On Starshaped Intuitionistic Fuzzy Sets

doi:10.4236/am.2011.28146

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

1The Mathematics an d Statistics, Chongqing University of Technology, Chongqing, China

2School of Management, Xi’an Jiaotong University, Xi’an, China

E-mail: chxuwh @gmail.com, liuyufeng@cqut.edu.cn, sunxuxin520@163.com

Received May 23, 201 1; 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. Th e core idea of in tuitionistic fuz zy 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 p resent, the research of in tuition fuzz y set theor y

is more active in international, especially the research of

some similar problems in fuzzy set. Through more than 20

years of d evelop ment, the in tuitionis tic fuzzy set has mad e

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 ,n

x

yR,





|

x

yzzx y





is a line segment,

where 0



,0



,1









. A set is simply said

to be starshaped relative to a point Sn

x

R, if

x

yS

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

ker SS

x

S such that

x

yS 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

AA

Ax xxxR



where , satisfy



:[0,

n

AR



01

1]



:[0,

n

AR



1]



AA









for all n

x

R,



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













|1,

n

AA

xRxx 0







is non-

empty.

Example 2.1. Let

















,,|

AA

A

xx xxR



,

where

W. H. XU ET AL.

1052

1,0]

0,1]

wise

ise





1[

1(

0other

A

xx

xx x











 









[1,0]

0otherw

,

A

xx

xx x

 





 



 





Then





A

FR.

Definition 2.2. An intuitionistic fuzzy set







n

A

FR

is called quasi-convex if















min ,

AAA

x

yyx y

 















max ,

AAA

x

yyx y

 

for all ,n

x

yR, [0,1]



Definition 2.3. Let





,n

A

BFR, the union, inter-

section and complement of



A

and



B are defined as

follows,



















,, |

n

ABAB

A

Bx xxxxxR 





















,, |

n

ABAB

A

Bx xxxxxR 













,1 ,1|

cn

AA

A

xx xx 



R

Definition 2.4. [11] Let





,n

A

BFR, ,[0,1]







,

the [,]





cut , [,)



cut, (,]



cut and





,







cut of

A

are defined as follows:













[,] |, ,

AA

AxxXx x 























[,) |, ,

AA

AxxXxx 























(,] |, ,

AA

AxxXx x 























(,) |, ,

AA

AxxXxx 











3. Starshaped Intuitionistic Fuzzy Sets

Definition 3.1 An intuitionistic fuzzy set







n

A

FR is

said to be i.f.s. relative to n

y

R if









AA

x

yy x

 

and









AA

x

yy x

 

for all n

x

R, [0,1]



,

Proposition 3.1 Let







n

A

FR is i.f.s. relative to

n

y

R, then











sup 1

n

AA

xR

yx



















inf 0

n

AA

xR

yx







Proof. Let



A

is i.f.s. relative to , then for all y

n

x

R,















AA

x

yy x

 

and















AA

x

yy x

 

are true for 01





.Thus, only take 0



, it can be

found that













AA

y

x







and

 







AA

y

x







are

true for all n

x

R.

Hence













sup 1

n

AA

xR

yx







and













inf 0

n

AA

xR

yx









Example 3.1. The intuitionistic fuzzy







A

FR

with





e(

e(0,

x

Ax

x

xx



 











,0]

)





1e( ,0]

1e(0, )

x

Ax

x

xx



















is i.f.s. relative to0y



.

Proposition 3. 2. An intuitionistic fuzzy set







n

A

FR is i.f.s. respect to iff its level sets

n

yR

are starshaped relative to y.

Proof. Suppose





,

A





is starshaped relative to

n

yR

for all],[0,1







. For n

x

R



, let





A

x





,







A

x





,

then



[,]

xy A





. That is, for any[0.1]



















AA

x

yy x

 

and















AA

x

yy x

 

Conversely, if for all n

x

R, [0,1]



,















AA

x

yy x



and















AA

x

yy x

 

hold. Since



[,]

A





, there exists



[,]

xA





, that

means



Ax()





 and



()

Ax





. Hence,















AA

xy yx









and

W. H. XU ET AL.

1053









AA

xy yx 





for any [0,1]



.

So



[,]

xy A





. Thus



[,]

A





is starshaped relative to

y.



