860





(119906) le ℎ120590(119895)




(1) sube

(2) supe


1 and ℎ



(1) ℎ119888 = cup120574isinℎ

1 minus 120574



1 ℎminus

2)



1 ℎ+

2)





(1) 119860119888 = ⟨119906 ℎ119888

119860(119906)⟩ | 119906 isin 119880

(2) 119860 cup 119861 = ⟨119906 ℎ119860(119906) cup ℎ

119861(119906)⟩ | 119906 isin 119880

(3) 119860 cap 119861 = ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880


1 and ℎ

2

(1) ℎ120582 = cup120574isinℎ

120574120582

(2) 120582ℎ = cup120574isinℎ

1 minus (1 minus 120574)120582


1205741+ 1205742minus 12057411205742


12057411205742




119865 119860 997888rarr (119880) (4)






02 03 04 06 07 02 04 03 05 06 06ℎ2

05 06 05 07 08 06 07 02 02 03 05ℎ3

03 06 08 08 09 05 05 07ℎ4

03 05 07 09 03 05 06 07 02 04ℎ5

04 05 03 04 05 04 06 05 06 05 07ℎ6

06 07 03 07 08 03 05

Consider

119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119865 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

119865 (1198904) =

ℎ1

03 05 06

ℎ2

02

ℎ3

05

ℎ4

06 07

ℎ5

05 06

ℎ6

08

119865 (1198905) =

ℎ1

06

ℎ2

02 03 05

ℎ3

05 07

ℎ4

02 04

ℎ5

05 07

ℎ6

03 05

(5)



(1) 119860 sube 119861

(2) For all 119890 isin 119860 119865(119890) sube 119866(119890)





1 1198902 1198903 =


119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119866 (1198901) =

ℎ1

03

ℎ2

06 08

ℎ3

04 05

ℎ4

03 04 05

ℎ5

05

ℎ6

07

119866 (1198902) =

ℎ1

04 05 06 07

ℎ2

07 08

ℎ3

09

ℎ4

08 09

ℎ5

05

ℎ6

05

119866 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

(6)

Then we have (119865 119860) sube (119866 119861)





119860 if 119865(119890) = 120601 for

all 119890 isin 119860


119860 if119865(119890) = 1 for all 119890 isin 119860



119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

02 03 02 03 02 03 03 04 05 06 07 02 04ℎ2

05 06 05 06 05 06 05 06 07 08 05 07 08 05 06 07ℎ3

03 03 03 04 05 06 08 06 08ℎ4

03 05 03 05 03 05 03 04 05 07 08 09 03 05ℎ5

03 04 05 04 05 04 05 03 04 05 03 04 05 03 04 05ℎ6

03 05 06 07 03 03 03




(119865 119860)119888

= (119865119888 119860) (7)


119888

for all 119890 isin 119860


)119888





lated as follows

119865119888(1198901) =

ℎ1

07 08

ℎ2

04 05

ℎ3

07

ℎ4

05 07

ℎ5

05 06

ℎ6

03 04

119865119888(1198902) =

ℎ1

03 04 06

ℎ2

02 03 05

ℎ3

02 04

ℎ4

01 03

ℎ5

05 06 07

ℎ6

07

119865119888(1198903) =

ℎ1

06 08

ℎ2

03 04

ℎ3

01 02

ℎ4

05 07

ℎ5

04 06

ℎ6

03

119865119888(1198904) =

ℎ1

04 05 07

ℎ2

08

ℎ3

05

ℎ4

03 04

ℎ5

04 05

ℎ6

02

119865119888(1198905) =

ℎ1

04

ℎ2

05 07 08

ℎ3

03 05

ℎ4

06 08

ℎ5

03 05

ℎ6

05 07

(8)









(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119888

= (119869 119860 times 119861)119888

= (119869119888 119860 times 119861) Similarly

(119865 119860)119888

or (119866 119861)119888

= (119865119888 119860) or (119866

119888 119861) = (119874 119860 times 119861) Now take


119869119888(120572 120573)

= (119869 (120572 120573))119888

= (119865 (120572) cap 119866 (120573))119888

= ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880

119888


+

119860 ℎ+

119861)⟩ | 119906 isin 119880

119888



119860 ℎ+

119861)⟩ | 119906 isin 119880


119861)



)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880



119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

03 04 05 06 07 02 03 04 04 06 07 04 05 06 07 04 06 07ℎ2

06 08 07 08 06 07 06 07 08 07 08 06 07 08ℎ3

04 05 09 08 09 06 08 09 08 09ℎ4

03 04 05 08 09 03 05 07 09 08 09 07 09ℎ5

05 05 04 05 06 05 05 04 05 06ℎ6

07 06 07 07 07 05 07

119874 (120572 120573)

= 119865119888(120572) cup 119866

119888(120573)

= ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880

119888

cup ⟨119906 ℎ119861(119906)⟩ | 119906 isin 119880

119888


119861(119906)⟩ | 119906 isin 119880


119861)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880

(9)


((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


(119865 119860) and ((119866 119861) and (119869 119862)) = ((119865 119860) and (119866 119861)) and (119869 119862)

(119865 119860) or ((119866 119861) or (119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)

(10)




(119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)


(119865 119860) and ((119866 119861) or (119869 119862))

= ((119865 119860) and (119869 119862)) or ((119866 119861) and (119869 119862))

(119865 119860) or ((119866 119861) and (119869 119862))

