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1/7

Fuzzy Sets ~md System s 31 (1989) 343 -349

N or t h - H o l l a nd

343

I N T E R V A L V A L U E D I N T U I T I O N | S T I C F U Z Z Y S E T S

K A T A N A S S O V

Institute o f Microsy~ tems, Sofia , B ulgaria

G G A R G O V

LM L, CIC T-P AS , S o f ia , Bu lg a ria

Rece ived Augus t 1987

Revised Janu a ry 1988

Abstract:

A genera l iza t ion of the not ion of intui t ionis t ic fuzzy ~ct is given in the spir i t of

ordinary interva l va lued fuzzy se ts . The new nat ion is ca l led interva l va lued intui t ionis t ic fuzzy

se t (IVIFS) . He re w e pre sent the bas ic pre l imina r ie s of IVIF S theory .

Keywords: Fuzzy

se t ; interva l va lued fuzzy se t ; interva l va lued intui t ionis t ic fuzzy se t ;

intui t ionis t ic fuzzy se t ; modal opera tor .

1 | n ~ d u ~ o n

A n i n t u i t i o n i s t i c f u z z y s e t ( I F S ) A , f o r a g i v e n u n d e r l y i n g s e t E i s r e p r e s e n t e d

b y a p a i r ( # a , v A ) o f f u n c t i o n s E - - * [0 , 1 ]. F o r x e E , # a ( x ) g i v e s t h e d e g r e e o f

m e m b e r s h i p t o A , v A (x ) g i ve s t h e d e g r e e o f n o n - m e m b e r s h i p . T h i s i n t e r p r e t a t io n

e n t a i l s t h e n a t u r a l r e s t r i c t i o n

C l e a r l y o r d i n a r y f u z z y s e t s ( F S ) o v e r E m a y b e v i e w e d a s s p e c i a l c a s e s o f

I F S ' s - h e r e t h e d e g r e e o f m e m b e r s h i p is t h e o n l y n e c es s a ry d a t a , i . e . o r d i n a r y

f u z z y s e t s a r e t o b e c o n s i d e r e d a s I F S ' s w i t h t h e a d d i t i o n a l c o n d i t i o n

v~ (x ) = 1 - #~ (x ) .

T h e t h e o r y o f I F S ' s i s d e v e l o p e d i n [ 1 ] . I n t h e p r e s e n t p a p e r w e b e g i n t h e

i n v e s t i g a t i o n o f a g e n e r a l i z a t i o n o f t h i s n o t i o n - t h e i n t e r v a l v a l u e d I F S ' s

( I V I F S ' s ) , b u t w e f i r s t c o n s i d e r t h e r e l a t i o n s h i p b e t w e e n I F S ' s a n d a n o t h e r

g e n e r a l i z a t i o n o f F S ' s - t h e i n t e r v a l v a l u e d F S ' s ( I V F S ' s , c f . e . g . [ 2 ] ) .

2 . I V F $ a n d I F S

A g e n e r a l i z a t i o n o f t h e n o t i o n o f f i j z z y s e t , p r o p o s e d b y s o m e r e s e a . r c h e r s i n

the a r e a in the se ve n t i e s ( c f . e . g . [2 ] ) , i s the so -c a l l e d in te rva l va lue d fuz z y se t .

H e r e w e g i v e a d e f i n i t i o n a n d e s t a b l i s h t h a t i n a s e n s e i n t e r v a l v a l u e d f u z z y s e t s

a r e a v e r si o n o f th e I F $ ( o r if y o u l i k e : - t h e o t h e r w a y a r o u n d ) .

0165-0114/89/$3.50 C~) 1989, Elsevier Sc ience P ubl ishers B.V . (No rth-H ollan d)


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344 K. Atanassov G. Gargov

D e f u ~ e n 1 . A n i n t er v a l v a l u e d f u z zy s e t A ( o v e r a b as ic s e t E ) i s g i v e n b y a

f unc t ion

MA(x)~

w h e r e

MA E--*

I N T ( [ 0 , 1 ]) , t h e s e t o f a ll s u b i n t e r v a l s o f t h e u n i t

i n t e r va l , i . e . f o r e ve r y x E, MA (x) i s a n i n t e r va l w i th in [ 0 , 1 ] .

T h e j u s t i f i c a t i o n o f t h i s g e n e r a l i z a t i o n l i e s i n t h e f o l l o w i n g o b s e r v a t i o n :

s o m e t i m e s i t i s n o t a p p r o p r i a t e t o a s s u m e t h a t t h e d e g r e e s o f m e m b e r s h i p f o r

c e r ta i n e l e m e n t s o f E a r e e x a c t l y d e f i n e d , s o w e a d m i t a k i n d o f f u r t h e r

u n c e r t a i n t y - t h e v a lu e o f

MA

i s n o t a n u m b e r a n y m o r e , b u t a w h o l e i n te r v a l. L e t

u s d e n o t e s u c h o b j e c t s b y I V F S .