Definition 3.2. An intuitionistic fuzzy set







n

A

FR

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







n

A

FR

is said to be i.f.q-s. relative to

if for all

n

yR

n

x

R,[0,1]



, the following hold,

















min ,

AAA

x

yx

 

y

A















max ,

AA

x

yyx x

 

Definition 3.4 An intuitionistic fuzzy set







n

A

FR

is said to be i.f.p-s. relative to n

y

R if for all

n

x

R,[0,1]



, the following are true,









 





1

AAA

x

yy xy 











 





1

AAA

x

yy xy 



Definition 3.4. Let







ker

A

(respectively,







kerqA,





kerpA) be the totality of such that

n

yR



A

is

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

y

Definition 3.5. The intuitionistic fuzzy hypograph of



A

denoted by





.fhpy A, is defined as



.().() .()

f

hpyAf hpyf hpy







where











.,|,0,

AA

fhpyxt x Rtx



















.,|,

AA

f hpyxsxR sx









,1

Theorem 3.1. Let





n

A

FRis i.f.g.s. and



[,]

keryA





iff





n

A

FR is i.f.s. relative to . y

Proof. “” Since







n

A

FR is i.f.g.s. and



[,]

,ker A



, that is



[,]

,keryA







. 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

FR is i.f.p-s. relative to

n

yR, then it is i.f.q-s. relative to. y

Proof. Since for all n

x

R, [0,1]



, the following

hold,













 

















1

min ,

AA

A

x

yy xy

xy

 



















 

















1

max ,

AA

A

x

yy xy

xy

 



 



Thus



A

is i.f.q-s. relative to. y

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

FR with





2

2[

[1,1]

2[1

0 otherwise

A

xx

xxx

2,1]

,2]















 











2

1[2

[1,1]

1[1,

1 otherwise

A

xx

xxx

,1]

2]



 



 



 



 





is i.f.q-s. relative to0y



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

0y



.

Theorem 3.3. Let





n



A

FR is i.f.q-s. relative to

, then

n

yR













sup 1

n

AA

xR

yx



















inf 0

n

AA

xR

yx







iff







n

A

FR is i.f.s. relative to. y

Proof. “” Since





n

A

FR is i.f.q-s. relative to

,

n

yR













sup 1

n

AA

xR

yx







and













inf 0

n

AA

xR

yx







then for all n

x

R, [0,1]





, we have



























min ,

AAAA

x

yyx yx 

 



























max ,

AAAA

x

yyx yx 

 

Hence



A

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

“



” Since



A

is i.f.s. relative to,that means y

n

x

R , [0,1]





,















AA

x

yy x

 

and

1054 W. H. XU ET AL.









AA

x

yy x



.

Take 0



, we get









AA

y

x



and













AA

y

x



for alln

x

R.

Thus,





















min ,

AA

xy yx

x

y





























max ,

AA

xy yx

x

y











Hence,

A

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











sup 1

n

AA

xR

yx

















inf 0

n

AA

xR

yx











Theorem 3.4. Let







n

A

FR is i.f.p-s. relative to

, if

n

yR











sup 1

n

AA

xR

yx







and











inf 0

n

AA

xR

yx







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







n

A

FR

with





e(

e(0,

x

Ax

x

xx



 



,0]

)















1e( ,0]

1e(0, )

x

Ax

x

xx



 











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

. 0y

0y

4. Basic Properties of Starshapedness of

Intuitionistic Fuzzy Sets

Proposition 4.1. If







n

A

FR



is i.f.s. relative to

, then

n

yR











sup 1

n

AA

xR

yx





and











inf 0

n

AA

xR

yx







Proof. Let



A

is i.f.s. relative to , then for all y

n

x

R, [0,1]



















AA

x

yy x



and















AA

x

yy x



Take 0





, we get











AA

y

x



and













AA

y

x



for all n

x

R.

So













sup 1

n

AA

xR

yx







and













inf 0

n

AA

xR

yx









Proposition 4. 2. An intuitionistic fuzzy set







n

A

FR is i.f.s. relative to iff for all

n

yRn

x

R,

[0,1]





, the following hold,











AA

x

yxy





 

and











AA

x

yxy





 

Proof. Supposed

A

is i.f.s. relative to , that is, for

all y

n

x

R,[0,1]





,















AA

x

yy x



and















AA

x

yy x



Replacing

x

by

x

y



in the above inequality, we

can get the desired result. Similarly, we can get the con-

verse.



