On Starshaped Intuitionistic Fuzzy Sets

(1)

Applied Mathematics, 2011, 2, 1051-1058

doi:10.4236/am.2011.28146 Published Online August 2011 (http://www.SciRP.org/journal/am)

On Starshaped Intuitionistic Fuzzy Sets

Weihua Xu1,2, Yufeng Liu1, Wenxin Sun1 1

The Mathematics and Statistics, Chongqing University of Technology, Chongqing, China 2

School of Management, Xi’an Jiaotong University, Xi’an, China E-mail: [email protected], [email protected], sunxuxin520@163.com

Received May 23, 2011; revised June 28, 2011; accepted July 6, 2011

Abstract

Intuitionistic fuzzy starshaped sets (i.f.s.) is a generalized model of fuzzy starshaped set. By the definition of i.f.s., the intuitionistic fuzzy general starshaped sets (i.f.g.s.), intuitionistic fuzzy quasi-starshaped sets (i.f.q-s.) and intuitionistic fuzzy pseudo-starshaped sets (i.f.p-s.) are proposed and the relationships among them are studied. The equivalent discrimination conditions of i.f.q-s. and i.f.p-s. are presented on the basis of their properties which are meaningful for the research of the generalized fuzzy starshaped sets. Moreover, the invariance of the two given fuzzy sets under the translation transformation and linear reversible transforma-tion are discussed.

Keywords:Intuitionistic Fuzzy Starshaped Sets, Cut Sets, Starshapedness, Convex Sets

1. Introduction

Since the theory of fuzzy set was proposed by Zadeh in 1965, it has been widely used such as fuzzy control, fuzzy recognition, fuzzy evaluation, system theory, information retrieval and so on. The intuitionistic fuzzy set theory, proposed by Atanassov [1], is an extension of the fuzzy set theory. The core idea of intuitionistic fuzzy sets is added a non-membership functions in the basis of membership functions, which can cooperate with the membership func-tion and describe the fuzzy object of the world more exqui-sitely. At present, the research of intuition fuzzy set theory is more active in international, especially the research of some similar problems in fuzzy set. Through more than 20 years of development, the intuitionistic fuzzy set has made a lot of important achievements [2-9].

With the deepening of the research work and expan-sion, another more worthy of concern is the emergence of fuzzy starshaped set theory [10] and its applications. Fuzzy starshaped sets have more abundant properties and characteristics, it is a directly extension of the fuzzy set theory and convex set, many useful results have been obtained.

In this paper, we define a new kind of fuzzy set which is intuition fuzzy set, by combining the fuzzy starshaped set and intuition fuzzy set. By the basic definition of in-tuitionistic fuzzy starshaped set, we introduce some new different types of starshapedness. We discuss the rela-tionships among these different types of starshapedness,

and obtain some important properties.

2. Preliminaries

Let x y, Rn, xy



z z| xy



is a line segment, where  0, 0,  1. A set is simply said to be starshaped relative to a point

S

n

xR , if xyS

for any point yS . A set is simply said to be star-shaped, which means that there exists a point

S

x in such that is starshaped relative to

n

R

S x. The kernel

of is the set of all points

kerS S xS such that

xyS for any yS.

Definition 2.1. Let denote an universe of dis-course. An intuitionistic fuzzy set

n

R

_A_{is an object} hav-ing the form



 



 







, , | n



A A

A x  x  x xR

where _A:Rn[0, , satisfy

0 1

1] A:Rn [0,1]

 

A A

   for all xRn, _A and _A are called the degree of membership and the one degree of non- membership of the element x to A respectively. Let

 

n

F R

n

R

be the classes of normal intuitionistic fuzzy sets

on , that is



xRn|_A

 

x 1,_A

 

x 0



is non- empty.

(2)

1, 0]

0,1]

wise

ise 

 

1 [

1 (

0 other

A

x x

x x x

      



       





 

[ 1, 0]

0 otherw

,

A

x x

x x x

    



     _   

 

Then AF R

 

.

