Proceedings of IEEE CCIS2011

AN EXTENDED SOFT SET MODEL: TYPE-2 FUZZY SOFT SETS

Xi’ao Ma1, Guoyin Wang2

1 School of Information Science & Technology, Southwest Jiaotong University, Chengdu 610031, PR China 2 Institute of Computer Science & Technology, Chongqing University of Posts and Telecommunications,

Chongqing 400065, PR China maxiao73559@163.com, wanggy_cupt@yahoo.com.cn

Abstract The soft set theory has been initiated by Molodtsov as a useful mathematical tool for dealing with uncertainty, fuzzy, not clearly defined objects. However, it is inappropriate to be used to deal with fuzzy parameters that involve uncertain words, linguistic terms and have a non-measurable domain. In this paper, we present an extended soft set model which is called type-2 fuzzy soft sets by combining the type-2 fuzzy set and soft set models. The complement, ”and”, “or”, union and intersection operations are defined on the type-2fuzzy soft sets. The related properties of type-2fuzzy soft sets are also presented and discussed.

Keywords: Soft sets; Fuzzy sets; Type-2 fuzzy sets; Type-2 fuzzy soft sets

1 Introduction Uncertainty exists almost everywhere in the whole world. Many theories and models for expressingand processing uncertainty have been proposed, such as probability theory, fuzzy set theory[1] and rough set theory[2]. The major difficulty of these theories is the inadequacy of the parameterization tool of the theories. Soft set theory developed by Molodtsov[3] has become a useful mathematical tool for dealing with uncertainty which cannot be handled by the above mathematical tools. Since its introduction, it has been applied in many research fields successfully, such as function smoothness, Riemann integration, decision making, measurement theory, game theory, and so on.

Later, Maji et al.[4]defined some basic operations and proved some related propositions on soft set operations, Ali et al.[5] analyzed the incorrectness of some theorems in [4], then they proposed some new soft set operations and proved that certain De Morgan’s laws hold in soft set theory with respect to these new definitions. Maji et al. also[6] give an application of soft set theory in a decision making problem by the method of attributes reduction in the rough set theory. But chen et al.[7] point out that the reduction of parameter sets in soft set theory and attributes reduction in rough set theory

are the different method, so they proposed a new definition of parameter reduction for soft set and applied it to the decision making problem. Kong etal.[8] discussed the problems of suboptimal choice and added parameter set in the parameterization reduction of soft sets, then they introduced the definition of the normal parameter reduction with respect to soft sets and proposed a heuristic algorithm of normal parameter reduction in soft sets.

Above work is based on classical soft set. In practice, however, the objects may not precisely satisfy the problem’s parameters, thus Maji et al.[9]put forward the concept of fuzzy soft set by combining the fuzzy set and soft set, then they[10]presented a theoretic approach of the fuzzy soft set in decision making problem. Majumdar[11] further generalised the concept of fuzzy soft sets, they also studied some relevant properties, and gave an application for solving a decision making problem on fuzzy soft set. By combining the interval-valued fuzzy set and soft set, yang et al.[12] proposed the interval-valued fuzzy soft set and then analyzed a decision making problem in the interval-valued fuzzy soft set. Xu et al.[13] proposed the notion of vague soft set which is an extension to the soft set by introducing vague sets into soft sets. Jiang etal.[14] presented the concept of interval-valued intuitionistic fuzzy soft sets which is based on an interval-value fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory. In the real world, however, we often encounter the situation that the evaluation of parameters is a fuzzy concept. For example, we can consider the beauty of house that we are going to buy the house. Then, we can give an evaluation about the beauty of house that may be high, middle or low and so on. Here, high, middle and low are fuzzy concepts which can be represented by fuzzy sets. If the classical soft set is used to represent the fuzziness of the above evaluation of parameters directly, it will be difficult since the evaluation of parameters of the object is a linguistic term that is a fuzzy concept. Type-2 fuzzy set proposed by

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Zadeh[15] can solve this problem very good. Hence, it is necessary to extend soft set theory using type-2 fuzzy set. The rest of this paper is structured as follows. In next Section, we review some definitions and properties on soft set and type-2 fuzzy set. In section 3, we introduce the notion and operation of type-2 fuzzy sets and discuss related properties. The conclusion is concluded in Section 4.

2 Preliminaries In this section, we review some basic concepts and properties such as type-2 fuzzy sets and soft sets.

2.1 Type-2 fuzzy sets

Definition 1 ([16]). IfU is a collection of objects denoted generically by x , then a fuzzy set �F in U is a set of ordered pairs:

��{( , ( )) | }FF x x x U .

