Fuzzy Relations on Sets and Fuzzy Sets

6 FUZZY RELATIONS AND FUZZY GRAPHS

6.1 Fuzzy Relations on Sets and Fuzzy Sets

Fuzzy relations are fuzzy subsets of Xx Y, that is, mappings from X~Y.They have been studied by a number of authors, in particular by Zadeh [1965, 1971], Kaufmann [1975], and Rosenfeld [1975]. Applications of fuzzy relations are widespread and important. We shall consider some of them and point to more possible uses at the end of this chapter. We shall exemplarily consider only binary relations . A generali zation ton-aryrelations is straightforward.

Definition 6-1

Let X, Y~ ~ be universal sets; then

R

^={«x,y), llii(X,y»l(x, y)EXX Y}

is called a fu zzy relation on X x Y.

Example 6-1

Let X= Y= ~and

R:

="considerably larger than." The membership function of the fuzzy relation, which is, of course, a fuzzy set on X x Y, can then be

H.-J. Zimmermann, Fuzzy Set Theory - and Its Applications

© Kluwer Academic Publishers 2001

{

o

^forx:5:Y

~ii(X,y)=

^(x-y) fory<x:5:11y lOy

1 for x>lly A different membership function for this relation could be

{ o

^forx:5:y

~ii(X,y)= z-I

(1+(y - x

r )

^{for x>y}

For discrete supports, fuzzy relations can also be defined by matrixes .

Example 6-2

Let X

=

{XI. xz, X3} and Y

=

{YI. Yz, Y3, Y4}

Yz

R

⁼"x considerably larger than y":Xz

.8 I .1 .7

0 .8 0 0

.9 I .7 .8

and

yz

XI

i

⁼"y very close to x" : Xz

.4 0 .9 .6

.9 .4 .5 .7

.3 0 .8 .5

In definition 6-1 it was assumed that~Rwas a mapping from X x Yto [0, 1]; that is, the definition assigns to each pair (x, y) a degree of membership in the unit interval. In some instances, such as in graph theory, it is useful to consider fuzzy relations that map from fuzzy sets contained in the universal sets into the unit interval. Then definition 6-1 has to be generalized [Rosenfeld 1975].

Definition 6-2 LetX, Yk ~ and

A

= {(x,IlA(x»lxEX},

B

= {(y, Ilii(y»!yEY} , two fuzzy sets.

ThenR= {[(x, y), IlR(X, Y)]I(x, y) E XX Y} isa j uzzy relation on

A

^and

B

^if

Ilk(X, y)::;IlA(x),V(x,y)EXXY and

Ilk(X, y) ::;Ilii(y),V(x,y)EXXY.

This definition will be particularly useful when defining fuzzy graphs: Let the elements of the fuzzy relation of definition6-2be the nodes of a fuzzy graph that is represented by this fuzzy relation. The degrees of membership of the elements of the related fuzzy sets define the "strength" of or the flow in the respective nodes of the graph, while the degrees of membership of the corresponding pairs in the relation are the "flows" or "capacities" of the edges. The additional require- ment of definition6-2(llii(X, y)::;min{Il,.i(x), Illi(Y)})then ensures that the "flows"

in the edges of the graph can never exceed the flows in the respective nodes.

Fuzzy relations are obviously fuzzy sets in product spaces. Therefore set- theoretic and algebraic operations can be defined for them in analogy to the def- initions in chapters 2 and 3 by utilizing the extension principle.

Definition 6-3

LetRand

Z

be two fuzzy relations in the same product space. Theunion/inter- sectionofRwith

Z

is then defined by

lliiuZ(X, y )= max{llk(x,y),Ilz(x, y)},(x,y)EX XY Ilknz(x, y)=min{llk(x, y),Ilz(x, y)},(x,y)EXXY

Example 6-3

LetRand

Z

be the two fuzzy relations defined in example 6-2.The union ofR and

Z,

which can be interpreted as "x considerably larger or very close to y," is then given by

R U Z :Xz

yz

.8 1 .9 .7

.9 .8 .5 .7

.9 1 .8 .8

The intersection of Rand

i

is represented by

YI Yz

R

n z.