Proposition 4. 3. An intuitionistic fuzzy set







n

A

FR is i.f.q-s. relative to iff

n

yR





,

A







is

starshaped set relative to for

y





Ay





0,



,







Ay,1













.

Proof. “” Supposed



A

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

that is for all y

n

x

R, [0,1]





,





















min ,

AAA

x

yyx y

 





















max ,

AAA

x

yyx y

 

For any







0, Ay













, , if





,1

Ay









[,]

xA





,

then we have that



[,]

,xyA





. From the above inequal-

ity we get that











Axy y









W. H. XU ET AL.

1055

and





Axy y





So



[,]

xy A





.

“”, For

n

x

R, [0,1]



, if











AA

x

y



,

then let



A





y



. Accordingly we have



[,]

xy A





,that

is,















min ,

AAA

x

yyx y

 

If









AA

x

y



, then let





A

x





. Accordingly

we have



[,]

xy A





 ,that is,















min ,

AAA

x

yyx y

 

If









AA

x

y



, then let





A

y





.

Accordingly we have





,

xy A





A

,that is,















max ,

AA

x

yyx y

 

If













AA

x

y



, then let





A

x





. Accordingly

we have



[,]

xy A





, that is,















max ,

AAA

x

yyx y

 

.

Thus



A

is i.f.q-s. relative to . 

n

yR

Proposition 4. 4. An intuitionistic fuzzy set





n

A

FR is i.f.p-s. relative to iff y







.A

f hpy



is

starshaped relative to







,A

y





y

and







.A

f hpy



is

starshaped relative to .





,A

yy



Proof. “” If



A

is i.f.p-s. relative to , y







,.A

x tfhpy





and







,.A

xs fhpy



. Since



A

is i.f.p-s. relative to for any

y[0,1]



 we have









 









1

AA

A

x

yy xy

ty

 















 









1

AA

A

x

yy xy

sy

 



 



.

Thus, we have















,1 ,.

A

xtyyfhyp A







 













,1 ,.

AA

xsyy fhyp



 



Hence,





.A

fhyp



is starshaped relative to









,A

yy



and





A

yp.fh



is starshaped relative to









,A

yy



.

“” Assume that











,.

A

xxfhpyA



and

















,.

A

xxfhpyA





. By the starshapedness of





.A

f hpy



and







A

.fhyp



we can have













 















1 .

AA

yfhpy1,xy x A























 















1 .

AA

yfhpy 1,xy x A





[0,1]

 

.

for any





. Thus



A

is i.f.p-s. relative to . y

A path in a set in is a continuous mapping

S

:[0,1]

n

R

f

SS. A set is said to be path connected if,

there exists a path

f

such that



0

f

x



and





1

f

y



for all ,

x

yS



[12]. A intuitionistic fuzzy

set



A

is 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.

Proposition 4. 5. If





n



A

FR



is i.f.s. relative to

n

yR, 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.



Proposition 4.6. If





n

A

FR is a intuitionistic

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







n

A

FR 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 ,n

x

yR, [0,1]





, we have











mi











n ,

AA

x

yy





A

x y

 











ma











x ,

AA

x

yy



,n

A

x y

 



Then for all

x

yR



, the following hold,









xy y











,

AA

x ymin A



 



















xyy











,

AA

x ymax A



 





So,



[,]

xy A





. In other words



A

is i.f.g.s..

Additionally if







n

A

FR 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.



Proposition 4.7. If





n

A

FR

n

yR

is an intuitionistic

fuzzy and the point satisfies that

















yx



infn

xR







AA

and . Then















sup

n

A

xR

yx







A



A

is intuitionistic fuzzy quasi- starshaped set relative to

, that is,

y







ker Ayq .

1056 W. H. XU ET AL.

Proof. According to Definition 3.3, we have











infn

AA

xR

yx







and













sup

n

AA

xR

yx









. So this

statement is true.



Proposition 4.8. If





12

,n



A

AFR

n

are i.f.q-s. (re-

spectively, i.f.p-s.) relative to . Then

yR



12

A

A is

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

y

Proof. Because that





12

,n



A

AFR

n

are i.f.q-s. rela-

tive to , for all

n

yR

x

R,[0,1]



, we have















1min,

ii

AA

i

A

x

yx















1max,

ii

AA

y



i

A

x

yxy

 

1,i,2



So,













 



















12

12 12

2

1

max1, 1,2

min max,,

min ,

i

AA

A

AA

AAAA

xy

xyi

xx y

x

y













 

 



 



and,









































12

112 2

12 12

1

max1,1,2

max max,,max,

max ,

i

AA

A

AA AA

xy

xyi

x

yx

xy













 



 

y

Hence,



12

A

A is i.f.q-s. relative to . y

If





12

,n

A

AFR

n are i.f.p-s. relative to .