� ( )F x is called the membership function of grade

of membership of x in �F , the set of all fuzzy sets over U is denoted by �( )P U . Definition 2 ([16]). A type-2 fuzzy set, denoted �A , is characterized by a type-2 membership functio � ( , )A x u , where x U and [0,1]xu J , i.e., �

�{(( , ), ( , )) | , [0,1]}xAA x u x u x U u J .

In which �0 ( , ) 1A x u . �A can also be expressed

as � � ( , ) / ( , )xx U u J AA x u x u , [0,1]xu J ,

or, as �

�{( , ( )) | }AA x x x U . the set of all type-2 fuzzy sets over U is denoted by�2( )T U . Definition 3 ([16]). A type-1 fuzzy set can also be expressed as a type-2 fuzzy set. Its type-2 representation is (1/ ( )) /F x x or 1/ ( )F x ,

x U for short. The notation 1/ ( )F x means that the secondary membership function has only one value in its domain, namely the primary membership ( )F x , at which the secondary grade equals 1. Definition 4 ([16]) Let �A and �B be two type-2 fuzzy sets over a universeU , where �

� ( ) / [ ( ) / ] /ux

xAx U x U x JA x x f u u x

, [0,1]u

xJ ,

�� ( ) / [ ( ) / ] /

wx

xBx U x U x JB x x g w w x

, [0,1]w

xJ ,

the union of type-2 fuzzy sets �A and �B is given as � �

� � � �( , ) ( ) /A B A Bx UA B x v x x

.

Here, � � � �( ) ( ) ( )A B A Bx x x � ( ) ( ) /

u wx x

x xu J w J

f u g w u w

, x U .

Definition 5 ([16]). Let �A and �B be two type-2 fuzzy sets over a universeU , where �

� ( ) / [ ( ) / ] /ux

xAx U x U x JA x x f u u x

, [0,1]u

xJ ,

�� ( ) / [ ( ) / ] /

wx

xBx X x X x JB x x g w w x

, [0,1]w

xJ ,

the intersection of type-2 fuzzy sets �A and �B is given as

� �� � � �( , ) ( ) /A B A Bx U

A B x v x x

Here, � � � �( ) ( ) ( )A B A Bx x x � ( ) ( ) /

u wx x

x xu J w J

f u g w u w

, x U .

Definition 6 ([16]). Let �A be a type-2 fuzzy set over U , where �

� ( ) / [ ( ) / ] /ux

xAx U x U x JA x x f u u x

, [0,1]u

xJ ,

the complement of type-2 fuzzy sets �A is given as �

� �( , ) ( ) /A Ax U

A x v x x

.

Here, � �( ) ( ) ( ) / (1 )ux

xAAu J

x x f u u

, x U .

Definition 7 ([17]). Let �A and �B be two type-2 fuzzy sets over a universe U , a partially order relation is defined as

� � � � �( ) ( ) ( ) ( ) ( )A B A B Ax x x x x�

� � , x U .

Similarly, let �� be a partially ordered relation

given as

� � � � �( ) ( ) ( ) ( ) ( )A B A B Bx x x x x�

� � , x U . Theorem 1([17]). Let �A , �B and �C be three type-2 fuzzy sets over a universe U , then we have the following properties: (1) � � �( ) ( ) ( ),A A Ax x x x U �

� � �( ) ( ) ( ),A A Ax x x x U � (2) � � � �( ) ( ) ( ) ( ),A B B Ax x x x x U � �

� � � �( ) ( ) ( ) ( ),A B B Ax x x x x U � � (3) � � � � � �( ( ) ( )) ( ) ( ) ( ( ) ( ))A B C A B Cx x x x x x � � � � , x U

� � � � � �( ( ) ( )) ( ) ( ) ( ( ) ( ))A B C A B Cx x x x x x � � � � , x U (4) � �( ) ( ),AA

x x x U

(5) � � � �( ( ) ( )) ( ) ( ),A B B Ax x x x x U � �

� � � �( ( ) ( )) ( ) ( ),A B B Ax x x x x U � � (6) � �( ) 1 / 0 ( ),A Ax x x U �

� �( ) 1/1 ( ),A Ax x x U �

Theorem 2([17]). Let �A , �B and �C be three convex type-2 fuzzy sets over a universeU , then we have � � � � � � �( ) ( ( )) ( )) ( ( ) ( )) ( ( ) ( )),A B C A B A Cx x x x x x x x U � � � � �� � � � � � �( ) ( ( )) ( )) ( ( ) ( )) ( ( ) ( )),A B C A B A Cx x x x x x x x U � � � � �

