35
Fuzzy relations
A classical relation can be considered as a set of tuples, where a
tuple is an ordered pair. A binary tuple is denoted by (x,y), an example of
a ternary tuple is (x,y,z) and an example of n-ary tuple is (x1,...,xn).
Example: Let U be the domain of man {John, Charles, James} and V the
domain of women {Diana, Rita, Eva}, then the relation ”married to” on U ×V is,
for example
{(Charles, Diana), (John, Eva), (James, Rita) }
Definition: (classical n-ary relation) Let X1,...,Xn be classical(crisp) sets. The
subsets of the Cartesian product X1 ×···× Xn are called n-ary relations. If X1
= ··· = Xn and R Un then R is called an n-ary relation (operation) in U.
Let R be a binary relation in R. Then the characteristic function of R is
defined as
1, ( x, y )  R
0, ( x, y )  R
 R  x, y   
Example: Consider the following relation
x, y   R  x  a, b  y  c, d
1, ( x, y )  a, b  c, d
0, ( x, y )  a, b  c, d
 R  x, y   
Let R be a binary relation in a classical set
X. Then
Fig.12: Graph relation R
Definition. (reflexivity) R is reflexive if (x,x)  R for all xU.
Definition. (anti-reflexivity) R is anti-reflexive if f (x,x)  R for all xU.
Definition. (symmetricity) R is symmetric if from (x,y)  R  (y,x) R for all
x,yU.
Definition. (anti-symmetricity) R is anti-symmetric if (x,y)  R and (y,x)  R
then x=y for all x,yU.
Definition. (transitivity) R is transitive if (x, y) R and (y,z)R R then (x, z)  R,
for all x,y,zU.
36
Example. Consider the classical inequality relations on the real line R. It is clear
that ≤ is reflexive, anti-symmetric and transitive, < is anti-reflexive,
antisymmetric and transitive.
Other binary relations are
Definition. (equivalence) R is an equivalence relation if R is reflexive,
symmetric and transitive
Example.
The relation = on natural numbers is equivalence relation.
Definition. (partial order) R is a partial order relation if it is reflexive,
antsymmetric and transitive.
Definition. (total order) R is a total order relation if it is partial order and
for all x,yU (x,y)R or (y,x)R.
Example. Let us consider the binary relation ”subset of”. It is clear that we have
a partial order relation.
The relation ≤ on natural numbers is a total order relation.
Fuzzy relation
Definition of fuzzy relation. Let U and V be nonempty sets.
relation R is a fuzzy subset of U × V .
A fuzzy
In other words, R F (U × V ),  R : U V  0,1
It is often used equivalence notation  R ( x, y)  R( x, y) .
If U =V then we say that R is a binary fuzzy relation in U.
Let R be a binary fuzzy relation on R. Then R(x,y) is interpreted as the
degree of membership of the ordered pair (x,y) in R.
Example. A simple example of a binary fuzzy relation on
U = {1, 2, 3},
called ”approximately equal” can be defined as
R(1, 1) = R(2, 2) = R(3, 3)=1,R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2)=0.8 ,
R(1, 3) = R(3, 1)=0.3
37
 1 0.8 0.3 


In matrix notation it can be represented as  0.8 1 0.8 
 0.3 0.8 1 


Operations on fuzzy relations
The intersection
Fuzzy relations are very important because they can describe nteractions
between variables. Let R and S be two binary fuzzy relations on X × Y .
Definition: The intersection of R and S is defined by
(R  S)(x,y) = min{R(x,y),S(x,y)}.
Note that R : U ×V → <0, 1>, i.e. R the domain of R is the whole Cartesian
product U × V .
Definition: The union of R and S is defined by
(R  S)(x,v) = max{R(x, z),S(x, z)}
Example: Let us define two binary relations


x
R = ”x is considerable larger than y”=  1
x
 2
x
 3
y1 y 2 y 3 y 4 

0.8 0.1 0.1 0.7 
0 0.8 0 0 

0.9 1 0.7 0.8 
y1
y 2 y3 y 4 



S = ”x is very close to y”=
 x1 0.4, 0 0.9 0.6 
x
0.9 0.4 0.5 0.7 
 2

x

 3 0.3 0 0.8 0.5 
The intersection of R and S means that ”x is considerable larger than y” and
„is very close to y”.
38