= ((119865 119860) or (119869 119862)) and ((119866 119861) or (119869 119862))

(11)




119880 1198901

1198902

1198903

ℎ1

03 04 05 06 07 02 04ℎ2

06 08 07 08 06 07ℎ3

04 05 09 08 09ℎ4

03 04 05 08 09 03 05ℎ5

05 05 04 06ℎ6

07 05 07




119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(12)

We write (119865 119860) cup (119866 119861) = (119869 119862)



119865(119890) cap 119866(119890)

We write (119865 119860) cap (119866 119861) = (119869 119862)



(1) (119865 119860) cup (119865 119860) = (119865 119860)(2) (119865 119860) cap (119865 119860) = (119865 119860)(3) (119865 119860) cup Φ

119860= (119865 119860)

(4) (119865 119860) cap Φ119860= Φ119860

(5) (119865 119860) cup 119860= 119860



119880 1198901

1198902

ℎ1

02 03 04 05 06 07ℎ2

05 06 05 07 08ℎ3

03 06 08ℎ4

03 05 07 08 09ℎ5

03 04 05 03 04 05ℎ6

03 03

(6) (119865 119860) cap 119860= (119865 119860)

(7) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)(8) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)


(1) ((119865 119860) cup (119866 119861))119888

sub (119865 119860)119888

cup (119866 119861)119888

(2) (119865 119860)119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(13)

Thus ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(14)


cup (119866 119861)119888

= (119865119888 119860) cup (119866

119888 119861) =


119874 (119890) = 119865119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cup 119866

119888(119890) if 119890 isin 119860 cap 119861

(15)



119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888


(1) (119865 119860)119888

cap (119866 119861)119888

sub((119865 119860) cup (119866 119861))119888

(2) ((119865 119860) cap (119866 119861))119888

sub(119865 119860)119888

cup (119866 119861)119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(16)

Then ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(17)


cap (119866 119861)119888

= (119865119888 119860) cap (119866

119888 119861) = (119874


119888(119890)


119869119888(119890) thus (119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cup (119866 119861))119888 is proved


119888

sub (119865 119860)119888

cup (119866 119861)119888


(1) ((119865 119860) cup (119866 119860))119888

= (119865 119860)119888

cap (119866 119860)119888

(2) ((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888


119869 (119890) = 119865 (119890) cup 119866 (119890) (18)

Then ((119865 119860) cup (119866 119860))119888

= (119869 119860)119888

= (119869119888 119860) and for all 119890 isin 119860

119869119888(119890) = 119865

119888(119890) cap 119866

119888(119890) (19)


cap (119866 119860)119888

= (119865119888 119860) cap (119866

119888 119860) =

(119874 119860) and for all 119890 isin 119860

119874 (119890) = 119865119888(119890) cap 119866

119888(119890) (20)

Hence119874(119890) = 119869119888(119890) the ((119865 119860) cup (119866 119860))

119888

= (119865 119860)119888

cap (119866 119860)119888


((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888





119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08


Let 119880 = 1199061 1199062 119906



1199062ℎ119860(1199062) 119906



119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this






119865= (Γ 119860)



119865with respect


119865mid) or the



119865 119905) or

119871(Δ119865mid) or 119871(Δ




119894119888119894


119895may

be chosen



119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075




119894(119894 = 1 2 5) to be


(1198902) rent price (119890






119865= (Γ 119860)



119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890










119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119860 = ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880 (3)









120574


1and ℎ2 if 119904(ℎ

1) gt 119904(ℎ

2) then

ℎ1gt ℎ2 if 119904(ℎ

1) = 119904(ℎ

2) then ℎ

1= ℎ2

Let ℎ1and ℎ



that is 119897(ℎ1) = 119897(ℎ


1) 119897(ℎ2)


2= 04 05 and let 119897(ℎ

1) gt 119897(ℎ

2) To


2= 04 04 05


ℎ2as ℎ1015840


ℎ1015840




let ℎ120590(119895)119860





(119906) le ℎ120590(119895)




(1) sube

(2) supe


1 and ℎ



(1) ℎ119888 = cup120574isinℎ

1 minus 120574



1 ℎminus

2)



1 ℎ+

2)





(1) 119860119888 = ⟨119906 ℎ119888

119860(119906)⟩ | 119906 isin 119880

(2) 119860 cup 119861 = ⟨119906 ℎ119860(119906) cup ℎ

119861(119906)⟩ | 119906 isin 119880

(3) 119860 cap 119861 = ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880


1 and ℎ

2

(1) ℎ120582 = cup120574isinℎ

120574120582

(2) 120582ℎ = cup120574isinℎ

1 minus (1 minus 120574)120582


1205741+ 1205742minus 12057411205742


12057411205742




119865 119860 997888rarr (119880) (4)






02 03 04 06 07 02 04 03 05 06 06ℎ2

05 06 05 07 08 06 07 02 02 03 05ℎ3

03 06 08 08 09 05 05 07ℎ4

03 05 07 09 03 05 06 07 02 04ℎ5

04 05 03 04 05 04 06 05 06 05 07ℎ6

06 07 03 07 08 03 05

Consider

119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119865 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

119865 (1198904) =

ℎ1

03 05 06

ℎ2

02

ℎ3

05

ℎ4

06 07

ℎ5

05 06

ℎ6

08

119865 (1198905) =

ℎ1

06

ℎ2

02 03 05

ℎ3

05 07

ℎ4

02 04

ℎ5

05 07

ℎ6

03 05

(5)