Theorem 3([17]). Let �A be a normal type-2 fuzzy set over a universeU , then we have

(1) � ( ) 1 /1 1/1,A x x U � (2) � / 0( 0) /1 1 ,A x x U � Theorem 4([15]). Let �A and �B be two normal convex type-2 fuzzy sets over a universeU , then we have (1) � � � �( ) ( ( )) ( )) ( ),A A B Ax x x x x U � � (2) � � � �( ) ( ( )) ( )) ( ),A A B Ax x x x x U � �

2.2 Soft sets

Let U be an initial universe, E is a set of parameters, and A E . Definition 8 ([3]). A pair ( , )F A is called a soft set over U , where F is a mapping given by

: ( )F A P U . In other words, a soft set over U is a parameterized family of subsets of the universe U . Every set ( )F , A , may be regarded as the set of α-approximate elements or α-elements of the soft set ( , )F A . Definition 9 ([4]). Let U be an initial universe and E be a set of parameters. Suppose that ,A B E , ( , )F A and ( , )G B are soft sets, we say that ( , )F A is a soft subset of ( , )G B if and only if (1) A B ; (2) A , ( )F and ( )G are identical approximations. This relationship is denoted by �( , ) ( , )F A G B . ( , )F A is said to be a soft superset of ( , )G B , if ( , )G B is a soft subset of ( , )F A . We denote it by �( , ) ( , )F A G B . Definition 10 ([4]). Let ( , )F A and ( , )G B be two soft sets over a universeU , ( , )F A and ( , )G B are said to be soft equal if and only if (1) ( , )F A is a soft subset of ( , )G B ; (2) ( , )G B is a soft subset of ( , )F A . We denote it by ( , ) ( , )F A G B . Definition 11 ([4]). Let 1 2{ , , , }nE e e e be a set of parameters. The not set of E denoted by E is defined by 1 2{ , , , }nE e e e where i ie not e . Definition 12 ([4]). The complement of a soft set ( , )F A is denoted by ( , )cF A and is defined by ( , ) ( , )c cF A F A where : ( )cF A P U is a mapping given by ( ) ( )cF X F , A . Definition 13 ([4]). If ( , )F A and ( , )G B are two soft sets then” ( , )F A and ( , )G B ”denoted by ( , ) ( , )F A G B is defined by ( , ) ( , ) ( , )F A G B H A B , where

( , ) ( ) ( )H F G , ( , ) A B . Definition 14 ([4]). If ( , )F A and ( , )G B be two soft sets then” ( , )F A or ( , )G B ”denoted by ( , ) ( , )F A G B

is defined by ( , ) ( , ) ( , )F A G B O A B , where ( , ) ( ) ( )O F G , ( , ) A B .

Definition 15 ([4]). The union of two soft sets of ( , )F A and ( , )G B over a universe U is the soft set ( , )H C , where C A B and C .

( ),( ) ( ),

( ) ( ),

F if A BH G if B A

F G if A B

We denote it by �( , ) ( , ) ( , )F A G B H C . Definition 16 ([13]). The intersection of two soft sets of ( , )F A and ( , )G B over a universeU is the soft set ( , )H C , where C A B and C .

( ),( ) ( ),

( ) ( ),

F if A BH G if B A

F G if A B

We denote it by �( , ) ( , ) ( , )F A G B H C .

3 Type-2 fuzzy soft sets In this section, we present an extended soft set model which is called type-2 fuzzy soft sets by combining the type-2 fuzzy set and soft set models. The related properties of type-2 fuzzy set will also be discussed. Definition 18. LetU be an initial universe and E be a set of parameters, A E . A pair �( , )F A is called a type-2 fuzzy soft set over U , where �F is a mapping given by

� �: 2( )F A T U . In other words, a type-2 fuzzy soft set over U is a parameterized family of type-2 fuzzy set of the universeU , thus, its universe is the set of all type-2 fuzzy sets ofU , i.e., �2( )T U . A type-2 fuzzy soft set is also a special case of a soft set because it is still a mapping from parameters to�2( )T U .