D e f i n i ti o n 2 . ( a ) T h e m a p f a s s i g n s t o e v e r y I V F S A a n I F S

B ~ f A )

gb ,e n by

/~8 (x ) = in f MA ( x ), v s ( x ) = sup

MA(x).

(~ j) T h e m a p g a ss ig n s t o e v e r y I ~ B a n I V F S

A f i g B )

g ive n by

M , , x ) ffi [ ~ , , x ) , 1 - v , , x ) ] .

T h e r e l a ti o n s h ip b e t w e e n t h e t w o g e n e r a li z at io n s m e n t i o n e d a b o v e - t h e I F S

a n d t h e I V F S - i s g i v e n i n t h e f o l l o w i n g l e m m a .

I ~ m m a 1 . (a )

For every IVFS A , g ( f (A ) ) = A .

b ) Fo~ et ,e,y I F S B , f g . ~ ) ) = e .

P r o e f . ( a ) L e t A b e a n I V F S . T h e n f o r e v e r y x E ,

ffi [i n f

MA(x) ,

1 - 1 + s u , MA ( x )]

ffi M ~ (x ),

s i n c e

MA x)

s a n i n t e r v a l

( b ) L e t B b e a n I F ' S . T h e n f o r e v e r y x ~ E,

/~f(g(s))(x ) ffi in f Ms(re(x)

ffi in f [ ~a (x ) , 1 - vs (x ) ]

= ~ . x ) ,

: 8 a , x ) = 1 - s u p

M g 8~ x

= 1 - sup [ ~8 ( x ) , 1 - va ( x ) ]

= ~ , , , x ) .

T h i s s h o w s I F S a n d I V F S t o b e e q u i p o l l e n t g e n e r a l i z a ti o n s o f t h e n o t i o n o f F S .

B u t t h e d e f in i ti o n o f I F S a l lo w s a f u r t h e r g e n e r a l i z a t i o n - t o b e c o n s i d e r e d i n t h e

n e x t s e c t io n .


3/7

n terval valued intuit ionistic fu zz y sets

3 . l [~e u |e t h eo ry o f I V I F S

345

Le t a se t E be f ixe d . An in t e r va l va lue d in tuRion i s t i c f uz z y se t s ( IVIFS) A ove r

E i s a n o b j e c t h a v i n g t h e f o r m

A f fi { x , M ~ x ) . N ~ x ) ) : x G E } ,

w h e r e M ~ ( x ) c [ 0 , 1 ] a n d N , ~ ( x ) c [ 0 , 1 ] a r e in t e r v a l s a n d f o r e v e r y x E ,

sup

M ~ x ) +

sup ~ . x ) ~< 1 .

F o r e v e r y t w o I V I F S s A a n d B t h e f o ll o w i n g r e la t io n s , o p e r a t i o n s a n d

o p e r a t o r s a r e v a l i d ( b y a n a l o g y f r o m [1]):

A = B i ff (Vx e E ) ( s up M A (x )


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346 K. Atanassov G. Gargov

d ) :

CILIA ffi I-l{(x, MA(X) , [inf NA(X), 1 -- sup MA(x)])" x ~ E}

= { (x,

MA(x) ,

[ in t tinf NA(X), 1 --s up MA (x)], 1 - sup M A(X)]) : X ~ E}

ffi { (x ,

MA(X) ,

[ inf NA(X), 1 --su p MA (X)]): X ~ E}

D A ;

( e ) - ( g )

are proved analogically.

Theorem 2 .

F o r e v er y tw o I V I F S ' s A a n d B :

a ) , 3 , N B ~ ) - - A U B ;

b ) , ~

u

e ~ -

= A n B.

~ r . a ) :

= { (x, [min(inf

NA(x) ,

inf N s(x)) , ra in(sup NA(x), sup N s(x))] ,

[max(inf MA(X) , inf M s(x)) , m ax(sup M A(x), sup M s ( x ) )] ; x e E } -

= { (x, [max(inf M ~(x) , inf M s(x)) , m ax(sup MA (x), sup M s(x))] ,

[min(inf NA (x), inf N s(x )), rain(sup N A(x), sup N ~ x ) ) ] ) : x ~ E }

=AUB

( b ) is p r o v e d a n a l o g i c a l l y .

~t~.orem 3.

For every two IVIFS ' s A and B:

a )

~ A u B ) - - D A u D B ;

( b ) i - 1 ( , 4 n B ) = C I A N l i B ;

c ) 0 A u B ) = O A U O D ;

( d )

O(A

f l B ) -

OA

n

On.

