PART 5Fuzzy Relations

1. Crisp and fuzzy relations2. Projections/Cylindric Ext.3. Binary fuzzy relations4. Relations on a single set5. Equivalence relations

FUZZY SETS AND

FUZZY LOGICTheory and Applications

6. Compatibility relations7. Ordering relations8. Fuzzy morphisms9. Sup-i compositions10. Inf-ωi compositions

FUZZY SETS AND

FUZZY LOGICTheory and Applications

3

Crisp/fuzzy relations

• Crisp Relation

A crisp relation represents the presence or absence of association, interaction or interconnectedness between the elements of two or more sets.

A relation among crisp sets X1, X2, ..., Xn is a subset of the Cartesian product It is denoted by R(X1, X2, ..., Xn).

.ii

Xn

N

form) ed(abbreviat )|(

) , , ,( 2121

ii

ni

nn

XiXR

XXXXXXR

nNN

4

Crisp/fuzzy relations

Using a characteristic function defines the crisp relation R :

A relation can be written as a set of ordered tuples. Another convenient way of representing a relation R(X1, X2, ..., Xn) involves an n-dimensional membership array:

otherwise

,, , , if

0

1), , ,( 21

21

RxxxxxxR n

n

otherwise

,, , , iff

0

1 21 , , , 21

Rxxxr n

iii n

].[ , , , 21 niiir R

5

Crisp/fuzzy relations

• Fuzzy relation

A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets X1, X2, ..., Xn where tuples 〈 x1, x2, ..., xn 〉 may have varying degrees of membership within the relation.

The membership grade indicates the strength of the relation present between the elements of the tuple.

6

Projections/Cylindric Ext.

• Projection

Given a relation R(X1, X2, ..., Xn), let [R↓Y] denote the projection of R on Y that disregards all sets in X except those in the family

Then, [R↓Y] is a fuzzy relation whose membership function is defined on the Cartesian product of sets in Y by the equation

). ,|| ,( nrrJ|iXJj|X ninj NN XY

).(max)]([ xRyRyx

Y

7

Projections/Cylindric Ext.

8

Projections/Cylindric Ext.

• Cylindric Extension

This operation on relations in some sense is an inverse to the projection.

Let R be a relation defined on the Cartesian product of sets in Y, and let [ R↑ X － Y ] denote the cylindric extension of R into

that are in X but are not in Y. Then,

[ R↑ X － Y ](x) = R(y)

for each x such that x y.

)( ni iX N

9

Projections/Cylindric Ext.

• Cylindric closure

The resulting relation of a relation which be exactly reconstructed from several of its projections by taking the set intersection of their cylindric extensions.

When projections are determined by the max operator, the min operator is normally used for the set intersection.

10

Projections/Cylindric Ext.

Given a set of projections of a relation on X, the cylindric closure, cyl{ Pi }, base on these projections is defined by the equation

for each where Yi denotes the family of sets on Pi is defined.

Xx

}|{ IiPi

)]([min)}({cyl xPxP iiIi

i YX

11

Projections/Cylindric Ext.

12

Binary fuzzy relations

• Domain : • Range :• Height :• Membership matrices :

) ,(max)( dom yxRxRYy

) ,(max)(ran yxRyR

Xx

) ,(maxmax)( yxRyRhXxYy

). ,( where],[R yxRrr xyxy

13

Binary fuzzy relations

• Sagittal diagram

14

Binary fuzzy relations

• Inverse relation:

The inverse of R(X, Y) is denoted R-1(Y, X).

by a membership matrix • Max-min composition

• Relational join

) ,() ,(1 yxRxyR

.)( ],[ 1111 RRR --yxr

)] ,( ), ,(min[max) ,]([) ,( zyQyxPzxQPzxRYy

)] ,( ), ,(min[) , ,]([) , ,( zyQyxPzyxQPzyxR

15

Binary fuzzy relations

16

Relations on a single set

• For crisp relations• R(X, X) is reflexive iff <x, x> R for each x X.

Otherwise, R(X, X) is called irreflexive.

If <x, x> R, R(X, X) is called antireflexive.• R(X, X) is symmetric iff <x, y>, <y, x> R for each

x, y X. Otherwise, R(X, X) is called asymmetric.

If <x, y> and <y, x> R implies x=y, then R(X, X) is called antisymmetric.