 x1
(R  S)(x,y) =min{R(x,y),S(x,y)}= 
x
 2
x
 3
y3 y 4 

0.1 0.6 
0 0.4 0 0 

0.3 0 0.7 0.5 
y1
0.4
y2
0
The union of R and S means that ”x is considerable larger than y” or ”x is very
close to y”.
y1 y 2 y 3 y 4 



 x1 0.8 0 0.9 0.7 
 x 0.9 0.8 0.5 0.7 
(R  S)(x, y)=
 2

 x 0.9 1 0.8 0.8 
 3

Projections of fuzzy relation
Consider a classical relation R on R.
Fig. 13. Projections on axis.
1, ( x, y )  a, b  c, d
R  x, y   
0, ( x, y )  a, b  c, d
It is clear that the projection (or shadow) of R on the X-axis is the
closed interval <a, b> and its projection on the Y -axis is <c,d>.
Definition: If R is a classical relation in U × V then
ΠX = {x U| y V :(x, y)  R}
ΠY = {yV |x U :(x, y)  R}
where ΠX denotes projection on U and ΠY denotes projection on V .
39
Definition: Let R be a fuzzy binary fuzzy relation on U × V . The projection of
R on U is defined as
ΠX(x) = sup{R(x, y) | y V }
and the projection of R on Y is defined as
ΠY (y) = sup{R(x, y) | x U}
Example: Consider the relation


x
R = ”x is considerable larger than y”=  1
x
 2
x
 3
y1 y 2 y 3 y 4 

0.8 0.1 0.1 0.7 
0 0.8 0 0 

0.9 1 0.7 0.8 
then the projection on X means that
•x1 is assigned the highest membership degree from the tuples (x1,y1),
(x1,y2), (x1,y3), (x1,y4), i.e. ΠX(x1)=1, which is the maximum of the first row.
•x2 is assigned the highest membership degree from the tuples (x2,y1),
(x2,y2), (x2,y3), (x2,y4), i.e. ΠX(x2)=0.8, which is the maximum of the second
row.
•x3 is assigned the highest membership degree from the tuples (x3,y1),
(x3,y2), (x3,y3), (x3,y4), i.e. ΠX(x3)=1, which is the maximum of the third
row.
Shadow of fuzzy relation.
~
Definition: The membership function of Cartesian product of A F (U) and
~
B F (V) is defined as
~ ~
( A × B )(x,y) = min{A(x),B(y)}.
for all xU and yV.
40
Fig.14. Cartesian product of fuzzy sets
Cartesian product of fuzzy sets
It is clear that
Cartesian product of two fuzzy sets is a fuzzy relation.
If A and B are normal then ΠY (A × B)= B and ΠX(A × B)= A.
Really,
ΠX(x) = sup{(A × B)(x, y) | y}
= sup{A(x) ∧ B(y) | y} =
min{A(x),sup{B(y)}| y}
= min{A(x), 1} = A(x).
Definition: The sup-min composition of a fuzzy set
relation R F (U × V ) is defined as
~
C F
(U) and a fuzzy
~
( C  R)(y) = sup {min{C(x),R(x, y)}}
xU
for all yV .
~
The composition of a fuzzy set C and a fuzzy relation R can be
~
considered as the shadow of the relation R on the fuzzy set C .
41
Fig. 15.
~
~
Example: Let A and B fuzzy sets and let
x  a
 b  a , x  a, b
 c  x
 A x   
, x  b, c
c

b

x  a, c
0,

 xe
 f  e , x  e, f

gx
 B x   
,x f ,g
g

f

x  e, g
 0,

~ ~
Let R = A × B Is fuzzy relation.
~
~
~ ~
~
Observe the following property of composition A  R = A  ( A × B )= A ,
~ ~ ~
~
~
B  R = B  ( A × B )= B .
~
Example: Let C be a fuzzy set in the universe of discourse {1, 2, 3} and let R
be a binary fuzzy relation in {1, 2, 3}. Assume that
 1 0.8 0.3 