(1) 119860 sube 119861

(2) For all 119890 isin 119860 119865(119890) sube 119866(119890)





1 1198902 1198903 =


119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119866 (1198901) =

ℎ1

03

ℎ2

06 08

ℎ3

04 05

ℎ4

03 04 05

ℎ5

05

ℎ6

07

119866 (1198902) =

ℎ1

04 05 06 07

ℎ2

07 08

ℎ3

09

ℎ4

08 09

ℎ5

05

ℎ6

05

119866 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

(6)

Then we have (119865 119860) sube (119866 119861)





119860 if 119865(119890) = 120601 for

all 119890 isin 119860


119860 if119865(119890) = 1 for all 119890 isin 119860



119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

02 03 02 03 02 03 03 04 05 06 07 02 04ℎ2

05 06 05 06 05 06 05 06 07 08 05 07 08 05 06 07ℎ3

03 03 03 04 05 06 08 06 08ℎ4

03 05 03 05 03 05 03 04 05 07 08 09 03 05ℎ5

03 04 05 04 05 04 05 03 04 05 03 04 05 03 04 05ℎ6

03 05 06 07 03 03 03




(119865 119860)119888

= (119865119888 119860) (7)


119888

for all 119890 isin 119860


)119888





lated as follows

119865119888(1198901) =

ℎ1

07 08

ℎ2

04 05

ℎ3

07

ℎ4

05 07

ℎ5

05 06

ℎ6

03 04

119865119888(1198902) =

ℎ1

03 04 06

ℎ2

02 03 05

ℎ3

02 04

ℎ4

01 03

ℎ5

05 06 07

ℎ6

07

119865119888(1198903) =

ℎ1

06 08

ℎ2

03 04

ℎ3

01 02

ℎ4

05 07

ℎ5

04 06

ℎ6

03

119865119888(1198904) =

ℎ1

04 05 07

ℎ2

08

ℎ3

05

ℎ4

03 04

ℎ5

04 05

ℎ6

02

119865119888(1198905) =

ℎ1

04

ℎ2

05 07 08

ℎ3

03 05

ℎ4

06 08

ℎ5

03 05

ℎ6

05 07

(8)









(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119888

= (119869 119860 times 119861)119888

= (119869119888 119860 times 119861) Similarly

(119865 119860)119888

or (119866 119861)119888

= (119865119888 119860) or (119866

119888 119861) = (119874 119860 times 119861) Now take


119869119888(120572 120573)

= (119869 (120572 120573))119888

= (119865 (120572) cap 119866 (120573))119888

= ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880

119888


+

119860 ℎ+

119861)⟩ | 119906 isin 119880

119888



119860 ℎ+

119861)⟩ | 119906 isin 119880


119861)



)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880



119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

03 04 05 06 07 02 03 04 04 06 07 04 05 06 07 04 06 07ℎ2

06 08 07 08 06 07 06 07 08 07 08 06 07 08ℎ3

04 05 09 08 09 06 08 09 08 09ℎ4

03 04 05 08 09 03 05 07 09 08 09 07 09ℎ5

05 05 04 05 06 05 05 04 05 06ℎ6

07 06 07 07 07 05 07

119874 (120572 120573)

= 119865119888(120572) cup 119866

119888(120573)

= ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880

119888

cup ⟨119906 ℎ119861(119906)⟩ | 119906 isin 119880

119888


119861(119906)⟩ | 119906 isin 119880


119861)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880

(9)


((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


(119865 119860) and ((119866 119861) and (119869 119862)) = ((119865 119860) and (119866 119861)) and (119869 119862)

(119865 119860) or ((119866 119861) or (119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)

(10)




(119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)


(119865 119860) and ((119866 119861) or (119869 119862))

= ((119865 119860) and (119869 119862)) or ((119866 119861) and (119869 119862))

(119865 119860) or ((119866 119861) and (119869 119862))

= ((119865 119860) or (119869 119862)) and ((119866 119861) or (119869 119862))

(11)




119880 1198901

1198902

1198903

ℎ1

03 04 05 06 07 02 04ℎ2

06 08 07 08 06 07ℎ3

04 05 09 08 09ℎ4

03 04 05 08 09 03 05ℎ5

05 05 04 06ℎ6

07 05 07




119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(12)

We write (119865 119860) cup (119866 119861) = (119869 119862)



119865(119890) cap 119866(119890)

We write (119865 119860) cap (119866 119861) = (119869 119862)



(1) (119865 119860) cup (119865 119860) = (119865 119860)(2) (119865 119860) cap (119865 119860) = (119865 119860)(3) (119865 119860) cup Φ

119860= (119865 119860)

(4) (119865 119860) cap Φ119860= Φ119860

(5) (119865 119860) cup 119860= 119860



119880 1198901

1198902

ℎ1

02 03 04 05 06 07ℎ2

05 06 05 07 08ℎ3

03 06 08ℎ4

03 05 07 08 09ℎ5

03 04 05 03 04 05ℎ6

03 03

(6) (119865 119860) cap 119860= (119865 119860)

(7) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)(8) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)


(1) ((119865 119860) cup (119866 119861))119888

sub (119865 119860)119888

cup (119866 119861)119888

(2) (119865 119860)119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(13)

Thus ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(14)


cup (119866 119861)119888

= (119865119888 119860) cup (119866

119888 119861) =


119874 (119890) = 119865119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cup 119866