(strictly antisymmetric: If <x, y> or <y, x> R

implies )

yx

17

Relations on a single set

• R(X, X) is transitive iff <x, z> R whenever both <x, y>, <y, z> R for at least y X. Otherwise, R(X, X) is called nontransitive.

If <x, z> R whenever both <x, y>, <y, z> R, R(X, X) is called antitransitive.

18

Relations on a single set

• For fuzzy relations

19

Relations on a single set

• For fuzzy relations

20

Relations on a single set

For fuzzy relation R(X, X) • Reflexive:

irreflexive: if not for some x X.

antireflexive:

ε-reflexive:• Symmetric:

asymmetric: if not for some x, y X.

antisymmetric:

. allfor ,1) ,( XxxxR

. allfor ,0) ,( XxxxR

.10 where,) ,( xxR. , allfor ), ,() ,( XyxxyRyxR

. allfor that imples

0) ,( and 0) ,(when

Xx, yyx

xyRyxR

21

Relations on a single set

• Transitive

(or, more specifically max-min transitive):

nontransitive: if not for some members of X.

antitransitive:

.pair each for

)] ,( ), ,(min[max) ,(

2Xx, z

zyRyxRzxRYy

. allfor

)] ,( ), ,(min[max) ,(

2Xx, z

zyRyxRzxRYy

22

Relations on a single set

• Transitive closure - RT(X, X)

– For crisp relations: RT(X, X) is defined as the relation that is transitive, contains R(X, X), and has the fewest possible members.

– For fuzzy relations: the elements of the transitive closure have the smallest possible membership grades that still allow the first two requirements to be met.

23

Relations on a single set

• Algorithm of finding transitive closure

.' :Stop .3

.1 step togo and ' make ,' If .2

).(' .1

TRR

RRRR

RRRR

24

Relations on a single set

Example 5.8• Determine max-min closure RT(X, X) for a fuzzy

relation R (X, X)

08.00

004.0

1000

005.7.

R

25

Relations on a single set

Stop. .'

4.8.4.0

4.4.4.0

18.4.0

5.5.5.7.

1. step back to go '' Since

'.

4.8.4.0

4.4.4.0

18.4.0

5.5.5.7.

4.4.4.0

4.4.4.0

4.8.4.0

5.5.5.7.

1. step back to go '' Since

'.

08.4.0

4.04.0

18.00

5.05.7.

004.0

4.000

08.00

5.05.7.

1. Step

TRR

RR ,R R

RR)(RRRR

RR ,R R

RR)(RRRR

26

Equivalence relations

• Equivalence relation: A crisp binary relation R(X, X) that is reflexive,

symmetric, and transitive.

• Equivalence class: Ax is a crisp subset of X, where R(X, X) is a

equivalence relation. Ax is referred to as a equivalence class of R(X, X) with respect to x.

27

Equivalence relations

• Similarity relation:

A fuzzy binary relation R(X, X) that is reflexive, symmetric, and transitive.

• Similarity class :

A fuzzy set in which the membership grade of any particular element represents the similarity of that element to the element x.

28

Equivalence relations

Example 5.10• Let X = {1, 2 , . . . , 10}. The Cartesian product X

X ×Y contains 100 members: 〈 1, 1 〉 , 〈 1, 2 〉 , 〈 1, 3 〉 ,… , 〈 10, 10 〉 . Let R(X, X) = { 〈 x, y 〉 |x and y have the same remainder when divided by 3 }. The relation is easily shown to be reflexive, symmetric, and transitive and is therefore an equivalence relation on X. The three equivalence classes denned by this relation are:

29

Equivalence relations

A1 = A4 = A7 = A10 = {1, 4, 7, 10 },

A2 = A5 = A8 = {2, 5, 8 },

A3 = A6 = A9 = {3, 6, 9}.

Hence, in this example, X / R = { {1, 4, 7,10 }, {2, 5, 8 }, {3, 6, 9 } }.

30

Equivalence relations

• Example 5.10• The fuzzy relation R(X, X) represented by the

membership matrix is a similarity relation on

X =(a, b, c, d, e, f, g).

31

Equivalence relations

• To verify that R is reflexive and symmetric is trivial. To verify its transitivity, we may employ the algorithm for calculating transitive closures introduced in Sec. 5.4.