Definition 2.2. Anintuitionistic fuzzy set AF R

 

n

is called quasi-convex if







min





 

, 

 



A xy y  A x A y

   







max





 

, 

 



A xy y  A x A y

   

for all x y, Rn, [0,1]

Definition 2.3. Let  _{A B}, __{F R}

 

n

, the union, inter- section and complement of A and B are defined as follows,

 



 



 



 



 







, , | n



A B A B

AB x x  x  x  x xR

 



 



 



 



 







, _A _B , _A _B | n



AB x x  x  x  x xR



 



 







,1 ,1 |



c _n

A A

A  x  x  x xR

Definition 2.4. [11] Let  A B, F R

 

n ,  , [0,1], the [ , ]  cut , [ , )  cut, ( , ]  cut and



 ,



 cut of A are defined as follows:



 



 



[ , ] | , ,

A A

A   x xX  x   x 





 



 



[ , ) | , _A , _A

A   x xX  x   x 





 



 



( , ] | , _A , _A

A   x xX  x   x 





 



 



( , ) | , _A , _A

A   x xX  x   x 



3. Starshaped Intuitionistic Fuzzy Sets

Definition 3.1 An intuitionistic fuzzy set AF R

 

n is said to be i.f.s. relative to yRn if









 

A xy y  A x

  

and

_A



xy



y



 _A

 

x

  

for all _x_Rn

, [0,1],

Proposition 3.1 Let AF R

 

n is i.f.s. relative to

n

yR , then



 

sup





 



1

n

A A

x R

y x



 

 



 

inf_n





 



0

A A

x R

y x



 

 

Proof. Let A is i.f.s. relative to y, then for all

n

xR ,









 

  

and









 

A xy y  A x

  

are true for 0  1.Thus, only take  0, it can be found that _A

 

y _A

 

x and _A

 

y _A

 

x are

true for all xRn. Hence



 

sup





 



1

n

A A

x R

y x



 

 

and

_A

 

inf_n



_A

 



0

x R

y x



 

 

Example 3.1. The intuitionistic fuzzy AF R

 

with



 

e (

e (0,

x

A _x

x x

x 

   

  

  



 , 0]

)



 

1 e ( , 0]

1 e (0, )

x

A _x

x x

x 

   

  

  





is i.f.s. relative toy0.

Proposition 3.2. An intuitionisticfuzzy set



 

n

AF R is i.f.s. respect to yRn iff its level sets are starshaped relative to y.

Proof. Suppose A , is starshaped relative toyRn

for all , [0,1]. For _x_Rn

, let  _A

 

x ,  _A

 

x ,

thenxyA[ , ]  . That is, for any [0.1]









 

A xy y   A x

   

and









 

A xy y   A x

   

Conversely, if for all _x_Rn

,  [0,1],









 

  

and









 

  

hold. Since A[ , ]   , there exists xA[ , ]  , that

means _A( )x  and A( )x . Hence,









 

A xy y  A x 

   

(3)

W. H. XU ET AL. ₁₀₅₃









 

A xy y  A x 

   

for any  [0,1].

So xyA[ , ]  . Thus A[ , ]  is starshaped relative to

y.

Definition 3.2. An intuitionistic fuzzy set 

 

n

AF R

is said to be i.f.g.s. if all level sets are starshaped sets in .

n

R

Definition 3.3. An intuitionistic fuzzy set AF R

 

n is said to be i.f.q-s. relative to yRn if for all

n

xR , [0,1], the following hold,







min





 

, 

 



A xy  A x A

    y

A







max





 

, 

 



A xy y  A x x

   

Definition 3.4 An intuitionistic fuzzy set AF R

 

n is said to be i.f.p-s. relative to yRn if for all

n

xR , [0,1], the following are true,









  

1





 

A xy y  A x   A y

    









  

1





 

A xy y  A x   A y

    

Definition 3.4. Let ker

 

A (respectively, qker

 

A ,



 

ker

p A ) be the totality of _y_Rn such that _A is i.f.s. (respectively, i.f.q-s., i.f.q-s.) relative toy.