119888(119890) if 119890 isin 119860 cap 119861

(15)



119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888


(1) (119865 119860)119888

cap (119866 119861)119888

sub((119865 119860) cup (119866 119861))119888

(2) ((119865 119860) cap (119866 119861))119888

sub(119865 119860)119888

cup (119866 119861)119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(16)

Then ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(17)


cap (119866 119861)119888

= (119865119888 119860) cap (119866

119888 119861) = (119874


119888(119890)


119869119888(119890) thus (119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cup (119866 119861))119888 is proved


119888

sub (119865 119860)119888

cup (119866 119861)119888


(1) ((119865 119860) cup (119866 119860))119888

= (119865 119860)119888

cap (119866 119860)119888

(2) ((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888


119869 (119890) = 119865 (119890) cup 119866 (119890) (18)

Then ((119865 119860) cup (119866 119860))119888

= (119869 119860)119888

= (119869119888 119860) and for all 119890 isin 119860

119869119888(119890) = 119865

119888(119890) cap 119866

119888(119890) (19)


cap (119866 119860)119888

= (119865119888 119860) cap (119866

119888 119860) =

(119874 119860) and for all 119890 isin 119860

119874 (119890) = 119865119888(119890) cap 119866

119888(119890) (20)

Hence119874(119890) = 119869119888(119890) the ((119865 119860) cup (119866 119860))

119888

= (119865 119860)119888

cap (119866 119860)119888


((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888





119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08


Let 119880 = 1199061 1199062 119906



1199062ℎ119860(1199062) 119906



119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this






119865= (Γ 119860)



119865with respect


119865mid) or the



119865 119905) or

119871(Δ119865mid) or 119871(Δ




119894119888119894


119895may

be chosen



119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075




119894(119894 = 1 2 5) to be


(1198902) rent price (119890






119865= (Γ 119860)



119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890










119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












02 03 04 06 07 02 04 03 05 06 06ℎ2

05 06 05 07 08 06 07 02 02 03 05ℎ3

03 06 08 08 09 05 05 07ℎ4

03 05 07 09 03 05 06 07 02 04ℎ5

04 05 03 04 05 04 06 05 06 05 07ℎ6

06 07 03 07 08 03 05

Consider

119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119865 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

119865 (1198904) =

ℎ1

03 05 06

ℎ2

02

ℎ3

05

ℎ4

06 07

ℎ5

05 06

ℎ6

08

119865 (1198905) =

ℎ1

06

ℎ2

02 03 05

ℎ3

05 07

ℎ4

02 04

ℎ5

05 07

ℎ6

03 05

(5)



(1) 119860 sube 119861

(2) For all 119890 isin 119860 119865(119890) sube 119866(119890)





1 1198902 1198903 =


119865 (1198901) =

ℎ1

02 03

ℎ2

05 06

ℎ3

03

ℎ4

03 05

ℎ5

04 05

ℎ6

06 07

119865 (1198902) =

ℎ1

04 06 07

ℎ2

05 07 08

ℎ3

06 08

ℎ4

07 09

ℎ5

03 04 05

ℎ6

03

119866 (1198901) =

ℎ1

03

ℎ2

06 08

ℎ3

04 05

ℎ4

03 04 05

ℎ5

05

ℎ6

07

119866 (1198902) =

ℎ1

04 05 06 07

ℎ2

07 08

ℎ3

09

ℎ4

08 09

ℎ5

05

ℎ6

05

119866 (1198903) =

ℎ1

02 04

ℎ2

06 07

ℎ3

08 09

ℎ4

03 05

ℎ5

04 06

ℎ6

07

(6)

Then we have (119865 119860) sube (119866 119861)





119860 if 119865(119890) = 120601 for

all 119890 isin 119860


119860 if119865(119890) = 1 for all 119890 isin 119860



119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

02 03 02 03 02 03 03 04 05 06 07 02 04ℎ2

05 06 05 06 05 06 05 06 07 08 05 07 08 05 06 07ℎ3

03 03 03 04 05 06 08 06 08ℎ4

03 05 03 05 03 05 03 04 05 07 08 09 03 05ℎ5

03 04 05 04 05 04 05 03 04 05 03 04 05 03 04 05ℎ6

03 05 06 07 03 03 03




(119865 119860)119888

= (119865119888 119860) (7)


119888

for all 119890 isin 119860


)119888





lated as follows

119865119888(1198901) =

ℎ1

07 08

ℎ2

04 05

ℎ3

07

ℎ4

05 07

ℎ5

05 06

ℎ6

03 04

119865119888(1198902) =

ℎ1

03 04 06

ℎ2

02 03 05

ℎ3

02 04

ℎ4

01 03

ℎ5

05 06 07

ℎ6

07

119865119888(1198903) =

ℎ1

06 08

ℎ2

03 04

ℎ3

01 02

ℎ4

05 07

ℎ5

04 06

ℎ6

03

119865119888(1198904) =

ℎ1

04 05 07

ℎ2

08

ℎ3

05

ℎ4

03 04

ℎ5

04 05

ℎ6

02

119865119888(1198905) =

ℎ1

04

ℎ2

05 07 08

ℎ3

03 05

ℎ4

06 08

ℎ5

03 05

ℎ6

05 07

(8)









(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119888

= (119869 119860 times 119861)119888

= (119869119888 119860 times 119861) Similarly

(119865 119860)119888

or (119866 119861)119888

= (119865119888 119860) or (119866

119888 119861) = (119874 119860 times 119861) Now take


119869119888(120572 120573)

