FUZZY SETS
AND
FUZZY LOGIC
PART 5
Fuzzy Relations
Theory and
Applications
1. Crisp and fuzzy relations
2. Projections/Cylindric Ext.
3. Binary fuzzy relations
4. Relations on a single set
5. Equivalence relations
FUZZY SETS
AND
FUZZY LOGIC
Theory and
Applications
6. Compatibility relations
7. Ordering relations
8. Fuzzy morphisms
9. Sup-i compositions
10. Inf-ωi compositions
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  X i . It is
iN
denoted by R(X1, X2, ..., Xn).
n
R ( X 1 , X 2 ,  , X n )  X 1  X 2   X n
R( X i | i  N n )   X i
iN n
(abbreviated form)
3
Crisp/fuzzy relations
Using a characteristic function defines the crisp
relation R :
1 if x1 , x2 , , xn  R,
R( x1 , x2 , , xn )  
0 otherwise
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: R  [ri1 , i2 , , in ].
ri1 , i2 , , in
1 iff x1 , x2 , , xn  R,

0 otherwise
4
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.
5
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
Y  X j| j  J  Nn  (X  X i|i  Nn , | J | r, r  n).
Then, [R↓Y] is a fuzzy relation whose
membership function is defined on the Cartesian
product of sets in Y by the equation
[ R  Y ]( y)  max R( x).
x y
6
Projections/Cylindric Ext.
7
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 X i (i  Nn )
that are in X but are not in Y. Then,
[ R↑ X － Y ](x) = R(y)
for each x such that x  y.
8
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.
9
Projections/Cylindric Ext.
Given a set of projections {Pi | i  I } of a relation
on X, the cylindric closure, cyl{ Pi }, base on
these projections is defined by the equation
cyl{Pi }( x)  min[ Pi  X  Yi ]( x)
iI
for each x  X where Yi denotes the family of
sets on Pi is defined.
10
Projections/Cylindric Ext.
11
Binary fuzzy relations
•
•
•
•
Domain : dom R( x)  max R( x, y )
yY
Range : ran R ( y )  max R ( x, y )
x X
Height : h R( y )  max max R( x, y )
yY
xX
Membership matrices :
R  [rxy ], whererxy  R( x, y).
12
Binary fuzzy relations
• Sagittal diagram
13
Binary fuzzy relations
• Inverse relation:
The inverse of R(X, Y) is denoted R-1(Y, X).
R 1 ( y, x)  R( x, y)
by a membership matrix R1  [ryx1 ], (R-1 )-1  R.
• Max-min composition
R( x, z )  [ P  Q]( x, z )  max min[ P( x, y ), Q( y, z )]
• Relational join
yY
R( x, y, z)  [ P  Q](x, y, z)  min[P( x, y), Q( y, z)]
14
Binary fuzzy relations
15
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 x  y )
16
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.
17
Relations on a single set
• For fuzzy relations
18
Relations on a single set
• For fuzzy relations
19
Relations on a single set
For fuzzy relation R(X, X)
• Reflexive: R( x, x)  1, for all x  X .
irreflexive: if not for some x  X.
antireflexive: R( x, x)  0, for all x  X .
ε-reflexive: R( x, x)   , where 0    1.
• Symmetric: R( x, y)  R( y, x), for all x, y  X .
asymmetric: if not for some x, y X.
antisymmetric: when R( x, y )  0 and R( y, x)  0
imples that x  y for all x, y  X .
20
Relations on a single set
• Transitive
(or, more specifically max-min transitive):
R( x, z )  max min[R( x, y), R( y, z )]
yY
for each pair x, z  X .
2
nontransitive: if not for some members of X.
antitransitive:
R( x, z )  max min[R( x, y), R( y, z )]
yY
for all x, z  X 2 .
21
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.
22
Relations on a single set
• Algorithm of finding transitive closure
1. R'  R  ( R  R).
2. If R'  R, make R  R' and go to step1.
3. Stop : R'  RT .
23
Relations on a single set
Example 5.8
• Determine max-min closure RT(X, X) for a fuzzy
relation R (X, X)
.7 .5
0 0
R
 0 .4

0 0
0
0
0
.8
0
1
0

0
24
Relations on a single set
Step 1.
.7 .5 0 .5
.7 .5 0 .5
 0 0 .8 0 
 0 0 .8 1 
 R  (R  R)  
  R '.
RR  
 0 0 0 .4
 0 .4 0 .4




0
.
4
0
0
0
.
4
.
8
0




Since R '  R , R  R ' go back tostep1.
.7 .5 .5 .5
.7
 0 .4 .8 .4
0
 R  (R  R)  
RR  
 0 .4 .4 .4
0



0
.
4
.
4
.
4


0
Since R '  R , R  R ' go back tostep1.
.7
0

0

0
.5 .5 .5
.4 .8 1 
 R'  R T .