^Xz

.4 0 .1 .6

0 .4 0 0

.3 0 .7 .5

So far, "min" and "max" have been used to define intersection and union. Since fuzzy relations are fuzzy sets, operations can also be defined using the alterna- tive definitions in section 3.2. Some additional concepts, such as the projection and the cylindrical extension of fuzzy relations, have been shown to be useful.

Definition~

Let

R

⁼ {[(x,y), ~R(X,^y)]

I

(x, y) E XX Y}be a fuzzy binary relation . The first projection of

R

is then defined as

R<I)

={(x, max~Rex,

y»lex,

y)EXXY}

y

The second projection is defined as

R<Z)={(y,maxujt,e,

y»lex,

y)EXXY}

.r

and the total projection as

R(T)=max maxlu sfx,

y)lex,

y)EXX Y}

x x

Example~

Let

R

be a fuzzy relation defined by the following relational matrix . The first, second , and total projections are then shown at the appropriate places below.

Y3 Ys

First projection [IlR(1)(X)]

.1 .2 .4 .8 I .8

.2 .4 .8 I .8 .6

.4 .8 I .8 .4 .2

Second projection:

.4 .8 .8 I

Total projection

The relation resulting from applying an operation of projection to another rela- tion is also called a "shadow" [Zadeh 1973a]. Let us now consider a more general space, namely,X = XI

x ...

XXn ; and let

R

qbe a projection on Xii

x ...

XXi"

where (i" .. . ,h) is a subsequence of(l, . .. ,n).Itis obvious that distinct fuzzy relations in the same universe can have the same projection. There must, however, be a uniquely defined largest relation

R

qL(XI> .. .,Xn ) withIlR_qL(XiI' . ..,Xi,)for each projection. This largest relation is called the cylindrical extension ofthe pro- jection relation.

Definition 6-5

R

qLk X is the largest relation in X of which the projection is

R

q,

R

qLis then called the cylindrical extension of

R

qand

R

qis the base of

R

qL•

Example 6-5

The cylindrical extension ofR(2)(example _~) is

Ys

.4 .8 I I I .8

.4 .8 I I I .8

.4 .8 I I I .8

Definition 6-6

- - -

LetR be a fuzzy relation onX =X I X ..• X X; andRI andR2be two fuzzy pro- jections onX I x . ..XX , andX,x . ..XXn, respectively, with s~r+I and

R

^IL,

R

u their respective cylindrical extensions.

_ The meet of

R

^{I and}

R

²is then defined as

R

^{I L}

n R

^u.and their join as

R

^{IL U}

«:

6. 1.1 Compositions of Fuzzy Relations

Fuzzy relations in different product spaces can be combined with each other by the operation "composition." Different versions of "composition" have been sug- gested, which differ in their results and also with respect to their mathematical properties. The max-min composition has become the best known and the most frequently used one . However, often the so-called max-product or max-average compositions lead to results that are more appealing.

Definition 6-7

- -

Max-min composition: Let RI(x,y), (x,y)E XX Yapd R2(y,z),

0',

^{z)E YX Z}

be two fuzzy relations. The max-min composition RI max-min R2is then the fuzzy set

R_I⁰ R₂={[(x ,z), maxlrninju j, (x,y),flii 2(y,z)}}]lxEX,yEY,zEZ}

y

fl R,oR2is again the membership function of a fuzzy relation on fuzzy sets (defini- tion 6-2).

A more general definition of composition is the "max-* composition."

Definition 6-8

~et

R

^{I and}

R

2be defined as in definition 6-7. The max-" composition of

R

^{I and}

R²is then defined as

RI~R2={[(x,z), max(flii, (x,y)*flii 2(y ,z))]lx^{E X,}yEY,z^EZ}

y

If * is an associative operation that is monotonically nondecreasing in each argu- ment, then the max- * composition corresponds essentially to the max-min com - position. Two special cases of the max- * composition are propo sed in the next definition.

Definition 6-9

[Rosenfeld 1975]: Let

R,

and

R

z, respectively, be defined as in definition 6-7.

Themax-prod compo sition

R,

^?