= (119869 (120572 120573))119888

= (119865 (120572) cap 119866 (120573))119888

= ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880

119888


+

119860 ℎ+

119861)⟩ | 119906 isin 119880

119888



119860 ℎ+

119861)⟩ | 119906 isin 119880


119861)



)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880



119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

03 04 05 06 07 02 03 04 04 06 07 04 05 06 07 04 06 07ℎ2

06 08 07 08 06 07 06 07 08 07 08 06 07 08ℎ3

04 05 09 08 09 06 08 09 08 09ℎ4

03 04 05 08 09 03 05 07 09 08 09 07 09ℎ5

05 05 04 05 06 05 05 04 05 06ℎ6

07 06 07 07 07 05 07

119874 (120572 120573)

= 119865119888(120572) cup 119866

119888(120573)

= ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880

119888

cup ⟨119906 ℎ119861(119906)⟩ | 119906 isin 119880

119888


119861(119906)⟩ | 119906 isin 119880


119861)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880

(9)


((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


(119865 119860) and ((119866 119861) and (119869 119862)) = ((119865 119860) and (119866 119861)) and (119869 119862)

(119865 119860) or ((119866 119861) or (119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)

(10)




(119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)


(119865 119860) and ((119866 119861) or (119869 119862))

= ((119865 119860) and (119869 119862)) or ((119866 119861) and (119869 119862))

(119865 119860) or ((119866 119861) and (119869 119862))

= ((119865 119860) or (119869 119862)) and ((119866 119861) or (119869 119862))

(11)




119880 1198901

1198902

1198903

ℎ1

03 04 05 06 07 02 04ℎ2

06 08 07 08 06 07ℎ3

04 05 09 08 09ℎ4

03 04 05 08 09 03 05ℎ5

05 05 04 06ℎ6

07 05 07




119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(12)

We write (119865 119860) cup (119866 119861) = (119869 119862)



119865(119890) cap 119866(119890)

We write (119865 119860) cap (119866 119861) = (119869 119862)



(1) (119865 119860) cup (119865 119860) = (119865 119860)(2) (119865 119860) cap (119865 119860) = (119865 119860)(3) (119865 119860) cup Φ

119860= (119865 119860)

(4) (119865 119860) cap Φ119860= Φ119860

(5) (119865 119860) cup 119860= 119860



119880 1198901

1198902

ℎ1

02 03 04 05 06 07ℎ2

05 06 05 07 08ℎ3

03 06 08ℎ4

03 05 07 08 09ℎ5

03 04 05 03 04 05ℎ6

03 03

(6) (119865 119860) cap 119860= (119865 119860)

(7) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)(8) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)


(1) ((119865 119860) cup (119866 119861))119888

sub (119865 119860)119888

cup (119866 119861)119888

(2) (119865 119860)119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(13)

Thus ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(14)


cup (119866 119861)119888

= (119865119888 119860) cup (119866

119888 119861) =


119874 (119890) = 119865119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cup 119866

119888(119890) if 119890 isin 119860 cap 119861

(15)



119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888


(1) (119865 119860)119888

cap (119866 119861)119888

sub((119865 119860) cup (119866 119861))119888

(2) ((119865 119860) cap (119866 119861))119888

sub(119865 119860)119888

cup (119866 119861)119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(16)

Then ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(17)


cap (119866 119861)119888

= (119865119888 119860) cap (119866

119888 119861) = (119874


119888(119890)


119869119888(119890) thus (119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cup (119866 119861))119888 is proved


119888

sub (119865 119860)119888

cup (119866 119861)119888


(1) ((119865 119860) cup (119866 119860))119888

= (119865 119860)119888

cap (119866 119860)119888

(2) ((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888


119869 (119890) = 119865 (119890) cup 119866 (119890) (18)

Then ((119865 119860) cup (119866 119860))119888

= (119869 119860)119888

= (119869119888 119860) and for all 119890 isin 119860

119869119888(119890) = 119865

119888(119890) cap 119866

119888(119890) (19)


cap (119866 119860)119888

= (119865119888 119860) cap (119866

119888 119860) =

(119874 119860) and for all 119890 isin 119860

119874 (119890) = 119865119888(119890) cap 119866

119888(119890) (20)

Hence119874(119890) = 119869119888(119890) the ((119865 119860) cup (119866 119860))

119888

= (119865 119860)119888

cap (119866 119860)119888


((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888





119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08


Let 119880 = 1199061 1199062 119906



1199062ℎ119860(1199062) 119906



119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this






119865= (Γ 119860)



119865with respect


119865mid) or the



119865 119905) or

119871(Δ119865mid) or 119871(Δ




119894119888119894


119895may

be chosen



119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075




119894(119894 = 1 2 5) to be


(1198902) rent price (119890






119865= (Γ 119860)



119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890










119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

02 03 02 03 02 03 03 04 05 06 07 02 04ℎ2

05 06 05 06 05 06 05 06 07 08 05 07 08 05 06 07ℎ3

03 03 03 04 05 06 08 06 08ℎ4

03 05 03 05 03 05 03 04 05 07 08 09 03 05ℎ5

03 04 05 04 05 04 05 03 04 05 03 04 05 03 04 05ℎ6

03 05 06 07 03 03 03




(119865 119860)119888

= (119865119888 119860) (7)