.4 .4 .4

.4 .8 .4
.5 .5 .5
.4 .8 1 
 R '.

.4 .4 .4

.4 .8 .4
Stop.
25
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.
26
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.
27
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:
28
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 } }.
29
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).
30
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 Λ
R and α >
0. Their refinement relationship can be
conveniently diagrammed by a partition tree, as
31
shown in Fig. 5.7.
Equivalence relations
32
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.
33
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.
34
Compatibility relations
Example 5.11
• Consider a fuzzy relation R(X, X) defined on X =
N9 by the following membership matrix:
35
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 Λ
R = {0, .4,
.5, .7, .8,1} are depicted in Fig. 5.9.
36
Compatibility relations
37
Compatibility relations
38
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
39
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.
40
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.
41
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.
42
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.
43
Ordering relations
44
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:
R[ x] ( y)  R( x, y) where y  X .
• Dominated class:
R[ x] ( y)  R( y, x) where y  X .
45
Ordering relations
• Undominated:
An elementx  X is undominated iff
R( x, y )  0.
for all y  X and x  y.
• Undominating:
An elementx  X is undominating iff
R( y, x)  0.
for all y  X and y  x.
46
Ordering relations
• Fuzzy upper bound :
For a crisp subset A of a set X on whicha fuzzy
partialorderingR is defined, the fuzzy upper bound
for A is thefuzzy set denotedby U ( R, A) and defined
by
U ( R, A)   R[ x ] ,
xA
where  denotesan appropriate fuzzy intersection.
47
Ordering relations
• Least upper bound :
If a least upper boundof theset A exists,it is the
unique elementx in U ( R, A) such that
U ( R, A)(x)  0 and R( x, y )  0
for all elementsy in thesupportof U ( R, A).
48
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.
49
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.
50
Ordering relations
51
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
x1, x2  R implies h( x1 ), h( x2 )  Q,
for all x1, x2  X.
52
Fuzzy morphisms
• Strong homomorphism:
x1 , x2  R implies h( x1 ), h( x2 )  Q
for all x1 , x2  X , and
y1 , y2  Q implies x1 , x2  R
for all y1 , y2  Y , where x1  h 1 ( y1 ) and x2  h 1 ( y2 ) .
53
Fuzzy morphisms
54
Fuzzy morphisms
55
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.
56
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
i
[ P  Q](x, z )  sup i[ P( x, y), Q( y, z )]
yY
for all x  X , z  Z .
57
Sup-i compositions
• Basic properties
58
Sup-i compositions
59
Sup-i compositions
• i-transitive:
We say that relation R on X2 is i-transitive iff
R( x, z)  i[ R( x, y), R( y, z)]
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.
60
Sup-i compositions
• Theorem 5.1
For any fuzzy relation R on X2, the fuzzy relation

RT (i )   R
(n)
n 1
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).
61
Inf-ωi compositions
• Operation ωi :
Given a continuous t-norm i, let
i (a, b)  sup{x [0, 1] | i(a, x)  b}
for every a, b  [0, 1].
62
Inf-ωi compositions
• Inf- ωi composition
Given a t - normi and theassociatedoperationi , theinf - i
i
composition, P  Q, of fuzzy relationP( X, Y) and Q(Y, Z) is
defined by theequation
i
( P  Q)(x, z )  inf i [ P( x, y), Q( y, z)]
yY
for all x  X , z  Z .
63
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:
64
Inf-ωi compositions
65
Inf-ωi compositions
• Theorem 5.4
Let P(X, Y), Q(Y, Z), R(X, Z), and S(Z, V) be
fuzzy relations. Then:
66
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,
67
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
68
Inf-ωi compositions
• Theorem 5.7
Let P(X, Y), Q(Y, Z), and R(X, Z) be fuzzy
relations. Then,
69
Exercise 5
•
•
•
•
•
•
5.1
5.4
5.6
5.11
5.19
5.20
70