Chapter 5 : Fuzzy Relation
5.1 Crisp and Fuzzy Relation
Crisp relation:
+ degrees or strengths of relation
Fuzzy relation
。Cartesian product :
 X i  {( x1 , , xn ) | xi  X i , i  N n }
iNn
Nn  {1,2, , n}
a subset of  X i
。n-ary relation:
i.e.,
iNn
R( X 1 , X 2 , , X n )  X 1  X 2  X n
|
|
a set
the universal set
Characteristic function:
1 if ( x1 , x1 ,  xn )  R
 R ( x1 , x1 , xn )  
otherwise
0
。Binary, Ternary, Quaternary, Quinary, n-ary
relations
· Representation of a relation


R( X1 , . . nX. ,

)

( ri1 ,i2 ,..,in )
ri1 ,i2 ,..,in
: n-D membership array
=
1
0
iff
( x1 ,..., xn )  R
otherwise
○ Example 5.1 :
 
R ( X Y ) Z )Y1Y2Y3Y4 Z 2 Z1Z 3 Z 4 Z 5







R ({ X 1 , X 3 } , X i J  N nY  X

{ X i | j  J  N n }  R2 





1
1
1
1
0.8
0.8
0.8
0.8







X , a,* X , a,$ Y , b,* Y , a,$ X , b,* X , b,$ Y , b,* Y , b,$
 y  { X 1 , X 3}
Y j  X j j  J


[ R  X  Y ] :[ R2  { X 1 , X 3}]








 y  { X 2 }, X  Y  { X 1 , X 3}  {(*,*), ( x,$), (Y ,*), (Y , s)}

( x)  R ( y )  [ R  { X i }](Y )  max R( x) Rij

x y

 Y j | j  J  X X j
jJ

1 0.7 0.4 0.8
R2,3 



.
a,* a,$ b,* b,$
0.9
0.4
1
0.7
0.8




X , a,* X , b,* Y , a,* Y , a,$ Y , b,$
0.9
0.4
1
0.8
0.9
0
1
0.8
 R1,2 



R1,3 



X , a X ,b Y, a Y,b
X ,* X ,$ Y ,* Y ,$
1
0
0.6
0.9
0.7
0  R( x1 , x2 ,..., xn )  1





( NY , Beijing ) ( NY , NY ) ( NY , London) ( Paris, Beijing ) ( Paris , NY ) ( Paris

X

={English , French} ,

Y
={ dollar , pound , franc ,
mark}
Z={US , France , Canada , Britain , Germary}
 
R( X Y Z )

={(English , dollar , US) , (French , franc ,
France)
(English , dollar , Canada) , (French ,
dollar , Canada)
(English , pound , Britain)}
Y1
Dollar
1
Dollar
Y2
Pound
0
0
0
Mark
Mark
0
0
Z2
Z3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Can
Z4
Can Brit Ger
Brit Ger
Z1
Z1
0
0
1
0
0
0
US Fran
US Fran
0
1
0
0
0
1
0
0
Franc
0
0
0
Franc
Y4
1
0
Pound
Y3
0
Z5
Z2
Z3
Z4
Z5
English
Franch
X1
X2
· Fuzzy Relations
Cartesian Product :
tuples :






X1 X 2  X n
( x1 , x2 ,..., xn )
membership grade :
0  R( x1 , x2 ,..., xn )  1
Example 5.2: Birary relation R : represents the
concept “ very far”

X


Y
= { New York , Paris}
={Beijing , New York , London}
Relation in list notation
 
R( X Y )

=
1
0
0.6
0.9
0.7
0.3





( NY , Beijing ) ( NY , NY ) ( NY , London) ( Paris, Beijing ) ( Paris, NY ) ( Paris, Londom)
Relation in membership array
NY
Paris
Beijing
1
0.9
NY
0
0.7
0.6
0.3
London
· Ordinary fuzzy relation
with valuation set [0,1]
L-fuzzy relation
With ordered valuation set L
5.2 Projection and Cyclindric Extensions

· set family X =
{X i | i  Nn }

Let X =
 xi | i  Nn 
Let Y =
 Yj | j  J 
Where

 X Xi
jJ


 X Xj
jJ

,|J|=r
J  Nn
Y a subsequence of X ,
iff
YX
Y j  X j j  J
⊙ Projection :






R( X1 , X 2 , , X n )

Y={X

[ R  y]
i
[ R  y]
the projection of R on Y
: a relation
| j  J  Nn }
: a fuzzy relation (set)
[ R  y](Y )  max R( x)
x y
※ max can be generalized by other t-conorms
· Example 5.3



R( X 1 , X 2 , X 3 )




X1



2

3
0.4
1
0.7
0.8
= X0.9




, a,* X , b,* Y , a,* Y , a,$ Y , b,$


Let R = [ R  {X , X }] ,
ij
 R1,2 

={X,Y}, X ={a,b}, X ={*,$}

i

j

Ri  [ R  { X i }]