A , � ( )F is referred as the fuzzy grade set of parameter , it is actually a type-2 fuzzy set ofU ,where x U and A , it can be written as:

�� ( )( ) { ( ) / | }FF x x x U

,

where, �( ) ( )F x

is the fuzzy grade that object x

holds on parameter . Theorem 5. Let �( , )F A be a type-2 fuzzy soft set over U , if A and x U , �( ) ( )F x

has only

one value, at which the secondary grade equals 1, then � ( )F will be degenerated to be an ordinary fuzzy set and �( , )F A will be degenerated to a traditional fuzzy soft set. Definition 19. Suppose that �( , )F A is a type-2 fuzzy soft set over U , � ( )F a is the fuzzy grade set of

parameter a , then all fuzzy grade sets in type-2 fuzzy soft set �( , )F A are referred to as the fuzzy grade set class of �( , )F A and is denoted by �( , )F AC ,

then we have �

�( , ) { ( ) | }F AC F A .

Example 1. Let �( , )F A be a type-2 fuzzy soft set, whereU is the set consist of six houses under the consideration of a decision maker to purchase, which is denoted by 1 2 6{ , , , }U x x x , and A is a set of parameters, where 1 2 3 4{ , , , }A {beautiful, expensive, in the green surroundings, in good repair}. Type-2 fuzzy soft set �( , )F A describes the ”attractiveness of the houses” to the decision maker. Suppose that �

� � � �1 2 3 4( , ) { ( ), ( ), ( ), ( )}F AC F F F F ,

where � 1( )F {low/ x1, high/ x2, middle/ x3, very high/ x4, very low/ x5, not high/ x6}; �

2( )F {low/ x1, middle/ x2, very high/ x3, high/ x4, very low/ x5, high/ x6};

�3( )F {middle/ x1, very good/ x2, good/ x3, poor/ x4,

not good/ x5, very good/ x6}; �

4( )F {good/ x1, middle/ x2, very good/ x3, very good/ x4, good/ x5, not poor/ x6}.

The tabular representation of a type-2 fuzzy soft �( , )F A is displayed in table 1, where

middle=0.3/0.3+0.7/0.4+1/0.5+0.7/0.6+0.3/0.7; low=poor=1/0+0.9/0.1+0.7/0.2+0.4/0.3; high=good=0.4/0.7+0.7/0.8+0.9/0.9+1/1; very low=very poor

=1/0+0.81/0.1+0.49/0.2+0.16/0.3; very high=very good

=0.16/0.7+0.49/0.8+0.81/0.9+1/1; not low=not poor

=0.1/0.1+0.3/0.2+0.6/0.3+1/(0.4+0.5+…+1); not high=not good

=1/(0+0.1+…+0.6)+0.6/0.7+0.3/0.8+0.1/0.9. Table 1 A type-2 fuzzy soft set �( , )F A

U α1 α2 α3 α4

x1 low low middle good

x2 high middle very good meddle

x3 middle very high good very good

x4 very high high poor very good

x5 very low very low not good good

x6 not high high very good not poor Obviously, we can see that the evaluation for each object on each parameter is a linguistic term which is a fuzzy concept. They can be represented by fuzzy sets. For example, we cannot give the precise evaluation of how beautiful house x1 is, but the beauty of house x1 is low, it is a fuzzy set.

Definition 20. LetU be an initial universe and E be a set of parameters. Suppose that ,A B E , �( , )F A and �( , )G B are two type-2 fuzzy soft sets, we say that �( , )F A is a type-2 fuzzy soft subset of �( , )G B if and only if (1) A B ; (2) For all x U and A , � � ( )( ) ( ) ( )GF x x

� �

or � � ( )( ) ( ) ( )GF x x�

� .

This relationship is denoted by � � �( , ) ( , )F A G B . �( , )F A is said to be a type-2 fuzzy soft superset of �( , )G B , if �( , )G B is a type-2 fuzzy soft subset of �( , )F A . We denote it by � � �( , ) ( , )F A G B .

Definition 21. Let �( , )F A and �( , )G B be two type-2 fuzzy soft sets over a universeU , �( , )F A and �( , )G B are said to be type-2 fuzzy soft equal if and only if (1) �( , )F A is a type-2 fuzzy soft subset of �( , )G B ; (2) �( , )G B is a type-2 fuzzy soft subset of �( , )F A . We denote it by � � �( , ) ( , )F A G B . Definition 22. A type-2 fuzzy soft set �( , )F A over U is said to be a null type-2 fuzzy soft set denoted by � A , if A , �( ) ( ) 1/ 0F x

, x U .

Definition 23. A type-2 fuzzy soft set �( , )F A over U is said to be an absolute type-2 fuzzy soft set denoted by � AU , if A , �( )( ) 1/1F x

, x U .