Definition 3.5. The intuitionistic fuzzy hypograph of 

A denoted by f hpy A.

 

 , is defined as 

. ( ) . ( ) . ( )

f hpy A  f hpy  f hpy

where



 



 



 



. , | , 0,

A A

f hpy   x t xR t   x _



 



 



 



. , | ,

A A

f hpy   x s xR s  x ,1_

Theorem 3.1. LetAF R

 

n is i.f.g.s. and 

[ , ]

ker

y A

    iff AF R

 

n is i.f.s. relative to y.

Proof. “” Since AF R

 

n is i.f.g.s. and 

[ , ] ,

kerA 

   , that is



[ , ] ,

ker

y A 

 

  . ThenA[ , ] 

is starshaped relative to . By Proposition 3.1 we get that

y

_A_{is i.f.s. relative to}_y_.

“ ” it follows directly from Definition 3.2 and Proposition 3.1.



Theorem 3.2. Let 

 

n

AF R is i.f.p-s. relative to

n

yR , then it is i.f.q-s. relative toy.

Proof. Since for all _x_Rn

, [0,1], the following hold,









  





 



 



 





1

min ,

A A

A

x y y x y

x y

    



    

 









  





 



 



 





1

max ,

A A

A

x y y x y

x y

    



    

 

ThusAis i.f.q-s. relative toy.

Remark 3.1. The inverse statements do not hold in general as shown in the following example.

Example 3.3. The intuitionistic fuzzy

 

n

AF R with



 

2

2 [

[ 1,1]

2 [1

0 otherwise

A

x x

x

x x

2, 1]

, 2]

   



    

 

   



  



  



 

2

1 [ 2

[ 1,1]

1 [1,

1 otherwise

A

x x

x

x x

, 1]

2]

      



    

 

    

    

is i.f.q-s. relative toy0. But it is not i.f.p-s. relative to 0

y .

Theorem 3.3. Let AF R



n



is i.f.q-s. relative to , then

n

yR



 

sup





 



1

n

A A

x R

y x



 

 



 

inf_n





 



0

A A

x R

y x



 

 

iff AF R

 

n is i.f.s. relative toy.

Proof. “” Since AF R

 

n is i.f.q-s. relative to ,

n

yR



 

sup





 



1

n

A A

x R

y x



 

 

and



 

inf_n





 



0

A A

x R

y x



 

 

then for all _x__Rn

, [0,1], we have







min





 

, 

 





 

A xy y  A x A y  A x

    







max





 

, 

 





 

A xy y  A x A y  A x

    

Hence Ait is i.f.s. relative toy.

“  ” Since A is i.f.s. relative to y ,that means

n

x R

  , [0,1],









 

  

(4)









 

   .

Take0, we get_A

 

y _A

 

x and _A

 

y _A

 

x

for allxRn. Thus,









 



 



 





min ,

A A

x y y x

x y

  



  

 









 



 



 





max ,

A A

x y y x

x y

  



  

 



Hence, Ais i.f.q-s. relative to y, 

 

sup





 



1

n

A A

x R

y x



 

 



 

inf_n





 



0

A A

x R

y x





  

Theorem 3.4. Let 

 

n

AF R is i.f.p-s. relative to , if

n

yR



 

sup





 



1

n

A A

x R

y x



 

 

and



 

inf_n





 



0

A A

x R

y x



 

 

then it is i.f.s. relative to y.

Proof. It follows from Theorem 3.2, Theorem 3.3. 

Remark 3.2. The inverse statements do not hold in general as shown in the following example.

Example 3.4. The intuitionistic fuzzy _A_{F R}

 

n

with



 

e (

e (0,

x

A _x

x x

x 

    

  , 0]

)

  

 



 

1 e ( , 0]

1 e (0, )

x

A _x

x x

x 

    

  

   

 

is i.f.s. relative to . But it is not i.f.p-s. relative to .