0.9
0.4
1
0.8



X , a X ,b Y, a Y,b
R1,3 
0.9
0
1
0.8



X ,* X ,$ Y ,* Y ,$
R2,3 
1 0.7 0.4 0.8



a,* a,$ b,* b,$
R1 
0.9 1

*
y
R2 
1 0.8

a b
1 0.8
R3  
* $
⊙Cyclindric Extension
[R  X  Y ]
the CE of R into
X-Y
X-Y : sets

Xi

that are in X but are not in Y
[ R  X  Y ]( x)  R ( y )
R: a relation defined on Y
·Example 5.4 ( Refer to example 5.3)

Let X =


{ X 1 , X 2 , X 3}


And R= R




 y  { X 1 , X 3}
1,2
∴ X-Y =


= {*,$}
X3

From example 5.3

∴ [R  X  Y ]  [R
1,2  { X 3 }]

0.9
0.4
1
0.8



X , a X ,b Y, a Y,b
R1,2 
=
0.9
0.9
0.4
0.4
1
1
0.8
0.8







X , a,* X , a,$ X , b,* X , b,$ Y , a,$ Y , a,$ Y , b,* Y , b,$


[ R  X  Y ]:[ R12  { X 3}]




Consider
2 } ][ R23  { X 1}]


[ R2  { X1 , X 3}]

[ R1 3 {X





[ R3  { X1 , X 2 }]
[R  X  Y ]






= [R




2  { X 1 , X 3}]
 y  { X 2 }, X  Y  { X1 , X 3}  {(*,*),( x,$),(Y ,*),(Y , s)}
{x,y} {x,$}
R= R
2
∴ [R

1 0.8

a b




2  { X 1 , X 3}]




[ R1  { X 2 , X 2 }]
=
1
1
1
1
0.8
0.8
0.8
0.8







X , a,* X , a,$ Y , b,* Y , a,$ X , b,* X , b,$ Y , b,* Y , b,$
5-7
Cyclindric closure
-A relation may be exactly reconstructed from
several of its projections by taking the set
intersection of their cyclindric extensions
Pi | i  I :a
set of projections of a relation on X
 cyl{Pi }( X )  min [ Pi  X  Yi ]( X )  R
iI
Yi
：The family of sets on which
Pi
is defined.
‧ Example：
Cyl{R1, 2 , R1,3 , R2,3 } 
0.9
0.4
1
0.7
0.4
0.8





x, a, x, b, y, a, y, a,$ y, b, y, b,
Refer to the original relation
_
_
_
R( X , X , X )
 1
 2
 3
in example
5.3.
It is not fully reconstructable from its projections
become of ignoramus of
R1
, R ,and
2
R3
.
5-8




5.3. Binary Relations R( X , Y )








X Y
X Y
：bipartite graph
：directed graph
‧ Representations
i, matrices
R  [ rij ]
, where
ii, sagittal diagrams
Examples：
i)
ii)
y1
y2
1
y3
y4
x1
.9
0
0
0
x2
0 .4 0
0
0
x3
0
0
1 .2
0
x4
0
0
0
0 .4
x5
0
0
0
0 .5
x6
0
0
0
0 .2
y5
rij  R ( xi , y j )
5-9
‧ Domain：dom R
Crisp – dom R =
{x  X | ( x, y )  R, y  Y }
Fuzzy – dom R(x) =
max R( x, y)
yY
The domain of a fuzzy relation R(x,y) is a fuzzy set
on X; dom R(x) is its membership function.
e.g. dom R( X ) = max(0.9, 1) = 1
1
‧ Range：ran R
Crisp – ran R =
{ y  Y | ( x, y )  R, x  X }
Fuzzy – ran R(y) =
max R( x, y )
xX
e.g. ran R( y ) = max(0.4, 0.5, 0.2) = 0.5
5
‧ Height： h(R)  max max R( x, y)
yY
e.g.,
h( R )  1
xX
normal fuzzy relation
5-10
‧ Inverse： R
1
(Y , X )
R 1 ( y, x)  R( x, y)
 R 1  RT , ( R 1 ) 1  R
e.g.
0.3 0.2
R   0
1 
0.6 0.4
0.3 0 0.6
R 1  R T  

0.2 1 0.4
‧ Composition： R( X , Z )  P( X , Y )  Q(Y , Z )
R( x, z)  [ P  Q]( x, z)  max min[ P( x, y), Q( y, z)]
yY
Max-min composition
Properties：
P。Q  Q。P


1
1
1
 ( P。Q)  Q 。P
( P。Q。
) R  P。(Q。R)

Matric form： [r ]  [ p
ij
Where
ik
。
] [qkj ]
[rij ]  max min( pik , qkj )
k
5-11
R( x, z)  [ P。Q]( x, z)  max [ P( x, y‧) Q( y, z)]
yY
max-product composition
matrix form
Where
[rij ]  [ pik 。
] [qkj ]
[rij ]  max ( pik , qkj )
‧ Example
0.3 0.5 0.8 0.9 0.5 0.7 0.7
0.0 0.7 1.0 。0.3 0.2 0.0 0.9



