On the Shadow of a Fuzzy Set

Copyright © IFAC Fuuy lnfonnation Marseille, France, 1983

ON THE SHADOW OF A FUZZY SE1 Liwen Pei and Mian Ouyang Department of Mathematics, Wuhan University, Wuhan, Hubei, People's Republic of China

Abstract. This paper is mainley concerned with certain properties of the ~hclvw of a fuzzy set proposed in [21, [3]. At the same time, several correc.ions are made for some results in [2] and formulae for the decomposition of a fuzzy set into the cylinder of the shadow are discussed in detail. The necessary and sufficient condition for the equality of the fuzzy sets are also discussed. Keywords. set theory; fuzzy set; convex fuzzy set; convex hull; shadow of fuzzy set; cylinder of shadow; optimization; fuzzy restraint.

The shadow concept of a fuzzy set is very im-

led orthogonal complementary point-shadow.

portant in the optimization problem with fuz-

Complementary point-shadow and point-shadow

zy restraint.

have similar property as follows:

Definition 1.

If A, B are fuzzy sets, then we have

Let A be a fuzzy set in Rn.

Let H be a hyperplane in Rn,

P.

~ Rn.

(1)

A fuz-

zy set S(A) in Rn is referred to be point-

K HO, 1),

shadow of a fuzzy set A from Po to H, if the

C( KA) - KC ( A) ,

Here KI. is a fuzzy set: }1KA(X)=K...(JA(x). (2) Monotonicity.

membership function of S(A) is as follows: suP)JA(x)

Homogeneity.

hfH

"~L

.uS(A) (h)

z

(1)

{

o

h~

(3) Distributivity and subdistributivity.

H

Here L shows linking line of p. and h.

C(A U B);;! C(A) U C(B)

From

C(AnB)

now on, S(A) is used to express a point-shadow.

=

C(A)n C(B).

(4) Nilpotentiality.

Definition 2.

Let C(A) be a fuzzy set.

C"(A) .. c( C(A) ) =et>

The

membership function of C(A) is as follows: inf J..1 (x)

llC(A)(h)= {

A

"~L

(5) Keep concavity.

hEH

If A is a concave fuz-

zy set in Rn, then orthogonal complementary

(2)

shadow of A is a concave fuzzy set in H.

o

Definition

Here L expresses linking line of h and

(6)

~

•

3.

Let A be a fuzzy set in Rn.

Let H be the hyperplane in Rn.

S"

Shows a

So that C(A) is called the complementary

orthogonal shadow of A to H.

point-shadow of a fuzzy set A from P. to H.

a orthogonal complementary shadow of A to H.

Therefore, on H the complementary point-sha-

If \;/

dow of A to H equals a complementary When Pa

=~,

Rn

fuzzy JJ.S*(x) ·.uS(A)(x*)

set of point-shadow of the complementary fuzzy set of A to H.

X f

CH expresses

.uC*(x) -'uC(A)(X*)'

L.LH, S(A)

is called orthogonal shadow and C(A) is cal-

Here x*is a orthogonal projection point of 433

Liwen Pei and Mian Ouyang

434

x to H.

The fuzzy sets

S*, C*

are

the cylined fuzzy sets generated from

called

----

SH'C~,

Clearly

T ..

u (8)

..uC*(x) = inf }J.A(h) ","L lt

Lx

Here

is line which is perpendicular to

H through x.

So that

(9)

UC .. *!;;AC{\S,,*

"

" Theorem 1.

If A is a strictly convex fuzzy

set, then (10 ) If A is a strictly co ncave fuzzy set, then

)( c" * = A ;2 Qs" * Proof:

is inside of

r.. ,

t ly small E > 0, then

Xo

When Xo

( 11 )

i;;+t {][ I AlA (x)~ c("'E}

E r:. . . ~

,

here

is a strictly convex

set. When

According to the known conclusion,

take a sufficien-

Rn =

R~,

i t is shown by the followine

figure.

in order to prove (10) it is sufficient to prove

A2~ S,,*, that is )J. A(x )?, )J.

n "

S". (x) = i~f(.uS ,,* (x))( 12)

So that we want only to prove that: VXoE Rn , :1 ding

H,,_,

rot

such that for correspon-

S,,:.

Let 'uA (xo) =cx < 1,

s ince A is a strictly

convex fuz<.y set, hence

r..

a strictly convex set.

lfuen

boundary of~ , through perplane

1T

={x I.uA(x)~O(} Xo

Xo

of supporting r;

is at the

we make a

,

II

is

hy-

Through

Hence

except for

,u A(xo) =0( •

Owing to

with

IT

in Rn, and

,

on the herperplane

B>

0

i s arbitrary, we obtai n

H".

is perpendicalar, we make orthogonal

projection, from A to obtain a fuzzy cylinder

H••

so that we can

S,,:.