0

y

0

y

4. Basic Properties of Starshapedness of

Intuitionistic Fuzzy Sets

Proposition 4.1. If AF R

 

n

 

is i.f.s. relative to , then

n

yR



 

sup





 



1

n

A A

x R

y x



 

and



 

inf_n





 



0

A A

x R

y x



 

 

Proof. Let A is i.f.s. relative to y, then for all

n

xR , [0,1]









 

  

and









 

  

Take 0, we get A

 

y A

 

x and



 



 

A y  A x

  for all _x_Rn

. So



 

sup





 



1

n

A A

x R

y x



 

 

and



 

inf_n





 



0

A A

x R

y x



 

 

Proposition 4.2. An intuitionistic fuzzy set



 

n

AF R is i.f.s. relative to yRniff for all xRn, [0,1]



 , the following hold,













A xy  A xy

  

and













A xy  A xy

  

Proof. Supposed A is i.f.s. relative to , that is, for all

y

n

xR ,[0,1],









 

  

and









 

  

Replacing x by xy in the above inequality, we can get the desired result. Similarly, we can get the con-verse.



 

n

AF R is i.f.q-s. relative to n iff

yR A , 

 is starshaped set relative to y for _0,_A

 

y _ ,



 

A y ,1

 

  

  .

Proof. “” Supposed A is i.f.q-s. relative to , that is for all

y

n

xR ,  [0,1],







min





 

, 

 



   







max





 

, 

 



   

For any 0,_A

 

y ,  _A

 

y ,1, if xA[ , ]  ,

then we have that x y, A[ , ]  . From the above

inequal-ity we get that







A xy y 

(5)

W. H. XU ET AL. ₁₀₅₅

and







A xy y 

  

So xyA[ , ]  .

“”, For n

xR , [0,1], if A

 

x A

 

y ,

then let  A

 

y . Accordingly we have

_{[ , ]}

xyA ,that is,







min





 

, 

 



   

If _A

 

x _A

 

y , then let   _A

 

x . Accordingly

we have xyA[ , ]  ,that is,







min





 

, 

 



   

If _A

 

x _A

 

y , then let   _A

 

y .

Accordingly we have xyA , 

A

,that is,







max





 

, 

 



A xy y  A x y

   

If _A

 

x _A

 

y , then let   _A

 

x . Accordingly

we have xyA[ , ]  , that is,







max





 

, 

 



    .

Thus A is i.f.q-s. relative to yRn. 



 

n

AF R is i.f.p-s. relative to y iff f hpy.

 

_A is starshaped relative to



y,_A

 

y



and f hpy.

 

_A is starshaped relative to



y,A

 

y



.

Proof. “” If A is i.f.p-s. relative to y,

 

x t, f hpy.



_A



and

 

x s, f hpy.

 

_A . Since A is i.f.p-s. relative to y for any  [0,1] we have









  





 







 

1

A A

x y y x y

t y

    

  

    

  









  





 







 

1

A A

x y y x y

s y

    

  

    

  

.

Thus, we have

  

x t, 1





y, _A

 

y



f hyp.

 

_A

     

  

x s, 1





y, _A

 

y



f hyp.

 

_A

     

Hence, f hyp.

 

A is starshaped relative to



y,A

 

y



and f h. yp

 

_A is starshaped relative to



y,_A

 

y



. “  ” Assume that



x,A

 

x



f hpy.

 

A and



 



x,A x



f hpy.

 

A



. By the starshapedness of

 

.

A

f hpy  and f hyp.

 

_A we can have



 



  





 



x 1  y ,A x  1  A y



f hpy.

 

A



 



  





 



x 1  y ,_A x  1 _A y



f hpy.

 

_A [0,1]

   .

for any   . Thus A is i.f.p-s. relative to y. A path in a set S in is a continuous mapping : [0,1]

n

R

f S. A set S is said to be path connected if, there exists a path f such that f

 

0 x and

 

1

f  y for all x y, S [12]. A intuitionistic fuzzy set Ais said to be a path connected intuitionistic fuzzy set if its level sets are path connected [13]. Since a star-shaped crisp set is path connected, one can easily prove the following proposition.



n





is i.f.s. relative to

n

yR , or is i.f.g.s., then A is a path connected in-tuitionistic fuzzy set.