119888

for all 119890 isin 119860


)119888





lated as follows

119865119888(1198901) =

ℎ1

07 08

ℎ2

04 05

ℎ3

07

ℎ4

05 07

ℎ5

05 06

ℎ6

03 04

119865119888(1198902) =

ℎ1

03 04 06

ℎ2

02 03 05

ℎ3

02 04

ℎ4

01 03

ℎ5

05 06 07

ℎ6

07

119865119888(1198903) =

ℎ1

06 08

ℎ2

03 04

ℎ3

01 02

ℎ4

05 07

ℎ5

04 06

ℎ6

03

119865119888(1198904) =

ℎ1

04 05 07

ℎ2

08

ℎ3

05

ℎ4

03 04

ℎ5

04 05

ℎ6

02

119865119888(1198905) =

ℎ1

04

ℎ2

05 07 08

ℎ3

03 05

ℎ4

06 08

ℎ5

03 05

ℎ6

05 07

(8)









(1) ((119865 119860) and (119866 119861))119888

= (119865 119860)119888

or (119866 119861)119888

(2) ((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


119888

= (119869 119860 times 119861)119888

= (119869119888 119860 times 119861) Similarly

(119865 119860)119888

or (119866 119861)119888

= (119865119888 119860) or (119866

119888 119861) = (119874 119860 times 119861) Now take


119869119888(120572 120573)

= (119869 (120572 120573))119888

= (119865 (120572) cap 119866 (120573))119888

= ⟨119906 ℎ119860(119906) cap ℎ

119861(119906)⟩ | 119906 isin 119880

119888


+

119860 ℎ+

119861)⟩ | 119906 isin 119880

119888



119860 ℎ+

119861)⟩ | 119906 isin 119880


119861)



)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880



119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

03 04 05 06 07 02 03 04 04 06 07 04 05 06 07 04 06 07ℎ2

06 08 07 08 06 07 06 07 08 07 08 06 07 08ℎ3

04 05 09 08 09 06 08 09 08 09ℎ4

03 04 05 08 09 03 05 07 09 08 09 07 09ℎ5

05 05 04 05 06 05 05 04 05 06ℎ6

07 06 07 07 07 05 07

119874 (120572 120573)

= 119865119888(120572) cup 119866

119888(120573)

= ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880

119888

cup ⟨119906 ℎ119861(119906)⟩ | 119906 isin 119880

119888


119861(119906)⟩ | 119906 isin 119880


119861)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880

(9)


((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


(119865 119860) and ((119866 119861) and (119869 119862)) = ((119865 119860) and (119866 119861)) and (119869 119862)

(119865 119860) or ((119866 119861) or (119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)

(10)




(119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)


(119865 119860) and ((119866 119861) or (119869 119862))

= ((119865 119860) and (119869 119862)) or ((119866 119861) and (119869 119862))

(119865 119860) or ((119866 119861) and (119869 119862))

= ((119865 119860) or (119869 119862)) and ((119866 119861) or (119869 119862))

(11)




119880 1198901

1198902

1198903

ℎ1

03 04 05 06 07 02 04ℎ2

06 08 07 08 06 07ℎ3

04 05 09 08 09ℎ4

03 04 05 08 09 03 05ℎ5

05 05 04 06ℎ6

07 05 07




119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(12)

We write (119865 119860) cup (119866 119861) = (119869 119862)



119865(119890) cap 119866(119890)

We write (119865 119860) cap (119866 119861) = (119869 119862)



(1) (119865 119860) cup (119865 119860) = (119865 119860)(2) (119865 119860) cap (119865 119860) = (119865 119860)(3) (119865 119860) cup Φ

119860= (119865 119860)

(4) (119865 119860) cap Φ119860= Φ119860

(5) (119865 119860) cup 119860= 119860



119880 1198901

1198902

ℎ1

02 03 04 05 06 07ℎ2

05 06 05 07 08ℎ3

03 06 08ℎ4

03 05 07 08 09ℎ5

03 04 05 03 04 05ℎ6

03 03

(6) (119865 119860) cap 119860= (119865 119860)

(7) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)(8) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)


(1) ((119865 119860) cup (119866 119861))119888

sub (119865 119860)119888

cup (119866 119861)119888

(2) (119865 119860)119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(13)

Thus ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(14)


cup (119866 119861)119888

= (119865119888 119860) cup (119866

119888 119861) =


119874 (119890) = 119865119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cup 119866

119888(119890) if 119890 isin 119860 cap 119861

(15)



119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888


(1) (119865 119860)119888

cap (119866 119861)119888

sub((119865 119860) cup (119866 119861))119888

(2) ((119865 119860) cap (119866 119861))119888

sub(119865 119860)119888

cup (119866 119861)119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(16)

Then ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(17)


cap (119866 119861)119888

= (119865119888 119860) cap (119866

119888 119861) = (119874


119888(119890)


119869119888(119890) thus (119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cup (119866 119861))119888 is proved


119888

sub (119865 119860)119888

cup (119866 119861)119888


(1) ((119865 119860) cup (119866 119860))119888

= (119865 119860)119888

cap (119866 119860)119888

(2) ((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888


119869 (119890) = 119865 (119890) cup 119866 (119890) (18)