When ~ =1, (12 ) is clear.

Because line

L (which is a perpendicalar li ne with HA" ) Xo 1 id in the hyperplane 1T , so we have

Generally, relati onsh ip "!;." (10) can't be changed into (in the same way, at (11) can't be changed into

Thus

IT

and one side of the herperplane .

such that

on one side oflT , atlT "uA(X )(ot ,

H".

we can make a hyperplane

such that ).lA (xo )~ O(Tf

AlA(x)
We make a hyperplane

Xo

..

A ={\ S,,*.

on the left of "="

relationship

"2"

on the right

"=").

Using the similar method we can prove (11), and also we can use (1 0) to prove (11) • Because

A=]R.'uA(x)/x

set, so that vex fuzzy set.

AS]

Rn

is a concave fuzzy

(1-J.1 (x ) )/x A

is a con-

435

On the Shadow of a Fuzzy Set Owing to keep convexity of convex, and

A=(' S:(A), A

SA(A):J

=j

sup

j

= H"

SA(A)

also

fixed hyperplane, this conclusion is not correct.

but

Remark 2.

(1-'uA(h))/x

H" h.~L" H~

SA'

The fuzzy set of theorem 1

be strictly convex (or concave).

(1- inf)l A(h))/x

Without

"strictly" condition, the conclusion may be

h~l."

not correct.

(1 - )l C (A) (x) )/x

3.

Remark

= C~(A)

(13)

then

(14 )

Remark 4.

If the fuzzy set A is not convex,

QS:

may not be convex. The convex hull of a fuzzy set A

(that is a mininum convex fuzzy set

therefore

ding A)

A= UC,,"

"

Inference 1.

If A,B

are two arbitrary

rictly convex fuzzy sets in

st-

Remark

may be not strictly convex.

5.

Whether A is a ordinary set

nn,

by convA.

**

S""(A)

'v'"

=

(16 )

SA "(B)

are two arbitrary strictly

concave fuzzy sets in

A,R

Suppose that

are two

fuzzy

Here

SeA),

S(B)

aretwo

point-shadows.

Rn, Remark 6.

then A= B

or a

sets, from convA= convB, we may not obtain S(A)=S(B).

Similarly, if A,B

inclu-

fuzzy set, the convex hull of A is denoted

then

A= B

must

** 'v'A

C~ "(A)

=

From what the arbitrary point-sh-

adow of fuzzy set

C.t(B)

A,B

is equal, we may not

obtain convA = convB. Here

S,,"

is a cylinder fuzzy set of a C~

thogonal shadow;

or-

is a cylinder fuzzy

Due to the space forbids, those examples have been omitted, useing them explains these

set of a orthogonal complementary shadow. In

remarks above mentioned.

this case there exists decomposition as follLet A be a fuzzy set in

Definition 4.

ows: A-B=0S: convex

fu~zy

A,E

and suppose the corresponding cutting

are the strictly

of orbitrary

(1A)

sets

A=B=~C:

A,B

is a convex closed set, then A is called a

are the strictly

concave fuzzy sets

set

0/ (O<"'~I)

(19)

convex closed fuzzy set.

If

roe

is a concave

closed set, then A is called a concave Inference 2.

If

A,B

are two arbitrary

closed fuzzy set.

strictly co nvex fuzzy sets in Rn , and HII

H:LI

then

•••••• ,

A= B

shadow in

<=~

Rn

Hn

are n coordinate planes,

for an arbitrary point-

A,B

C(A) = C(E) Here a point-shadow is made from corresponding coordinate plane.

then we have A =V C,,*

In (2) the conditions of the

for n coordinate planes, it is only for one

(21 )

A

Proof:

Because A is a convex closed

set,

V

set in

Rn.

take

cX~(

f,~0

O,1},

Let

~

is a convex closed O~

'uA(x O ) =01",

such that

fuzzy

0(. T

0(.

<

1,

£ <. 1, and

is a convex closed set, and Since

theorem 2 of the fourth chapter § 2 are not

the distance from than zero.

FIKR-O

(20)

A

for an arbitrary point-

shadow we have

Remark 1.

A=ns:

Rn, then

A = B <==>

If A is a convex closed fuzzy set

Rn, then we have

If A is a concave closed fuzzy set in

are two arbitrary strictly concave

fuzzy sets in

in

we have

SeA) = S(B) If

Theorem 2.