• If the algorithm is applied to R and terminates after the first iteration, then, clearly, R is transitive. The level set of R is ΛR = {0, .4, .5, .8, .9,1}. Therefore, R is associated with a sequence of five nested partitions π (αR), for α ΛR and α > 0. Their refinement relationship can be conveniently diagrammed by a partition tree, as shown in Fig. 5.7.

32

Equivalence relations

33

Compatibility relations

R(X, X) is a reflexive and symmetric fuzzy relation, it is sometimes called a proximity relation.

• Compatibility class : Given a crisp compatibility relation R(X, X), a compatibility class is a subset A of X such that 〈 x, y 〉 R for all x, y A.

• Maximal compatibility class : a compatibility class that is not properly contained within any other compatibility class.

34

Compatibility relations

• Complete cover : The family consisting of all the maximal compatibles induced by R on X is called a complete cover of X with respect to R.

• α-compatibility class : A subset A of X such that R(x, y) ≧α for all x, y A.

35

Compatibility relations

Example 5.11• Consider a fuzzy relation R(X, X) defined on X =

N9 by the following membership matrix:

36

Compatibility relations

• Since the matrix is symmetric and all entries on the main diagonal are equal to 1, the relation represented is reflexive and symmetric; therefore, it is a compatibility relation.

• The graph of the relation is shown in Fig. 5.8; its complete α-covers for α > 0 and α ΛR =

{0, .4, .5, .7, .8,1} are depicted in Fig. 5.9.

37

Compatibility relations

38

Compatibility relations

39

Ordering relations

• Partial ordering : A crisp binary relation R(X, X) that is reflexive, antisymmetric, and transitive is called a partial ordering.

• Minimum, maximum, minimal, maximal

40

Ordering relations

• Lower bound, greatest lower bound: Let X be a set on which a partial ordering is

defined, and let A be a subset of X(A X). If x X and x y for every y A, then x is called a lower bound of A on X with respect to the partial ordering.

If a particular lower bound succeeds every other lower bound of A, then it is called the greatest lower bound, or infimum, of A.

41

Ordering relations

• Upper bound, least upper bound: If x X and y x for every y A, then x is called

an upper bound of A on X with respect to the relation.

If a particular upper bound precedes every other upper bound of A, then it is called the least upper bound, or supremum, of A.

• Lattice: A partial ordering on a set X that contains a

greatest lower bound and a least upper bound for every subset of two elements of X is called a lattice.

42

Ordering relations

• Hasse diagram: Each element of X is expressed by a single node

that is connected only to the nodes representing its immediate predecessors and immediate successors. The connections are directed in order to distinguish predecessors from successors; the arrow ← indicates the inequality

. Diagrams of this sort are called ≦ Hasse diagrams.

43

Ordering relations

Example 5.12• Three crisp partial orderings P, Q, and R on the

set X = {a, b, c, d, e} are defined by their membership matrices (crisp) and their Hasse diagrams in Fig. 5.10. The underlined entries in each matrix indicate the relationship of the immediate predecessor and successor employed in the corresponding Hasse diagram. P has no special properties, Q is a lattice, and R is an example of a lattice that represents a linear ordering.

44

Ordering relations

45

Ordering relations

• Fuzzy partial ordering : A fuzzy binary relation R on a set X is a fuzzy

partial ordering iff it is reflexive, antisymmetric, and transitive under some form of fuzzy transitivity.

• Dominating class:

• Dominated class:

. where) ,()(][ XyyxRyR x

. where) ,()(][ XyxyRyR x

46

Ordering relations

• Undominated:

• Undominating:. and allfor

.0) ,(

iff dundominate is element An

yxXy

yxR

Xx

. and allfor

.0) ,(

iff ngundominati is element An

xyXy

xyR

Xx

47

Ordering relations

• Fuzzy upper bound :

on.intersectifuzzy eappropriatan denotes where

,) (

by

defined and ) (by denotedset fuzzy theis for

thedefined, is ordering partial

fuzzy aon which set a of subset crisp aFor

][

xAx

RAR,U

AR,UA

r boundfuzzy uppeR

XA

48

Ordering relations

• Least upper bound :

).,( ofsupport in the elements allfor

0) ,( and 0))( ,(

such that ),(in element unique

theisit exists, set theof a If

ARUy

yxRxARU

ARUx

Ar boundleast uppe

49

Ordering relations

Example 5.13• The following membership matrix defines a fuzzy partial

ordering R on the set X = {a, b, c, d, e}:

• The columns of the matrix give the dominated class for each element. Under this ordering, the element d is undominated, and the element c is undominadng.