R

z and themax-QV compo sition

R.

a~

R

z are then defined as follows:

R,

^?

Rz(x,

^{z)=max[jl RI(x,y)' IlR2(y ,}

z)lx

^{E X ,YEY,zEZ]}

y

R,

a~

Rz(x,

z)=

+.

^{max[jlRI(x,y )' IlR2(y ,}

^z)lx

^{EX,YE Y,zEZ]}

1

Example 6-6

- -

Let RI(x, y) and R z(y, z) be defined by the following relational matrixes [Kaufmann 1975, p. 62]:

Ys

.1 .2 0 1 .7

.3 .5 0 .2 I

.8 0 1 .4 .3

Ys

.9 0 .3 .4

.2 1 .8 0

.8 0 .7 1

.4 .2 .3 0

0 1 0 .8

We shall first compute the min-max-composition

R,

⁰

R

z(x, z). We shall show in detail the determination forx =xls

z

=

z,

and leave it to the reader to verify the total results shown in the matrix at the end of the detailed computations. We first perform the min operation in the minor brackets of definition 6-7:

Letx=xlsz=zlsandY=Yi,i= 1, . . . , 5:

min{lliiI(x" Yt), IlR2(Y"

z,n

= min{.I, .9} =.1 min{ll iiI(x" yz ), IlR²(yz,

zln

= m in{.2 , .2} =.2 min{ll iiI(XI,Y3),Il ~(Y3,

z,n

= min{O, .8} = 0 min{ll iiJ(x" Y4), IlR2(Y4,

zln

= min{I,.4} = 4 min{ll iiJ(x" Ys),IlR2(Ys,

z,n

= min{.7, O} = 0

R

^I0

Rz(x"

^z.)=

«x" z,),

IlR,oR2^{( X I ,}

z,»

=

«x"

z.),max{.I , .2, 0, .4, O}) =

«x"

z.),.4)

Inanalogy to the above computation we now determine the grades of member- ship for all pairs (Xj, 2),i= 1, ... , 3,j = 1, . .. , 4 and arrive at

2z

.4 .7 .3 .7

.3 1 .5 .8

.8 .3 .7 1

For the max-prod, we obtain

x=x.,2=^2.,Y=Yi,i= 1, . . . ,5:

IlRI (Xl,YI)' Il R2(YI,21)=.1·.9 = .09 IlRI(XI,yz)' Il R2(yz,21)=.2· .2 = .04 IlRI(XI, Y3)'IlR2(Y3,21)

=

0 ·.8

=

0

IlRI(XI,Y4)'IlR2(Y4,21)=1·.4=4

IlRI(XI, Y5)' Il R2(Y5,21)= .7 ·0 = 0 Hence

R,

^?

k,

(XI,21)=«XI,21),(IlRl oR2(X" 21»)

=«XI, 21),max{.09, .04, 0,.4,O})

=«XI, 21)'.4)

After performing the remaining computations, we obtain

2z

.4 .7 .3 .56

.27 1 .4 .8

.8 .3 .7 1

The max-av composition finally yields

- - - - Il(X.,

yJ

+Il(y j, 2,)

1 1

2 .4

3 .8

4 1.4

5 .7

Hence

X'

^max{llk,_y ^(Xl>y;)+11hz (Yi,Zl )}=

X'

^{(1.4)= .7}

Zz

.7 .85 .65 .75

.6 I .65 .9

.9 .65 .85 I

6. 1.2 Properties of the Min-Max Composition

(For proofs and more details see, for instance, Rosenfeld 1975.) Associativity. The max-min composition is associative, that is,

(R

₃⁰

R

_z)⁰

R

₁=

R

₃⁰

(R

_z⁰

R,).

Hence

RIo RI oR

^{I =}R~,and the third power of a fuzzy relation is defined . Reflexivity

Definition 6-10

Let

R

be a fuzzy relation in X x X.

1.

R

is calledreflexive [Zadeh 1971] if

llk(x, x)=1V^X EX 2.

R

is callede-reflective [Yeh 1975] if

llk(x, x)~^EV^XEX 3.