~ o r . a ) :

E](A U B ) - El{ (x, [m ax(in f

MA ( x ) ,

inf

Ma ( x ) ) ,

max(sup

MA ( x ) ,

sup M a(x))] ,

[min(inf

NA(x) ,

inf N s(x)) , ra in(sup

NA(x) ,

sup

N a ( x ) )] ) : x ~ E }

= { ( x ,

[maxOnf

MA ( x ) ,

inf

Ma ( x ) ) ,

max(sup

MA ( x ) ,

sup M a(x))] ,

[inf[min(inf NA(X), inf N s ( x ) ) , rain(sup NA(x) , sup N s(x) ) ],

1 sup[m ax(inf M A(~), inf

M s ( x ) ) ,

max(sup MA ( x ) , sup M a(x)) ]]) x ~ E }

= { (x, [max(inf MA(x), inf Ma ( x ) ) , max(sup MA ( x ) , sup M a(x))] ,

[min (inf N.4(x), inf

N a ( x ) ) ,

1 - max(sup

MA(x) ,

sup M ~(x)) ] ): x E}

= { (x, [max(inf M a(x) , inf

MD(x)) ,

max(sup

MA(x) ,

sup M s(x) ) ],

[min(inf NA(x) , inf NB(x) ) ,

min~l - - sup MA(x) , 1 - - su p M s(x) )]) x ~. E }


5/7

ln,:erval valu ed intuiaonistic uz zy sets

=

{ x , M ,~ ( x ) , [ i n f N , ~ x ) , 1 - su p M , ~ ( x ) l ) x ~ E }

U { ( x , M n ( x ) , [inf N n ( x ) , 1 - sup M F,(x)])" x E}

= F 'i A U E ] B ;

( b ) is p r ov e d a na log ic a l ly ;

( d ) : F r o m ( a ) a n d f r om Th~-.o re ms l ( a , b ) a nd 2 f o l low s ~ha t:

0 , 4 n ~ ) = O A n D ) - ) - = 0 ~ u l i ) ) -

=

0 ~ u o i l y = o ~ ) - n o i l ) - = O A n O a ;

c ) i s p r o v e d a n a l o g i c a l l y

347

A n o p e r a t o r w h i c h a s s o c ia t e s t o e v e r y I V I F S a n d I F S c an b e d e f i n e d . L e t A b e

a n I V I F S . T h e n w e s e t

* A -

{ (x , inf M A ( x ) , inf N A ( x ) ) : x ~ E} .

T h e o r e m 4 . F o r e v er y I V I F S A

a ) . [ : I A = . A ;

b )

0 . A = A ;

c ) * A = * A .

P r o o f . a ) :

O A = [ x , M , , x ) , [ i n f N , , x ) , 1 - s u p M , ( x ) l ) x ~ ~ )

= { ( x , i n f M , , x ) , i n f N A ( x ) ) : x ,~. ~ }

ffi * A ;

( b ) is p r o ve d a na log ic a l ly ;

c):

~= ,{ x,~A x),MA x)):xGE}

= { (x, inf NA(X), inf M A ( x ) ) ' x ~ E )

= { x , i n f M A x ) , i n f N ~ ( x ) ) x G E } -

~A

T h e o r e m $ . F o r e v e r y t w o I V I F S ' s A a n d B :

a ) , A U B ) f , a u , a ~

(b ) , ( A N B ) - - , A N * B .

P r o o f . a ) :

*( A U B) = * { ( x , [ma x( in f M a ( x ) , i n f M n ( x ) ) , ma.~(sup M a ( x ) , s u p M n ( x ) ) l,

[min( in f

N a ( x ) ,

inf

N n ( x ) ) ,

r a in ( sup

N a ( x ) ,

sup N n( x ) ) ] ): x ~ E}

= { ( x , m a x( in f Ma ( x ) , i n f M n ( x ) ) , m i n ( i n f N A ( x ) , inf N n ( x ) ) ) x ~ E }

= { ( x , i n f M a ( x ) , i n f N a ( x ) ) ' x ~ E } U { ( x , i n f M n ( x ) ) , inf N n ( x ) ) ' x ~ E }

= . A U . B .


6/7

348

D e f m ~ e n 3 . L e t A

# ( A ) - - ( B :

v A ) = { B :

K. Atanassov G. Gargov

b e a n I F S . W e s h a l l d e f i n e :

B = ( ( x , M . ( x ) , N ~ ( x )) : x G E }

(Vx e E ) ( sup M s x ) + sup N s x ) ~ 1 )

& (Vx e

E) ( in f Ms ( x ) > ~ . ~a ( x ) &

su p N s( x ) .~- vA(:~))},

B = { x . M . x ) , ~ . x ) ) : x ~ E )

,~ V x ~ E ) s u p M s x ) + sup N . x ) ~ 1

& ( V x e E ) ( s u p

M s( x)


7/7

Interval value d intuitionistic fu zz y sets

C c D . T h e n

(V x E ) ( i n f

Mc(x)

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