Definition 24. The complement of a type-2 fuzzy soft set �( , )F A is denoted by �( , )cF A and is defined by � �( , ) ( , )

ccF A F A where � �: 2( )c

F A T U is a mapping given by

� � ( )( )( ) ( )c FFx x

, A , x U .

Definition 25. If �( , )F A and �( , )G B are two type-2 fuzzy soft sets over a universe U then” �( , )F A and �( , )G B ”denoted by � �( , ) ( , )F A G B is

defined by � � �( , ) ( , ) ( , )F A G B H A B , where � � �( , ) ( ) ( )H F F , ( , ) A B , that is �

� � ( )( )( , )( ) ( ) ( )GFH x x x � , ( , ) A B , x U .

Definition 26. If �( , )F A and �( , )G B are two type-2 fuzzy soft sets over a universe U then” �( , )F A or �( , )G B ”denoted by � �( , ) ( , )F A G B is defined

by � � �( , ) ( , ) ( , )F A G B H A B , where � � �( , ) ( ) ( )H F F , ( , ) A B , that is � � � ( )( )( , )( ) ( ) ( )GFH x x x

� ,

( , ) A B , x U .

Theorem 6. Let �( , )F A be a type-2 fuzzy soft set over a universeU , then we have � �(( , ) ) ( , )C CF A F A .

Proof. Since � �( , ) ( , )CCF A F A , then we have

� �(( , ) ) ( , )CC C CF A F A . Suppose that �( , )

C CF A � �( , ) ( , )

CCH C H C , where C A A .

� � � �( ) ( ) ( )( )( ) ( ) ( ) ( )C H F FHx x x x

,

A , x U .

Hence, �C

H and �F are the same operators.

So � �(( , ) ) ( , )C CF A F A . Theorem 7. Let �( , )F A and �( , )G B be two type-2 fuzzy soft sets over a universeU , then we have the following properties: (1) � � � �(( , ) ( , )) ( , ) ( , )C C CF A G B F A G B ; (2) � � � �(( , ) ( , )) ( , ) ( , )C C CF A G B F A G B . Proof. (1) Suppose that � � �( , ) ( , ) ( , )F A G B H A B . Then we have � � � �(( , ) ( , )) ( , ) ( , ( ))

CC CF A G B H A B H A B .

Since � �( , ) ( , )CCF A F A and � �( , ) ( , )

CCG B G B ,

then we have � � � �( , ) ( , ) ( , ) ( , )C CC CF A G B F A G B .

Suppose that � � �( , ) ( , ) ( , ( ))C C

F A G B O A B ,

where � � �( , ) ( ) ( )C C

O F G , ( , ) A B , x U . Now, we take ( , ) ( )A B , here, � � �( , ) ( )( )

( ) ( ) ( )C CO GFx x x

� ,

( , ) A B , x U .

� � � � ( )( , ) ( )( , )( ) ( ) ( ( ) ( ))C GH FHx x x x

�

� �( )( ) ( ) ( )GF x x �

� � ( )( )( ) ( )C C

GFx x

� ,

( , ) A B , x U .

Hence, �C

H and �O are the same operators.

So � � � �(( , ) ( , )) ( , ) ( , )C C CF A G B F A G B . (2) The proof is similar to that of (1). Theorem 8. Let �( , )F A , �( , )G B and �( , )H B be three type-2 fuzzy soft sets over a universeU , then we have the following properties: (1) � � � � � �( , ) (( , ) ( , )) (( , ) ( , )) ( , )F A G B H C F A G B H C ; (2) � � � � � �( , ) (( , ) ( , )) (( , ) ( , )) ( , )F A G B H C F A G B H C . Definition 27. The union of two type-2 fuzzy soft sets of �( , )F A and �( , )G B over a universeU is a type-2 fuzzy soft set �( , )H C , where C A B and C .

�

�

�

� �

( )

( )( )

( )( )

( ), ,

( ) ( ), ,

( ) ( ), ,

F

GH

GF

x if A B x U

x x if B A x U

x x if A B x U

�

We denote it by � � � �( , ) ( , ) ( , )F A G B H C . Definition 28. The intersection of two type-2 fuzzy soft sets of �( , )F A and �( , )G B over a universe U is the soft set �( , )H C , where C A B and

C .