R

is calledweakly reflexive [Yeh 1975] if

llk(x, y)

s

llk(x, x)}^\-I X

v X,YE . llk(Y, x)~11k(x, x)

Example 6-7

Let X= {X., XZ, X3, X4} andY= {y.,Yz, Y3, Y4}'

The following relation"y is close tox"is reflexive:

Yz

R:^Xz

I 0 .2 .3

0 I .1 I

.2 .7 I .4

0 I .4 I

IfR^I and

k,

are reflexive fuzzy relations, then the max-min compositionRI oRz is also reflexive.

Symmetry Definition 6-11

A fuzzy relationRis calledsymmetric ifR(x,y)= R(y, x) 'ifx,yE X.

Definition 6-12

Arelation is calledantisymmetric if for

x

*

^y either IlR(X, y) *Il R(y,x ) }

'ifX,Y E X or IlR(X,y)=IlR(X,x)=0

[Kaufmann 1975, p. 105].

A relation is calledperfectly antisymmetric if for x

*

^{y whenever}

IlR(X, y)>0 then IlR(y,x)=0'ifx,Y^EX [Zadeh 1971].

Example 6-8

Xz

.4 0 .1 .8

.8 I 0 0

0 .6 .7 0

0 .2 0 0

.4 0 .7 0

0 I .9 .6

.8 .4 .7 .4

0 .1 0 0

.4 .8 .1 .8

.8 I .0 .2

.1 .6 .7 .1

0 .2 0 0

R

^I is a perfectly antisymmetric relation, while

R

2is an antisymmetric, but not perfectly antisymmetric relation.

R

3is a nonsymmetric relation, that is, there exist x,y E X with IlR(X,y) ;tIlR(y, x), which is not antisymmetric and therefore also not perfectly antisymmetric.

One could certainly define other concepts, such as an n-antisymmetry (1IlR(x,y) - IlR(y,x)1 _~aVx, yE X).These concepts would probably be more in line with the basic ideas of fuzzy set theory. Since we will not need this type of definition for our further considerations, we will abstain from any further definition in this direction .

Example 6-9

Let X and Y be defined as in example 6-8. The following relation is then a sym- metric relation:

R(x, y):X2

0 .1 0 .1

.1 1 .2 .3

0 .2 .8 .8

.1 .3 .8 1

Remark 6-1

For max-min compositions, the following properties hold:

1. It

R

^I~s refle~iveand

R

2is an arbitrary fuzzy relation, then

RIo R

2 :2

R

2and R_{2oR 1} :2R_2•

2. If

R

is reflexive, then

R s R ° R.

3. If

R

^{I and}

R

2are reflexive relations, so is

RIo R

2•

4. If

R

1and

R

2are symmetric, then

RIo R

2 is symmetric if

RIo R

2

= R

2°

R

I •

5. If

R

is symmetric, so is each power of

R.

Transitivity Definition6-13

A fuzzy relation

R

is called (max-min) transitiveif

Example 6-10

Let the fuzzy relation

R

be defined as

X2

R:X2

.2 1 .4 .4

0 .6 .3 0

0 1 .3 0

.1 1 1 .1

ThenR

°

^R is

X2

X2

.2 .6 .4 .2

0 .6 .3 0

0 .6 .3 0

.1 1 .3 .1

Now one can easily see thatlliioii(X,y)~llii(X,y)holds for allx,y E X.

Remark 6-2

Combinations of the above properties give some interesting results for max-min compositions :

1. If

R

is symmetric and transitive, thenllii(X,y) ~^llii(X,x) for allx,y E X.

2. If

R

is reflexive and transitive, then

R

⁰

R

=

R.

3. If

R

^{I and}

R

2are transitive and

RIo R

2=

R

2⁰

R

^I>then

RIo R

2is transitive.

The properties mentioned in remarks 6-1 and 6-2 hold for the max-min compo- sition.For the max-prod composition, property 3 of remark 6-2 is also true but not properties 1 and 3 of remark 6-1 or property 1 of remark 6-2. For themax- av composition,properties 1 and 3 of remark 6-1 hold as well as properties 1 and 3 of remark 6-2. Property 5 of remark 6-1 is true for any commutative operator.