0.4 0.6 0.5 1.0 0.0 0.5 0.5
Max-min =
Max-prod =
 0 .8 0 . 3 0 .5 0 .5 
1.0 0.2 0.5 0.7 


0.5 0.4 0.5 0.6 
0.8 0.15 0.4 0.45
1.0 0.14 0.5 0.63


0.5 0.2 0.28 0.54
‧ Relational join： R( X , Y , Z )  P( X , Y ) * Q(Y , Z )
R( x, y, z )  [ P * Q]( x, y, z )  min[ P( x, y ), Q( y, z )]
※The max-min composition can be obtained by
aggregating appropriate elements of the
corresponding join.
5-12
‧ Example
※[P。Q]( x, z)  max[P * Q]( x, y, z)
yY
5.4Binary Relation on a Simple Set
‧ Representations
5-14
◎characteristic Properties (Crisp case)
i,
reflexive
ii,
irrflexive

antiflexive

symmetric
asymmetric

antisymmetric
( x, y )  R, ( y, x)  R  x  y
strictly antisymmetric
iii,
transitive
nontransitive

x  y, ( x, y )  R
or
( y, x)  R
antitransitive

◎ Fuzzy Relations
i,
reflexive
---
x
,
irreflexive ---
x
,
R ( x, x )  1
antiflexive ---
x
,
R ( x, x )  1
 -reflexive
,
---
x
R ( x, x )  1
R ( x, x )  
5-14
ii, symmetric
asymmetric
--
x, y,
--
x, y,
antisymmetric --
R ( x, x )  R ( y , x )
R( x, x)
 R ( x, x )  0

 R( y, x)  0
iii, max-min transitive
--
R( y, x)
x y
x, z
R( x, z )  max
min  R( x, y ), R( y, z ) 

yY
max-product transitive --
x, z
R( x, z )  max
min  R( x, y )  R( y, z ) 

yY
nontransitive
--
x, z
R( x, z )  max
min  R( x, y ), R( y, z ) 

yY
antitransitive --
x, z
R( x, z )  max
min  R( x, y ), R( y, z ) 

yY
◎ Example 5.7

R:very near
reflexive, symmetric, nontransitive
5-15
Crisp:
equivalence;
Fuzzy:
similarity
Quasi-equiv
alence
Compatibilit
y or
Tolerance
Partial
ordering
Preordering
or
Quasi-orderi
e
metric
Transitiv
ic
Antisym
xive
Symmetr
Antirefle
Reflexive
◎Summary
ng
Strict
ordering
Figure3.6 Some important types of binary
relation R(X,X)
◎ transitive closure:

RT ( X )

Algorithm for computing
1.
RT
R /  R  ( R R)
2. If
R/  R
3. Stop,
Where
:
, Let
R/  R
, go to step1
RT  R /
component-wise max
5-16
◎Example 5.8
0.7 0.5 0.0 0.0 
 0.0 0.0 0.0 1.0 

R
 0.0 0.4 0.0 0.0 


 0.0 0.0 0.8 0.0 
0.7 0.5 0.0 0.5
 0.0 0.0 0.8 0.0 

R R
 0.0 0.0 0.0 0.4 


 0.0 0.4 0.0 0.0 
0.7 0.5 0.0 0.5
 0.0 0.0 0.8 1.0 
  R/
R  ( R R)  
 0.0 0.4 0.0 0.4 


 0.0 0.4 0.8 0.0 
Step1:
Step2:
R/  R
, Let
R  R/
repeat step1
0.7
 0.0
R R
 0.0

 0.0
0.7 0.5
 0.0 0.4
R  ( R R)  
 0.0 0.4

 0.0 0.4
Step3:
R/  R
0.5 0.5 0.5
0.4 0.8 0.4 
0.4 0.4 0.4 

0.4 0.4 0.4 
0.5 0.5
0.8 1.0 
 R/

0.4 0.4

0.8 0.0 
, Let
R  R/
repeat step1
0.7
 0.0
R'  
 0.0

 0.0
Step4: Stop
0.5 0.5 0.5
0.4 0.8 1.0 
R
0.4 0.4 0.4 

0.4 0.8 0.4 
RT  R /
5-17
5.5
Fuzzy Equivalence Relation
◎Crisp binary relation
equivalence: reflexive, symmetric, and
transitive
equivalence classes
partition: X/R
◎ Example 5.9:

X  1, 2,





R( X  X ) 
,10
{ ( x, y) | x, y have the same remainder when
divided by 3}
R: reflexive, symmetric, transitive
 equivalence
partition

X / R  (1, 4, 7,10), (2,5,8), (3, 6,9)

5-18
◎Fuzzy Binary Relation
。 Fuzzy
Similarity relation
equivalence relation
Similarity classes
equivalence classes
2 Interpretations of a similarity relation:
1.Group similar elements into crisp classes
whose members are similar to each other
to some specified degree.
2.

x  X ,

associate a fuzzy set
defined on
。a fuzzy relation

X

Ax
.
(Theorem 2.5,
 R
R
 [0,1]
Eqs.(2.1)(2.2))
If R: Similarity relation,
 R :
。Let
equivalence relation
 (  R) :
the partition of
  ( R )   (R ) | 
 0 ,

X

w.r.t.