�

�

�

� �

( )

( )( )

( )( )

( ), ,

( ) ( ), ,

( ) ( ), ,

F

GH

GF

x if A B x U

x x if B A x U

x x if A B x U

�

We denote it by � � � �( , ) ( , ) ( , )F A G B H C . Theorem 9. Let E be a set of parameters, A E . If � E is a null type-2 fuzzy soft set, � EU an absolute type-2 fuzzy soft set, and �( , )F A a type-2 fuzzy soft set over U , then we have the following properties: (1) � � � �( , ) ( , ) ( , )F A F A F A ;

(2) � � � �( , ) ( , ) ( , )F A F A F A ;

(3) � � � �( , ) ( , )EF E U F E ;

(4) � � � �( , ) ( , )EF E F E . Theorem 10. Let �( , )F A and �( , )G B be two type-2 fuzzy soft sets over a universeU , then we have the following properties: (1) � � � � � �(( , ) ( , )) ( , ) ( , )C C CF A G B F A G B ;

(2) � � � � � �(( , ) ( , )) ( , ) ( , )C C CF A G B F A G B . Theorem 11. Let �( , )F A , �( , )G B and �( , )H C be three type-2 fuzzy soft sets over a universeU , then we have the following properties: (1) � � � � � � � � � �( , ) (( , ) ( , )) (( , ) ( , )) ( , )F A G B H C F A G B H C ;

(2) � � � � � � � � � �( , ) (( , ) ( , )) (( , ) ( , )) ( , )F A G B H C F A G B H C . Theorem 12. Let �( , )F A and �( , )G B be two type-2 fuzzy soft sets over a universeU , then we have the following properties: (1) � � � � � �( , ) ( , ) ( , ) ( , )F A G B G B F A ;

(2) � � � � � �( , ) ( , ) ( , ) ( , )F A G B G B F A ;

(3) � � � � � � � � � �( , ) (( , ) ( , )) ( , ) (( , ) ( , ))F A F B G C F A F A G B .

Definition 29. A type-2 fuzzy soft set �( , )F A over U is said to be convex, if A , �( )( )F x

is

a convex fuzzy set, x U . Definition 30. A type-2 fuzzy soft set �( , )F A over U is said to be normal, if A , �( )( )F x

is a

normal fuzzy set, x U . Otherwise, it is subnormal.

Furthermore, a type-2 fuzzy soft set which is convex and normal is referred to as a normal convex type-2 fuzzy soft.

Theorem 13. Let �( , )F A , �( , )G B and �( , )H C be three type-2 fuzzy soft sets over a universeU , then we have the following properties: (1) � � � � � � � � � � � �( , ) (( , ) ( , )) (( , ) ( , )) (( , ) ( , ))F A G B H C F A G B F A H B ;

(2) � � � � � � � � � � � �( , ) (( , ) ( , )) (( , ) ( , )) (( , ) ( , ))F A G B H C F A G B F A H B . Theorem 14. Let E be a set of parameters, A E .If � E is a null type-2 fuzzy soft set, � EU an absolute type-2 fuzzy soft set, and �( , )F E a normal type-2fuzzy soft set over U , then we have the following properties: (1) � � � �( , ) E EF A U U ;

(2) � � � �( , ) E EF E ;

Theorem 15. Let �( , )F A and �( , )G B be two normal convex type-2 fuzzy soft sets over a universe U ,then we have the following properties: (1) � � � � � �( , ) (( , ) ( , )) ( , )F A F A G B F A ;

(2) � � � � � �( , ) (( , ) ( , )) ( , )F A F A G B F A .

4 Conclusion The soft set theory has been conceived by Molodtsov as a new mathematical tool for dealing with uncertainty, fuzzy, not clearly defined objects. In the real world, however, we are often faced with the situation that the evaluation of parameters of the object is usually uncertain words or linguistic terms. For instance, the beauty of house may be high, middle or low and so on, where high, middle and low are linguistic terms. If the classical soft set is used to represent the vagueness of the evaluation parameters directly, it will be inappropriate since the evaluation of parameters of the object is a fuzzy concept. Type-2 fuzzy set can solve this problem effectively. Hence, it is necessary to introduce the notion of type-2 fuzzy soft set as an extension to the soft set.

In this paper, we present an extended soft set model which is called type-2 fuzzy soft sets by introducing the type-2 fuzzy set into soft set model. On this basis, we defined the complement, ”and”,“or”, union and intersection operations on the type-2 fuzzy soft sets. Finally, we presented and discussed the related properties of type-2 fuzzy soft sets.

Acknowledgements This work is partially supported by National Science Foundation of P. R. China under Grant No. 61073146

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