Proof. It follows from Definition 3.1, Definition 3.2. 

 

n is a intuitionistic fuzzy quasi-convex set, then it is i.f.g.s.. Furthermore, if



 

n

AF R is i.f.g.s. then A is i.f.s., and is a fuzzy quasi-convex set.

Proof. If A is a intuitionistic fuzzy quasi-convex set, that for all x y, Rn, [0,1], we have







min





 

, 

 



A xy y A  



A

x y

 











max





 

, 

 



A xy y A  

, n

A

x y

 



Then for all x yR



, the following hold,







x y y







 

, 

 



A   min A x A y 

    





x y



y







 

, 

 



A   max A x A y 

    

So, xyA[ , ]  . In other words A is i.f.g.s..

Additionally if AF R

 

n is i.f.g.s., then A , 

y

is starshaped. Thus they are path convex. In other words, they are intervals and there is at least one point in

_{ }_1,0

A . It is well known that intervals are convex sets in , so we have that

R A is i.f.s. relative to y and is a fuzzy quasi-convex set.

 

n

yR

is an intuitionistic fuzzy and the point satisfies that



 

y inf_n





 

x



x R 

_A _A and 

 

sup





 

. Then

n

A

x R

y x



_A





A is intuitionistic fuzzy quasi- starshaped set relative to , that is,

(6)

Proof. According to Definition 3.3, we have



 

inf_n





 



A A

x R

y x





  and 

 

sup





 

n A A x R y x  

 



. So this

statement is true.

Proposition 4.8. If  A A1, 2F R



n



n

are i.f.q-s. (respectively, i.f.p-s.) relative to yR . Then A1A2 is

i.f.q-s. (respectively, i.f.p-s.) relative to y.

Proof. Because that  A A1, 2F R



n



n

are i.f.q-s. relative to yRn, for all xR , [0,1], we have

_i



1



min



_i

 

, 

 



A x  y  A x Ai

    







1



max





 

, 

 



i i

A A

y  x  y  x _A_i y

     ,i1,2

So,  





















 







 





 





 



 



 





1 2 1 2

2

1

max 1 , 1, 2

min max , ,

min ,

i

A A

A

A A

A A A A

x y

x y i

x x y

x y                           and,  





















 



 









 



 







 





 



 



 





1 2

1 1 2 2

1 2 1 2

1

max 1 , 1, 2

max max , , max ,

max ,

i

A A

A

A A A A

x y

x y i

x y x

x y                            y

Hence, A1A2 is i.f.q-s. relative to y.

If  A A1, 2F

 

Rn

n

are i.f.p-s. relative to . Then all

n

yR xR , [0,1], we have

_i





_i

  

1





 

A xy y  A x   Ai y

     ,

_i





_i

  

1





 

A xy y  A x   Ai y

     , i1, 2

So,  





















  





 







 



 













 



 



 





  





 



 

1 2

1 2 1 2

1

min 1 , 1, 2

min , 1 min ,

1 i i i A A A A A

A A A A

x y

x y i

x x y

x y                                           y and  





















  





 







 



  





 



 





 

  





 



 

1 2

1 2 1 2

1

max 1 , 1, 2

max , } 1 max{ ,

( ) 1

i

i i

A A

A

A A

A A A A

x y

x y i

x x y

x y                                           y

Hence A1A2 is i.f.p-s. relative to y.

Proposition 4.9. If  A A1, 2F R



n



n

are i.f.q-s. (re-spectively, i.f.p-s.) relative to yR and

₁

 

₂

 

A y  A y

  , _A₁

 

y _A₂

 

y . Then A1A2 is

i.f.q-s. (respectively, i.f.p-s.) relative toy.