1
nested, i.e.,
(  R ) :
a redefinement of
(  R ) :
 (  R)
iff

R
Prove that :A fuzzy relation
relation, then

R
R: X  X
is a similarity
is a equivalent relation
Pf : ∵ R : a similar relation
∴
R:
reflexive, i.e.,
x  X , R ( x, x)  1
symmetric, i.e.,
x, y  X , R( x, y )  R( y, x)
transitive, i.e.,
x, z  X 2 , R( x, z )  max[ R( x, y), R( y, z )]
yY
i,

R
: reflexive
x  X , R( x, x)  1   [0,1],( x, x)   R

ii,

R
R
: reflexive
: symmetric
∵ R : symmetric
x, y  Z , R( x, y )  R( y, x)
Let
R ( x, y )  R ( y , x )  
Then
iii,

R
 
or
 
a, if
 
=>
( x, y),( y, x)   R
b, if
 
=>
( x, y),( y, x)   R
: transitive
∵ R : transitive
x, z  X 2 , R( x, z )  max[ R( x, y), R( y, z )]
yY
Let
R( x, y )  1
Assume
Then
a. if
,
R( y, z )   2
1  2
  1  2
  1  2
=>
,
1    2
, or
( x, y)  R,( y, z )   R
1  2  
--- (A)
R( x, z)  max[ R( x, y), R( y, z)]  min[ 1, 2 ]  1  
yY
 R( x, z )   ,( x, z )   R
(A) , (B) =>
b.
if

--- (B)
R : transitive
1    2
 ( x , y )  R , (y  ,z )
c.
if
, Rdon’t care
( x, z )
, don’t care
( x, z )
1  2  
 ( x, y)   R,( y, z )   R
Example 5.10 :
R( X , X )
: a fuzzy relation
R : reflexive , symmetric ,
transitive (
R '  R  ( R R)  R
)
∵ level set :
R  {0.0, 0.4, 0.5, 0.8, 0.9,1.0}
There are five nested partition

 's
 The similarity class for each element is a fuzzy
set defined by the row of the membership matrix
corresponds to that element
Example : see Example 5.10
For c :
0 0 1 0 1 0.9 0.5
    

a b c d e f
g
For e :
0 0 1 0 1 0.9 0.5
    

a b c d e f
g
∴ c and e are similar at any level

5.6 Compatibility Relations ---- reflexive ,
symmetric
compatibility
Alternatives :
tolerance
relation
proximity
 Crisp case :
Maximal compatibility classes – not properly
contained within any other compatibility class
Complete cover – all the maximal
compatibility classes
 Fuzzy case :
α-compatibility class ---- a subset A of X ,
s.t.
x, y  A, if R( x, y )  R( x, y )   , R :
fuzzy compatibility
relation
maximalα ---- compatibility classes
completeα-cover
Example 5.11 :
R( X , X )
: a fuzzy relation
∵ R : reflexive , symmetric
∴ a compatibility relation
∵
R  {0.0, 0.4, 0.5, 0.7, 0.8,1.0}
=> the completeα-covers
5-25
  0.5
0.7 0.9

b
d
a  1 0.7
b  0
1
c 0.5 0.7

d 0
0
e  0 0.1
e.q.
0 1 0.7 
0 0.9 0 
1 1 0.8 

0 1
0 
0 0.9 1 
1 1 0 0 0 0 0 0
1 1 0 0 0 0 0 0

0 0 1 1 1 0 0 0

0 0 1 1 1 1 1 0
0.5
R  0 0 1 1 1 1 1 1

0 0 0 1 1 1 0 0
0 0 0 1 1 0 1 0

0 0 0 0 1 0 0 1
0 0 0 0 0 0 0 0


0
0 
0

0
0

0
0

0
1 
 x  y  xx  yy  xx  X ( x, y )  XS  {x1 , x2 }  Xy  Ax  y
(x,y)y  X ( x  y, or  y  xAx  XA  Xx  yR[ x ] ( y )  R ( y , x )
x  U ( R, A)( x) 
1
1

0

0
0.4
R  0

0
0

0
0

x A
R[ x ]y 
1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 
0 1 1 1 0 0 0 0

0 1 1 1 1 1 0 0
0 1 1 1 1 1 1 0

0 0 1 1 1 1 0 0
0 0 1 1 1 1 0 0

0 0 0 1 0 0 1 0
0 0 0 0 0 0 0 1 
  0.4
Look for complete subgraphs
(1,2) , (3,4,5),(4,5,6,7),(5,8),(9)
(34,),(4,5,6),(4,5,7),(3,5),(5,6)
(4,5),(5,6,7),(4,6,7),(4,6),(6,7)

maximal compatible classes (the complete
0.4-cover):
(1,2),(3,4,5),(4,5,6,7),(5,8),(9)
These do not partition X.
5-26
e.q.
  0.5
1
1