Then all

n

yR

x

R[0,,1]



, we have









 







1

ii

AA

i

A

x

yy xy 

 

,







 





1

ii

AA

i

A

x

yy xy 

 

, 1,2

i



So,













 

































 





12

12 12

1

min1, 1,2

min,1 min,

1

i

ii

AA

A

AA

AA AA

xy

xyi

x

xy















 

 

 

  

 

y

and





















 













 













 





12

12 12

1

max1, 1,2

max,} 1max{,

()1

i

ii

AA

A

AA

AA AA

xy

xyi

x

xy















 

 

 

 

  

y

Hence



12

A

A is i.f.p-s. relative to . y

Proposition 4.9. If





12

,n



A

AFR

n

are i.f.q-s. (re-

spectively, i.f.p-s.) relative to

y

R and













12

AA

yy



,











12

AA

yy



. Then



12

A

A is

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

y

Proof. Since





12

,n



A

AFR

n

are i.f.q-s. relative to

n

yR, for all

x

R,[0,1]





, we have



















1min,

ii

AA

i

A

x

yx



y

and



















1max,

ii

AA

i

A

x

yx

 

y1, 2i,



By













12

AA

yy



and , we can

get









12

AA

y



y





















 



















12

12 12

2

1

max1,1,2

min max,,

min ,

i

AA

A

AA

AAAA

xy

xyi

xx y

x

y













 

 



 



and













































12

121

112

2

1

min1, 1,2

max min,,

max ,

i

AA

A

AA A

AAA

xy

xyi

x

xy

x

y













 



 

 

Hence



12

A

A is i.f.q-s. relative to . y

Since



12 (

n

, )

A

AFR

n are i.f.p-s. relative to ,

for all

n

yR

x

R[0,1],





, we have













 







11

ii

AA

i

A

x

yx

 

y

and















 





1

ii

AA

i

A

x

yy xy 

 

, 1, 2i



W. H. XU ET AL.

1057

From and









12

AA

yy















12

AA

yy



we can

find













 











 





12

12 12

1

max1,1,2

max1, 1,2

1

i

ii

AA

A

AA

AA AA

xy

xyi



x

y







 





 

 



 

and

























 





1

12 12

21

min1, 1,2

min(1), 1,2

1

i

ii

A

AA

AA AA

xy

xyi

x

y





 

 



 

 

 

 



Hence



12

A

A is i.f.p-s. relative to . y

Let 0n

x

R,





n

A

FR, then the translation of



A

by 0

x

is the intuitionistic fuzzy



0

x

A defined as









00

x

AxAxx.

Proposition 4.10. If







n

A

FRy is i.f.s. (respectively,

i.f.p-s., i.f.q-s.) relative to and 0n

x

R. 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 n

x

R, since





n

A

FR is i.f.s. relative to

. We have that

y





















00

0

A

AA

A

xxyxyx

0

x

yx yxx

xx









 



and





















00

0

A

AA

A

xxyxyx

0

x

yx yxx

xx









 



So,



0

x

A is i.f.s. relative to 0

x

y.



Let be a linear invertible transformation on ,

Tn

R





n

A

FR. Then by the Extension Principle we have

that









1

xAT x



TA .

Proposition 4.11 If







n

A

FR is i.f.s. (respectively,

i.f.p-s., i.f.q-s.) relative to and T is a linear invert- y

ible transformation on . Then

n

R





TA is i.f.s. (re-

spectively, i.f.p-s., i.f.q-s.) relative to .



Ty

Proof. We only give the proof for the case of in-

tuitionistic fuzzy quasi-starshapedness. Similarly, the

others can be prove d.

For any n

x

R, since





n



A

FR is i.f.q-s. relative

to . We have that

y















 

























1

11

min ,

min() ,

TAxTyATxy

ATx Ay

TA xTATy









  

Hence,







TA is i.f.q-s. relative to .



Ty 

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).

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