Then ((119865 119860) cup (119866 119860))119888

= (119869 119860)119888

= (119869119888 119860) and for all 119890 isin 119860

119869119888(119890) = 119865

119888(119890) cap 119866

119888(119890) (19)


cap (119866 119860)119888

= (119865119888 119860) cap (119866

119888 119860) =

(119874 119860) and for all 119890 isin 119860

119874 (119890) = 119865119888(119890) cap 119866

119888(119890) (20)

Hence119874(119890) = 119869119888(119890) the ((119865 119860) cup (119866 119860))

119888

= (119865 119860)119888

cap (119866 119860)119888


((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888





119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08


Let 119880 = 1199061 1199062 119906



1199062ℎ119860(1199062) 119906



119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this






119865= (Γ 119860)



119865with respect


119865mid) or the



119865 119905) or

119871(Δ119865mid) or 119871(Δ




119894119888119894


119895may

be chosen



119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075




119894(119894 = 1 2 5) to be


(1198902) rent price (119890






119865= (Γ 119860)



119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890










119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 1198901 1198901

1198901 1198902

1198901 1198903

1198902 1198901

1198902 1198902

1198902 1198903

ℎ1

03 04 05 06 07 02 03 04 04 06 07 04 05 06 07 04 06 07ℎ2

06 08 07 08 06 07 06 07 08 07 08 06 07 08ℎ3

04 05 09 08 09 06 08 09 08 09ℎ4

03 04 05 08 09 03 05 07 09 08 09 07 09ℎ5

05 05 04 05 06 05 05 04 05 06ℎ6

07 06 07 07 07 05 07

119874 (120572 120573)

= 119865119888(120572) cup 119866

119888(120573)

= ⟨119906 ℎ119860(119906)⟩ | 119906 isin 119880

119888

cup ⟨119906 ℎ119861(119906)⟩ | 119906 isin 119880

119888


119861(119906)⟩ | 119906 isin 119880


119861)⟩ | 119906 isin 119880


119861) | (1 minus ℎ)



)⟩ | 119906 isin 119880

(9)


((119865 119860) or (119866 119861))119888

= (119865 119860)119888

and (119866 119861)119888


(119865 119860) and ((119866 119861) and (119869 119862)) = ((119865 119860) and (119866 119861)) and (119869 119862)

(119865 119860) or ((119866 119861) or (119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)

(10)




(119869 119862)) = ((119865 119860) or (119866 119861)) or (119869 119862)


(119865 119860) and ((119866 119861) or (119869 119862))

= ((119865 119860) and (119869 119862)) or ((119866 119861) and (119869 119862))

(119865 119860) or ((119866 119861) and (119869 119862))

= ((119865 119860) or (119869 119862)) and ((119866 119861) or (119869 119862))

(11)




119880 1198901

1198902

1198903

ℎ1

03 04 05 06 07 02 04ℎ2

06 08 07 08 06 07ℎ3

04 05 09 08 09ℎ4

03 04 05 08 09 03 05ℎ5

05 05 04 06ℎ6

07 05 07




119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(12)

We write (119865 119860) cup (119866 119861) = (119869 119862)



119865(119890) cap 119866(119890)

We write (119865 119860) cap (119866 119861) = (119869 119862)



(1) (119865 119860) cup (119865 119860) = (119865 119860)(2) (119865 119860) cap (119865 119860) = (119865 119860)(3) (119865 119860) cup Φ

119860= (119865 119860)

(4) (119865 119860) cap Φ119860= Φ119860

(5) (119865 119860) cup 119860= 119860



119880 1198901

1198902

ℎ1

02 03 04 05 06 07ℎ2

05 06 05 07 08ℎ3

03 06 08ℎ4

03 05 07 08 09ℎ5

03 04 05 03 04 05ℎ6

03 03

(6) (119865 119860) cap 119860= (119865 119860)

(7) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)(8) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)


(1) ((119865 119860) cup (119866 119861))119888

sub (119865 119860)119888

cup (119866 119861)119888

(2) (119865 119860)119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(13)

Thus ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(14)


cup (119866 119861)119888

= (119865119888 119860) cup (119866

119888 119861) =


119874 (119890) = 119865119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cup 119866

119888(119890) if 119890 isin 119860 cap 119861

(15)



119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888


(1) (119865 119860)119888

cap (119866 119861)119888

sub((119865 119860) cup (119866 119861))119888

(2) ((119865 119860) cap (119866 119861))119888

sub(119865 119860)119888

cup (119866 119861)119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(16)

Then ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(17)


cap (119866 119861)119888

= (119865119888 119860) cap (119866

119888 119861) = (119874


119888(119890)


119869119888(119890) thus (119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cup (119866 119861))119888 is proved


119888

sub (119865 119860)119888

cup (119866 119861)119888


(1) ((119865 119860) cup (119866 119860))119888

= (119865 119860)119888

cap (119866 119860)119888

(2) ((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888


119869 (119890) = 119865 (119890) cup 119866 (119890) (18)

Then ((119865 119860) cup (119866 119860))119888

= (119869 119860)119888

= (119869119888 119860) and for all 119890 isin 119860

119869119888(119890) = 119865

119888(119890) cap 119866

119888(119890) (19)


cap (119866 119860)119888

= (119865119888 119860) cap (119866

119888 119860) =

(119874 119860) and for all 119890 isin 119860

119874 (119890) = 119865119888(119890) cap 119866

119888(119890) (20)

Hence119874(119890) = 119869119888(119890) the ((119865 119860) cup (119866 119860))