0

0
0.5
R  0

0
0

0
0


1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 
0 1 1 1 0 0 0 0

0 1 1 1 1 1 0 0
0 1 1 1 1 1 1 0

0 0 1 1 1 0 0 0
0 0 1 1 0 1 0 0

0 0 0 1 0 0 1 0
0 0 0 0 0 0 0 1 
maximal compatible classes
( The complete 0.5-cover)
(1,2),(3,4,5),(4,5,6),(4,5,7),(5,8),(9)
5-27
5.7. Ordering Relations
˙ partial ordering: reflexive , antisummetric ,
transitive
X Y
: X : predecessor
precedes Y : successor
if exist
First member : if
x y
y  X
Last member : if
yx
y  X
(minimum)
unique
(maximam)
may not
Minimal member : if
be unique Maximal member : if
˙properties :
1, if

, at most one first member
y xx y
x yx y
if

, at most one last member
2, There may be several maximal and minimal
member
3, if

a first member X ,

only one minimal
member Y exists and x=y
4, if

a last member x ,

only one maximal
member Y exists and x=y.
5, partial
member
ordering
member
the first member

the last
inverse
the last member
partial ordering

the first
5-28
※ In a partial ordering , it does not guarantee that
(x,y)
,
( x  y, or
If
,


yx
).
(x,y) : comparable (total ordering)
Otherwise (x,y) : non comparable
˙
A X
If
x X
, and

x: lower bound of A on X
If ……… ,

y  A
,
x y
,
x y
x : upper bound of A on X
˙greatest lower bound ( or infimum ) GLB
- a lower bound which succeeds every other lower
bound
Least upper bound ( or supermum ) LUB
- a upper bound which preceeds every other upper
bound
˙Lattice – A partial ordering on X contains GLB
and LUB ,
S  {x1 , x2 }  X
5-29
˙ Connected – a partial ordering is said to be
connected
iff
x, y  X
,
x y

x<y or y>x
˙ Linear ordering (total ordering , simple ordering ,
complete ordering )
- when a partial ordering is connected , then
( x , y )
: comparable
˙ Hasse diagrams – representing partial orderings in
which

indicates

˙ Example 5.12 : Crisp partial orderings
5-30
˙ Fuzzy partial ordering
- reflexive , antisymmetric , and transitive under
some form of transitivity.
※ any fuzzy partial ordering can be resolved into a
series of crisp partial ordering .
i.e. taking a series of  cut that produce increasing
levels of refinement
˙ In a fuzzy partial ordering , R
x  X
R[ x ]
, two fuzzy sets are associated with
: dominating class
R[ x ] ( y )  R( x, y )
R[ x ]
: dominated class
R[ x ] ( y )  R( y, x)
5-31
˙ x undominated
iff R(x,y) = 0
y  x
X undominating iff R(y,x) = 0
˙ Fuzzy upper bound for
U ( R, A) 
xA
A X
y  x
is a fuzzy set
R[ x ]
※ If a least upper bound of A exists , it is the
unique element
x  U ( R, A)
s.t. 1,
2,
U ( R, A)( x)
y 
>0
R(x,y) > 0 ,
support [ U(R,A) ]
˙ Example 5.13
a
c
d
e
Fuzzy partial ordering
R:
b
a  1 0.7 0 1 0.7 
b  0
1 0 0.9 0 
c 0.5 0.7 1 1 0.8 


d 0
0 0 1
0 
e  0 0.1 0 0.9 1 
1. row : dominating class for each element
column : dominated class for each element
2. d : undominated , C : undominating
3. For A = {a,b} , U(R,A) = the intersection of
The dominating classes of a and b =
4,
LUB(A) =b
0.7 0.9

b
d
5-32
5. Crisp ordering captured by the fuzzy ordering
e.g.
  0.5
1
0


R  1

0
0
1 0 1 1
1 0 1 0
1 1 1 1

0 0 1 0
0 0 1 1
# is → 2 3 1 5 3
※The ordering become weaken with the increasing
α
5-33
Fuzzy preordering – reflexive and transitive
Fuzzy weak ordering –
i, an ordering satisfying the proportion of a fuzzy
total ordering except antisymmetry.
ii, a fuzzy preordering in which
x  y
, either
R(x,y)>0 or R(y,x)>0
Fuzzy strict ordering –
Antireflexive
Antisymmetric
Transitive
5.8. Morphisms
‧ Crisp homomorphism h from (X,R) to (Y,Q)
Where R(X,X), Q(Y,Y)：binary relations
( x1 , x2 )  R  (h( x1 ), h( x2 ))  Q
‧ Fuzzy homomorphism h
If R(X,X), Q(Y,Y)：Fuzzy binary relations
And
R( x1 , x2 )  Q[h( x1 ), h( x2 )]
※ It’s possible that a relation
( x1 , x2 )  R
(h( x1 ), h( x2 ))  Q
which
.
※ If this is never the case h is called a strong
homomorphism.
5-34
‧ Crisp strong homomorphism h
If
( x1 , x2 )  R  (h( x1 ), h( x2 ))  Q
And
( y1 , y2 )  Q  (h 1 ( y1 ), h 1 ( y2 ))  R
※ where h : many to one →
h 1 ( y)
Xs
‧ Fuzzy strong homomorphism h
H imposes a partition
h
on X
Let
A  {a1 , a2 , , an }
B  {b1 , b2 , , bn }   h
R,Q：fuzzy relations
h : strong homomorphism
iff
max ( R(ai , b j ))  Q( y1 , y2 )
i, j
1
 h(ai )ai  A
2
 h(b j )b j  B
where  yy