Proof. Since  A A1, 2F R



n



n

are i.f.q-s. relative to

n

yR , for all xR , [0,1], we have

_i



1



min



_i

 

, 

 



A x  y  A x Ai

     y

and

_i



1



max



_i

 

, 

 



A x  y  A x Ai

     y , i1, 2

By _A₁

 

y _A₂

 

y and , we can get

₁

 

₂

 

A y  A

  y

 





















 







 





 





 



 



 





1 2 1 2

2

1

max 1 , 1, 2

min max , ,

min ,

i

A A

A

A A

A A A A

x y

x y i

x x y

x y                           and  





















 



 







 











 



 



 





1 2

1 2 1

1 2 1 2

1

min 1 , 1, 2

max min , ,

max ,

i

A A

A

A A A

x y

x y i

x x y

x y                         

Hence A1A2 is i.f.q-s. relative to y.

Since  A A1, 2F(Rn) n

are i.f.p-s. relative to , for all

n

yR xR ,[0,1], we have

_i



1



_i

  

1





 

A x  y  A x   Ai

      y

and

_i





_i

  

1





 

A xy y  A x   Ai y

(7)

W. H. XU ET AL. ₁₀₅₇

From _A₁

 

y _A₂

 

y and _A₁

 

y _A₂

 

y we can find

 



















  





 





 





  





 



 

1 2

1 2 1 2

1

max 1 , 1, 2

1

i

i i

A A

A

A A

A A A A

x y

x y i



x y

 

    

   

   



 

   

  

  

     

and























 



 





 





  





 



 

1

1 2 1 2

2 1

min 1 , 1, 2

min (1 ) , 1, 2

1

i

i i

A

A A

A A A A

x y

x y i

x y

 

   

   

  



   

  

  

     

Hence A1A2 is i.f.p-s. relative to y.

Let x0Rn, 

 

n

AF R , then the translation of A

by x0 is the intuitionistic fuzzy x0A defined as







 



_

_

0 0

x A x A xx .

 

n y

is i.f.s. (respectively, i.f.p-s., i.f.q-s.) relative to and x0Rn . Then



0

x A

0

is i.f.s. (respectively, i.f.p-s., i.f.q-s.) relative to

x y.

Proof. We only give the proof for the case of in-tuitionistic fuzzy starshapedness. Similarly, the others can be proved.

For any _x__Rn

, since 

 

n

AF R is i.f.s. relative to . We have that

y





























 

0 0

0

0 A

A A

A

x x y x y x₀

0

x y x y x x

x x

    

     

 

 

  



and





























 

0 0

0

A

A A

A

x x y x y x0

x y x y x x

x x

    

     

 

 

  



So, x0A is i.f.s. relative to x0y.

Let T be a linear invertible transformation on n,

R 

 

n

AF R . Then by the Extension Principle we have that

 

T A

 



 

x A T



1

 

x



.

Proposition 4.11 If AF R

 

n is i.f.s. (respectively,

i.f.p-s., i.f.q-s.) relative to y and T is a linear invertible transformation on n. Then

R T A

 

 is i.f.s. (respectively, i.f.p-s., i.f.q-s.) relative to T y

 

.

Proof. We only give the proof for the case of in-tuitionistic fuzzy quasi-starshapedness. Similarly, the others can be proved.

For any xRn, since AF R



n



is i.f.q-s. relative to y. We have that



 



  





  







_{ }





_{ }











 



_

_{ }

_





1

1 1

min ,

min ( ) ,

T A x T y A T x y

A T x A y

T A x T A T y



    





   

Hence, T A

 

 is i.f.q-s. relative to T y

 

.

5. Conclusions

Intuitionistic fuzzy set and fuzzy starshaped set are some special fuzzy sets. In this article, we introduce some new different types of intuitionistic fuzzy starshaped set. By discussing the relationships among these different types of starshapedness, and obtained some important proper-ties. Deepening people’s understanding of intuitionistic fuzzy sets, enrich and perfect the theories of fuzzy sets.

6. Acknowledgements

The project is supported by the postdoctoral Science Foundation of China (NO.20100481331) and the Na-tional Natural Science Foundation of China (NO. 71071124, 11001227).

7. References

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