Xo

rO(o T t.

is closed,

to r",.-t-t. is longer

So that through

Xo we can make

436

Liwen Pei and Mian Ouyang

a hyperplane n , such that not be intersected with

A-A S,,·

and on the hyperplane we have 'uA(x)
Proof:

Because,U A(x)

MD

Rn.

*(X O)
AS"

£)

Owing to

°

is arbitrary, we can obtain

is a closed set in

From convexity (or concavity), according

to theorem 2 we can obtain this decomposition.

4.

Theorem From

is continuous,

(0
r..

arbitrary

so that

(22)

(or

"

On one side of the hyperplane,

rot_Tt

Let

A,B

be two arbitrary fuzzy

sets, and suppose'uA(x), .LlB(x)

(9),

we have tinuous,

are

con-

"l2-~)J A(x) =,,~~..u. B(x) - 0,

then

All point-shadows according to (9) it

As for

,uA(x O ) = 1,

clear.

Therefore, we all have

is

convex hull convA = convB

SeA) = S(B) =*

j

All complementary point-shadows C(A) = C(B) =~

conca're kernel

concave fuzzy set including A)

(maxinum concA = concB.

To repeat inference from (13) to (15), we can know that if A is a concave closed fuzzy set,

Proof:

(21) will be also correct. Inference 1.

If

A,B

are two arbitrary con-

vex closed fuzzy sets in

convA =

1 A conv r.. j 0'"

convB

1 B conv r.. J0'"

=

Rn, then

(23)

here A = B 4=~

S~

V"

(A)

=

S: (B)

I:tA _ {XI

and t here exists decomposition

If

A,B

Because

are two concave closed fuzzy set, A = B 4=~ Y 1\

C: (A) - C,,· (B),

conv

r:.

B

Inference 2.

If

A,B

,uA(x),

are continuous,

,uB(x)

are two closed sets.

.. ~kxk

conv r""A={ x I x=~,

are arbitrary two conRn,

}

conv

r:.A,

are their convex hull.

and there exists decomposition

vex closed fuzzy sets in

r.,B = { x I )l B ( x ) ~ 0(

r..B

r.;A,

then

.tlA (x)~O(}

B

'"

convf: ={xlx=L:Al x ~=I

then

n n

I

,

'" L.)..l 'I1=l

n

= 1,

B

).1n~ 0, xnE-\": , n=1, ••• , m}

for any point-shadow in relatio n to n coordinate planes always have SeA ) = S(B ) If

A,B

then ~~

XO

E conv r.;B,

pl-

:.3)J..°, x o n n

ane always have =

'" 0 LJ..l = n _::.I

C(B)

1,

that Theorem

3.

If A is a convex (or concave

but

O

e: convr,:A.

o xn

1'\=1

since is continuous in Rn, then we have decomposi-

one of

( n=1,

€or:B

m ) n= 1, ••• , m )

such

'" "").10 x o =L x0 •

fu-

zzy set in Rn, and membership function t!A(x)

tion

X

Since

for any complementary

point-shadow in relation to n coordinate

C(A)

conv r:/= convr: •

Because otherwise, at least exist a point

are two co ncave closed fuzzy sets, A-B

B

so that we obtain

XO

n

n

~ conv r;."A, there exists at least x~, x~, ••• , x~

(denoted by

xo ) g

On the Shadow of a Fuzzy Set O

x E= conv CA

such that XO

(": otherwise

g

~ conv r..A ).

So that we can make a XO

perplane through

hy-

such that will not

g

be

intersected with convex hull, and have

11 A(x) < 0( on the hyperplane and on one side of the hyperplane.

Since

Hm }J-A(x) - 0, we

x ..... 00

have

SA(A)<<< for the hyperplane's pointB shadows. But x;E-C , hence S,,(B)~o( •

This shows that there exists at least one point-shadow such that

S(A)= S(B), that

contradictory to known conditions.

is

Hence

convA = convB. After proving the first conclusion, we can obtain the second conclusion. Since

C(A)

=

C(B),

so that

s(i) - s(ii) conv(i)- conv(B) conv(i) = conv(B) concA= concB

References 1.

Frederick, A. Valentine Convex Sets.

2.

Mizumoto, M.

(1964).

New York. (1970-1973).

algebra and its application. Sciences,

3.

Fuzzy Mathematical

8-11.

Zadeh, L.A. and Control,

(1965). ~,

Fuzzy Sets. Inform.

338-353.

437