contains a set of
5-35
‧ Example 5.14
R(X,X)
0
0 0.5 0
0 0 0.9 0 

R
1 0
0 0.5


0
0 0.6 0
Q(Y,Y)
0.5 0.9 0 
Q   1
0 0.9
 1 0.9 0 
→h：ordinary fuzzy homomorphism (one way)
 R( x1 , x2 )  Q(h( x1 ), h( x2 ))  strong
But
i,e,
Q( ,  )  0.9
( ,  )  Q
R(d , c)  0
while
(d , c)  R
where
h( d )   , h(c )  
5-36
R(X,X)
0
0
0
0.8 0.4 0
 0 0.5 0 0.7 0
0 

0
0 0.3 0
0
0


0 0.9 0.5
 0 0.5 0
0
0
0
1
0
0


0
0
0
1 0.8
 0
Q(Y,Y)
 0 .7 0 0 .9 
 0 .4 0 .8 0 


 1
0
1 
→h：strong fuzzy homomorphism (two way)
5-37
※Q represents a simplification of R
‧ Isomorphism : (congruence)
h:1-1, onto
 X Y
Endomorphism : (subgraph)
h:X→Y,
YX
Automorphism :
Isomorphism and End Endomorphism
i.e.m X=Y nad R=Q
5-38
5.9 SUP-i Compositions of Fuzzy Relations
Generalize max-min Composition
i : t-norm
sup : t-conorm
‧ P(X,Y), Q(Y,Z)：fuzzy relations
i
P o Q( X  Z )
:sup-i composition
i
[ P o Q]( X , Z )  sup i[ P( x, y ), Q( y, z )]
yY
‧ Properties
i
i
i
i
1.
( P o Q ) o R  P o( Q o R )
2.
P o( Q j )  ( P o Q j )
3.
P o( Q j )  ( P o Q j )
4.
( Pj ) o Q  ( Pj o Q)
5.
( Pj ) o Q  ( Pj o Q)
6.
( P o Q) 1  Q 1 o R 1
i
i
j
j
i
i
j
j
i
j
i
j
i
j
i
j
i
i
5-39
i
3. Show Eq.(5.16), i.e.,
jJ

Where

P( X , Y )


and
i
Qj ) 
P (
Q(Y , Z )
( P Q j ),
jJ
are fuzzy relations.
 i 
 P Q  ( x, z )  sup i  P( x, y ), Q( y, z )
yY
pf. From Eq.(5.13), i.e.,
 i



  P ( Q j )  ( x, z )  sup i  P( x, y), Q j ( y, z ) 

jJ
jJ



yY 
Let
Q
Qj
jJ
 Q  Q1 , Q  Q2 ,
i.e.,
, Q  QJ
Q( y, z)  Q1 ( y, z),
( y, z ),
, Q( y, z)  Q J ( y, z)
i is monotonically increasing
i[ P ( x, y ), Q j ( y , z )]  i[ P ( x, y ), Q1 ( y , z )]

jJ

 ...........

i[ P ( x, y ), Q j ( y , z )]  i[ P ( x, y ), Q J ( y , z )]
jJ

Q j )( y, z )] 
i[ P( x, y),(
jJ


i[ P( x, y), Q j ( y, z )]
jJ
 sup i[ P( x, y ), (
Q j )( y, z )]  sup

jJ
yY
( x ,y )
i[ P( x, y ), Q j ( y, z )]
yY jJ
s u ipP [x y( Q, j y) z, ( x ,y ) y] z,

( ,
jJ yY
i
 i



  P ( Q j )  ( x, z )   ( P Q j )  ( x, z ), ( x, z )
jJ


 jJ

i,e.,
i
i
Qj ) 
P (
jJ
(P Qj )
jJ
) , (
,
)
5-40
。Sup-i composition monotonically increases
i
i
P Q1  P Q2    (5.20)
i.e.,
i
i
if
Q1 P  Q2 P    (5.21)
。Identity of
i
1



0
1, x  y
E ( x, y )  
0, x  y
i
i.e.,
Q1  Q2
0 


1
i
E PP EP
。Relation R on
iff
 2
X

: i-transitive

R( x, z )  i  R( x, y ), R( y, z )  , x, y, z  X

i
R RR
。i-transitive closure
RT ( i )
--- The smallest i-transitive relation containing R
。Theorem 5.1: R: any fuzzy relation
 RT (i ) 