50

Ordering relations

• For the subset A ={a, b}, the upper bound is the fuzzy set produced by the intersection of the dominating classes for a and b. Employing the min operator for fuzzy intersection, we obtain

• The unique least upper bound for the set A is the element b. All distinct crisp orderings captured by the given fuzzy partial ordering R are shown in Fig. 5.11. We can see that the orderings became weaker with the increasing value of α.

51

Ordering relations

52

Fuzzy morphisms

• Homomorphism :

If two crisp binary relations R(X, X) and Q(Y, Y) are defined on sets X and Y, respectively, then a function h : X → Y is said to be a homomorphism from 〈 X, R 〉 to 〈 Y, Q 〉 if

for all x1, x2 X.

,)( ),( implies , 2121 QxhxhRxx

53

Fuzzy morphisms

• Strong homomorphism:

. )( and )( where, , allfor

, implies ,

and , , allfor

)( ),( implies ,

21

211

121

2121

21

2121

yhxyhxYyy

RxxQyy

Xxx

QxhxhRxx

54

Fuzzy morphisms

55

Fuzzy morphisms

56

Fuzzy morphisms

• Isomorphism :

If h : X → Y is a homomorphism from 〈 X, R 〉 to 〈 Y, Q 〉 , and if h is completely specified, one-to-one, and onto, then it is called an isomorphism.

57

Sup-i compositions

• Sup-i composition :

Given a particular t-norm i and two fuzzy relations P(X, Y) and Q(Y, Z), the sup- i composition of P and Q is a fuzzy relation on X x Z denned by

for all

)] ,( ), ,([sup) ,]([ zyQyxPizxQPYy

i

. , ZzXx

58

Sup-i compositions

• Basic properties

59

Sup-i compositions

60

Sup-i compositions

• i-transitive: We say that relation R on X2 is i-transitive iff

for all x, y, z X.

• i-transitive closure: When a relation R is not i-transitive, we define its

i-transitive closure as a relation RT(i) that is the smallest i-transitive relation containing R.

)] ,( ), ,([) ,( zyRyxRizxR

61

Sup-i compositions

• Theorem 5.1 For any fuzzy relation R on X2, the fuzzy relation

is the i-transitive closure of R.

• Theorem 5.2 Let R be a reflexive fuzzy relation on X2, where |

X| = n 2. Then,≧ RT(i)=R(n-1).

1

)()(

n

niT RR

62

Inf-ωi compositions

• Operation ωi :

Given a continuous t-norm i, let

for every a, b [0, 1].

}) ,(|]1 ,0[sup{) ,( bxaixbai

63

Inf-ωi compositions

• Inf- ωi composition

. , allfor

])( ),([inf) ,)( (

equation by the defined

is )( and )(relation fuzzy of , n,compositio

-inf the,operation associated theand norm- aGiven

ZzXx

y, zQx, yPzxQP

Y, ZQX, YPQP

it

iYy

ii

i

i

64

Inf-ωi compositions

• Theorem 5.3 For any a, aj, b, d [0, 1], where j takes values

from an Index set J, operation ωi has the following properties:

65

Inf-ωi compositions

66

Inf-ωi compositions

• Theorem 5.4

Let P(X, Y), Q(Y, Z), R(X, Z), and S(Z, V) be fuzzy relations. Then:

67

Inf-ωi compositions

• Theorem 5.5

Let P(X, Y), Pj(X, Y), Q(Y, Z), and Qj(Y, Z) be fuzzy relations, where j takes values in an index set J. Then,

68

Inf-ωi compositions

• Theorem 5.6 Let P(X, Y), Q1(Y, Z), Q2(Y, Z), and R(Z, V) be

fuzzy relations. If Q1 Q2, then

69

Inf-ωi compositions

• Theorem 5.7 Let P(X, Y), Q(Y, Z), and R(X, Z) be fuzzy

relations. Then,

70

Exercise 5

• 5.1

• 5.4

• 5.6

• 5.11

• 5.19

• 5.20