119888

= (119865 119860)119888

cap (119866 119860)119888


((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888





119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08


Let 119880 = 1199061 1199062 119906



1199062ℎ119860(1199062) 119906



119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this






119865= (Γ 119860)



119865with respect


119865mid) or the



119865 119905) or

119871(Δ119865mid) or 119871(Δ




119894119888119894


119895may

be chosen



119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075




119894(119894 = 1 2 5) to be


(1198902) rent price (119890






119865= (Γ 119860)



119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890










119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 1198901

1198902

ℎ1

02 03 04 05 06 07ℎ2

05 06 05 07 08ℎ3

03 06 08ℎ4

03 05 07 08 09ℎ5

03 04 05 03 04 05ℎ6

03 03

(6) (119865 119860) cap 119860= (119865 119860)

(7) (119865 119860) cup (119866 119861) = (119866 119861) cup (119865 119860)(8) (119865 119860) cap (119866 119861) = (119866 119861) cap (119865 119860)


(1) ((119865 119860) cup (119866 119861))119888

sub (119865 119860)119888

cup (119866 119861)119888

(2) (119865 119860)119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(13)

Thus ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(14)


cup (119866 119861)119888

= (119865119888 119860) cup (119866

119888 119861) =


119874 (119890) = 119865119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cup 119866

119888(119890) if 119890 isin 119860 cap 119861

(15)



119888

cap (119866 119861)119888

sub ((119865 119860) cap (119866 119861))119888


(1) (119865 119860)119888

cap (119866 119861)119888

sub((119865 119860) cup (119866 119861))119888

(2) ((119865 119860) cap (119866 119861))119888

sub(119865 119860)119888

cup (119866 119861)119888



119869 (119890) = 119865 (119890) if 119890 isin 119860 minus 119861

= 119866 (119890) if 119890 isin 119861 minus 119860

= 119865 (119890) cup 119866 (119890) if 119890 isin 119860 cap 119861

(16)

Then ((119865 119860) cup (119866 119861))119888

= (119869 119862)119888

= (119869119888 119862) and

119869119888(119890) = 119865

119888(119890) if 119890 isin 119860 minus 119861

= 119866119888(119890) if 119890 isin 119861 minus 119860

= 119865119888(119890) cap 119866

119888(119890) if 119890 isin 119860 cap 119861

(17)


cap (119866 119861)119888

= (119865119888 119860) cap (119866

119888 119861) = (119874


119888(119890)


119869119888(119890) thus (119865 119860)

119888

cap (119866 119861)119888

sub ((119865 119860) cup (119866 119861))119888 is proved


119888

sub (119865 119860)119888

cup (119866 119861)119888


(1) ((119865 119860) cup (119866 119860))119888

= (119865 119860)119888

cap (119866 119860)119888

(2) ((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888


119869 (119890) = 119865 (119890) cup 119866 (119890) (18)

Then ((119865 119860) cup (119866 119860))119888

= (119869 119860)119888

= (119869119888 119860) and for all 119890 isin 119860

119869119888(119890) = 119865

119888(119890) cap 119866

119888(119890) (19)


cap (119866 119860)119888

= (119865119888 119860) cap (119866

119888 119860) =

(119874 119860) and for all 119890 isin 119860

119874 (119890) = 119865119888(119890) cap 119866

119888(119890) (20)

Hence119874(119890) = 119869119888(119890) the ((119865 119860) cup (119866 119860))

119888

= (119865 119860)119888

cap (119866 119860)119888


((119865 119860) cap (119866 119860))119888

= (119865 119860)119888

cup (119866 119860)119888





119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08


Let 119880 = 1199061 1199062 119906



1199062ℎ119860(1199062) 119906



119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this






119865= (Γ 119860)



119865with respect


119865mid) or the



119865 119905) or

119871(Δ119865mid) or 119871(Δ




119894119888119894


119895may

be chosen



119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075




119894(119894 = 1 2 5) to be


(1198902) rent price (119890






119865= (Γ 119860)



119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890










119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 1198901

1198902

1198903

1198904

1198905

1199091

05 06 03 05 03 03 04 06 091199092

07 04 02 04 05 04 05 05 061199093

04 06 06 07 05 06 06 02 03 051199094

03 05 04 06 05 06 07 02 04 051199095

05 06 08 03 03 04 02 03 05 07 08


Let 119880 = 1199061 1199062 119906



1199062ℎ119860(1199062) 119906



119865(119890)= 119906

1

119904(ℎ119860(1199061)) 1199062119904(ℎ119860(1199062)) 119906

119899119904(ℎ119860(119906119899)) By using this






119865= (Γ 119860)



119865with respect


119865mid) or the



119865 119905) or

119871(Δ119865mid) or 119871(Δ




119894119888119894


119895may

be chosen



119880 1198901

1198902

1198903

1198904

1198905

1199091

055 04 03 035 0751199092

07 04 037 045 0551199093

05 065 055 06 0331199094

04 05 05 065 0371199095

063 03 035 033 075




119894(119894 = 1 2 5) to be


(1198902) rent price (119890






119865= (Γ 119860)



119865is as mid

Δ119865

= (1198901 0556) (119890

2 045)

(1198903 0414) (119890

4 0476) (119890










119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880 1198901

1198902

1198903

1198904

1198905

Choice value1199091

0 0 0 0 1 11199092

1 0 0 0 0 11199093

0 1 1 1 0 31199094

0 1 0 1 0 21199095

1 0 0 0 1 2

5 Conclusion





Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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