, where
R(n)
n 1
i
R ( n ) R R n(
1 )
5-41
By (5.15) (5.17)
proof:
i,
i

i 
 
RT (i ) RT (i )   R ( n )   R ( m)  
 n 1
  m1
 n 1


R(k ) 
k 2
i.e.,
RT ( i )


m 1

 ( n) i ( m) 
R 
R ( n m)
R

 n,m1
R ( k )  RT (i )
k 1
: i-transitive (
i
RT (i ) RT (i )  RT (i )
)
(5.20)(5.21) monotonically increasing
RS
ii, Let S: i-transitive,
i
i
 R( 2 ) R R  S S  S
If
R(n)  S ,
mathematical
induction
i-transitive
i
i
( )
 R( n 1 ) R R n 
S SS
 R( k )  S , k
 RT (i ) 

R(k )  S
k 1
i.e.,
RT ( i )
: smallest
。Theorem 5.2:
 2
X

,
R: reflexive fuzzy relation on

X n

 RT (i )  R
( n 1)
 R ( m )  R m 1 
  n
 m
n 1
R  R

5-42
proof : i,
R : reflexive,
i
i
 E  R, R  E R  R R  R (2)
(By repetition)
 R( n1)  R( n)
ii, show
R( n1)  R( n)
proof: If
i
If
x  y,  R( n1) ( x, x)  1
reflexive
x  y, 
Extension of definition
R( n ) ( x, y)  sup i  R( x, z1 ), R( z1 , z2 ),
Z1 , , Z n1

X n

 X  Z0 , Z1 ,
, Zn  y
contains
at least 2 identical element.
Say
 i  R( x, z1 ),
Z r  Z s (r  s )
, R( zr 1, zr ),
, R( zs , zs 1 ),
, R( zn1, y)  R( k ) ( x, y),(k  n 1)

x, y  X , R ( n ) ( x, y )  R ( n 1) ( x, y ),

R
(n)
 R( n1)    ( B)
 RT (i )  R( n1)
1 )
 R( n ) R n( 
(A B
, )
, R( zn1 , y)
5-43
5.10 INF- w Compositions of Fuzzy Relations
i
。 w operation:
a b b
i
a  b   1
wi (a, b)  sup x [0,1]| i(a, x)  b
where
※ If
, i : continuous t-norm
: logical conjunction (i.e.,
i
 wi
a, b  [0,1]
: logical implication (i.e.,
。Theorem 5.3
1,
i ( a, b)  d
iff
2,
wi (wi (a, b), b)  a
3,
wi (i(a, b), d )  wi (a, wi (b, d ))
4,
a  b,  wi (a, d )  wi (b, d )
wi (a, b)  b
---- i
wi (d , a)  wi (d , b)
5,
i(wi (a, b), wi (b, d ))  wi (a, d )
6,
wi (inf a j , b)  sup wi (a j , b)
7,
j
j
wi (sup a j , b)  inf wi (a j , b)
j
j
8,
wi (b,sup a j )  sup wi (b, a j )
j
9,
j
wi (b,inf a j )  inf wi (b, a j )
j
j
--- ii


, and)
, if
then)
10,
i(a, wi (a, b))  b
5-44
i(a, b)  d ,  b x | i(a, x)  d
proof: (1) i , If
 b  sup x | i(a, x)  d  wi (a, d )
( )
ii, If
d
( )
b  wi (a, d )
i: continuous monotone
i: monotone increasing
 i(a, b)  i(a, wi (a, d ))  i(a,sup x | i(a, x)  d )  sup i(a, x) | i(a, x)  d   d
b
a d
(3)
By (1)<=
b
i(a, x)  wi (b, d )  i(b, i(a, x))  d
a
b
d
By (1)=>
Associativity
communitation
 i(i(a, b), x)  d  x  wi (i(a, b)d )
By (A)
wi (i(a, b), d )
 wi (a, wi (b, d ))  sup x | i(a, x)  wi (b, d )  sup x | x  wi (i(a, b), d )  wi (i(a, b), d )
(7) Let
S  sup a j
---(B)
 a j  s, j
j
By(4)
 wi ( s, b)  wi (a j , b), j
 wi (s, b)  inf wi (a j , b)
j
---- (C)
inf wi (a j , b)  wi (a j0 , b), j0  J
j
By(1)
 i(a j0 ,inf wi (a j , b))  b, j0
j
i(s,inf wi (a j , b))  sup i(a j0 ,inf wi (a j , b))  b
j
j
j0
By(1)
 wi (s, b)  inf wi (a j , b)
j
By(B)(C)(D)
--- (D)
 wi (sup a j , b)  wi ( s, b)  inf wi (a j , b)
j
j
5-45
(2)Show
(Theorem 5.3 (2))
wi (wi (a, b), b)  a
proof :
wi (a, b)  Sup x [0,1]| i(a, x)  b
and by Theorem
3.10
imin (a, b)  i(a, b)  min(a, b)
i, If a>b
wi (a, b)  Sup x [0,1]| i(a, x)  b  Sup x [0,1]| min(a, x)  b  b
i ( wi ( a, b), a )

 i (b, a )
 i (b,1)
b
 i(wi (a, b), a)  b
i, If
By Axiom i2 wi (a, b)  b
By Axiom i2
By Axiom i1
( a  1)
ab
wi (a, b)  Sup x [0,1]| i(a, x)  b  Sup x [0,1]| min(a, x)  b  1
 i(wi (a, b), a)  i(wi (a, b), b)
 i (1,By
a) Axiom i2 wi (a, b)  1
 i (b,1)
By Axiom i2
 b By Axiom i1
 i(wi (a, b), a)  b
By Theorem 5.3 property 1(i.e.,
iff
wi (a, d )  b
)
i(wi (a, b), a)  b  wi ( wi (a, b), b)  a
i ( a, b)  d
(4) prove Theorem 5.3 (4) :
ab
=>
i,
Wi (a, d )  Wi (b, d )
ii,
Wi (d , a)  Wi (d , b)
proof : i,
Wi (a, d )  Wi (b, d )
Wi (a, d )  sup{x | i(a, x)  d}
---- (A)
Wi (b, d )  sup{x | i(b, x)  d}
---- (B)
a, if
d ab
=> (A)=d , (B)=d ,
∴ (A)=(B) ----- (1)
b, if
ad b
=> (A)=1 , (B)=d
∴ (A)  (B) ----- (2)
c, if
abd
=> (A)=1 , (B)=1
∴ (A)= (B) ----- (3)
(1),(2),(3) => (A)  (B)
i.e.,
Wi (a, d )  Wi (b, d )
ii, see i
5. show
Proof : ∵
i(Wi (a, d ),Wi (b, d ))  Wi (a, d )
if
if
a  b  Wi (a, b)  b
a  b  Wi (a, b)  1
A, if
=>
a  b  i(Wi (a, b),Wi (b, d ))  i(b,Wi (b, d ))
i(b,Wi (b, d ))  i(b, d )  min(b, d )  d
Wi (a, d )  d
=>
i(b,Wi (b, d ))  i(b,1)  b
Wi (a, d )  1
=>
i(b,Wi (b, d ))  i(b,1)  b
Wi (a, d )  1
B, if
a  b  i(Wi (a, b),Wi (b, d ))  i(1,Wi (b, d ))  Wi (b, d )
=>
Wi (b, d )  d
Wi (a, d )  d
Wi (b, d )  d
Wi (a, d )  1
Wi (b, d )  1
Wi (a, d )  1
10. show
i(a,Wi (a, b)  b
Proof : ∵
a  b  Wi (a, b)  b
a  b  Wi (a, b)  1
A, if
a b
 i(a,Wi (a, b))  i(a, b)  min(a, b)  b
B, if
ab
 i(a,Wi (a, b))  i(a,1)  a  b

composition
inf  Wi
Wi
inf
( P Q)( x, z)  y  Y Wi ( P( x, y), Q( y, z))
 Theorem 5.4 :
Wi
i
Wi
(1)( P Q  R)  (Q  P 1 R)  ( P  (Q R 1 ) 1 )
Wi
Wi
i
Wi
(2)( P (Q S )  ( P Q) S
 Theorem 5.5 :
Wi
Wi
( Pj ) Q  ( Pj Q)
j
j
Wi
Wi
( Pj ) Q)  ( Pj Q)
j
j
Wi
Wi
P ( Q j )  ( Pj Q j )
j
j
Wi
Wi
P ( Q j )  ( Pj Q j )
j
j
 Theorem 5.6 : if
Wi
=>
Q1  Q2
Wi
P Q1  P Q2
Wi
Wi
Q1 R  Q2 R
Proof :
=>
Q1  Q2
∵ (P
Wi
Wi
Wi
1
Wi
i
Wi
P 1 ( P Q)  Q
2.
R  P ( P 1 R )
3.
P  ( P Q ) Q 1
4.
R  ( R Q 1 ) Q
Wi
Wi
Wi
i
Wi
Wi
Wi
Wi
Wi
Wi
Wi
Wi
R)  (Q2 R)  (Q1  Q2 ) R  Q2 R
1
1.
Q1  Q2  Q2
P Q1  P Q2
=> Q
 Theorem 5.7 :
,
Q1 )  ( P Q2 )  P (Q1  Q2 )  P Q1
=>
∵ (Q
Q1  Q2  Q1
Wi
Wi
R  Q2 R
Proof :
(1)
Wi
Wi
---- (A)
P Q  ( P 1 ) 1 Q
(5.26)  (5.25)
Wi
i.e.,
Wi
let
i
(Q  P 1 R)  ( P Q  R)
,
P QQ
P 1  P
,
QR
i
 ( A)  P 1 ( P Q)  Q
(2)
i
i
---- (B)
P1 R  P1 R
Let
P 1  P
,
QR
Wi
,
i
 ( B)  R  P ( P 1 R)
(3) by (5.33) ,
Wi
i
[ P 1 ( P Q)]1  Q 1
i
 ( P Q) 1 P  Q 1
Let
Wi
---- (C)
Wi
P Q  P, P  Q, Q 1  R
Wi
Wi
 (C )  P  ( P Q) 1 Q 1
(4) follows (3)
i
P 1 R  R