~
C ={(1,0.2),(2,1)(3,0.3)} and R=  0.8 1 0.8 
 0,3 0,8 1 


Using the definition of sup-min composition we get
 1 0.8 0.3 


~
C  R=(0.2,1,0.3)  0.8 1 0.8  =(max{min{0.2,1},min{1,0.8},min{0.3,0.3}},
 0,3 0,8 1 


max{min{0.2,0.8},min{1,1},min{0.3,0.8}},max{min{0.2,0.3},min{1,0.8},min{0.3,1}}
42
=
=(0.8,1,0.8).
~
Example: Let C be a fuzzy set in the universe of discourse <0, 1> and let R be
a binary fuzzy relation in <0, 1>. Assume that C(x)= x and R(x, y)=1-|x-y|.
Using the definition of sup-min composition we get
1 y
~
. ( C R)(y)= sup min x,1  x  y  
2
x 0 ,1
for all y<0,1>
Sup-min composition of fuzzy relations
Definition: (sup-min composition of fuzzy relations) Let R F (U × V ) and S
F (V × T). The sup-min composition of R and S, denoted by RS is defined as
(R S)(x,z) = sup min Rx, y , S  y, z 
yV
It is clear that R S is a binary fuzzy relation in U×T.


 x1
R = ”x is considerable larger than y”=  x
 2
x
 3
Example: Consider two fuzzy relations
S = ”y is very close to z” =
y1 y 2 y3 y 4 

0.8 0.1 0.1 0.7 
0 0.8 0 0 

0.9 1 0.7 0.8 
z1
z2


 y1 0.4 0.9
y 0
0.4
 2
 y3 0.9 0.5

 y4 0.6 0.7
Then their composition is
RS=


 x1
x
 2
x
 3
z1

y4  
  y1 0.4
0.7  
 y2 0
0 
  y 0.9
0.8   3
 y 4 0.6
max 0.8,0.1,0.1,0.7
y1 y 2 y3
0.8 0.1 0.1
0 0.8 0
0.9 1 0.7
z3 

0.9 0.3 
0.4 0  

0.5 0.8 

0.7 0.5 
max 0.3,0,0.1,0.5
z2
 max 0.4,0,0.1,0.6

  max 0,0,0,0
max 0,0.4,0,0
max 0,0,0,0
 max 0.4,0,0.7,0.6 max 0.9,0.4,0.5,0.7 max 0.3,0,0.7,0.5






z3 

0.3 
0 

0.8 

0.5 
43
 0.6 0.8 0.5 


=  0 0.4 0 
 0.7 0.9 0.7 


i.e., the composition of R and S is nothing else, but the classical product of the
matrices R and S with the difference that instead of addition we use
maximum and instead of multiplication we use minimum operator.
Sup-product composition of fuzzy relations
Definition: (sup-product composition of fuzzy relations) Let R F (U × V )
and S F (V × T). The sup-product composition of R and S, denoted by RS is
defined as
(R S)(x,z) = sup Rx, y .S  y, z 
yV
It is clear that R S is a binary fuzzy relation in U×T.
Example: Consider two fuzzy relations
R = ”x is considerable larger than y”=
S = ”y is very close to z” =
Then their sup-product composition is
z1

y1 y 2 y3 y 4  


  y1 0.4
RS=  x1 0.8 0.1 0.1 0.7  
 y2 0
x
0 0.8 0 0  
2


 x 0.9 1 0.7 0.8   y3 0.9
 3

 y 4 0.6
z2
0.9
0.4
0.5
0.7
z3 

0.3 
0 

0.8 

0.5 
z1
z2


 y1 0.4 0.9
y 0
0.4
 2
 y3 0.9 0.5

 y4 0.6 0.7
z3 

0.3 
0 

0.8 

0.5 
44
 max 0.32,0,0.09,0.42 max 0.72,0.04,0.5,0.49 max 0.24,0,0.08,0.35

  max 0,0,0,0
max 0,0.72,0,0
max 0,0,0,0
 max 0.36,0,0.63,0.48 max 0.81,0.4,0.35,0.56 max 0.27,0,0.56,0.4






 0.42 0.72 0.35 


0.72
0 
 0
 0.63 0.81 0.56 


If possible to define composition fuzzy of relations in another manner.
For instance, operator max we can replace any t-conorm and min any t-norm.
Fuzzy relation is
Reflexive if R(x,x)=1 for all xU.
Symmetric if R(x,y)=R(y,x) for all (x,y)R
Transitive if
R(x, y)  sup R( x, z ).R( z, y)
zU
Total if for all xU R(x,y) >0 or R(y,x)>0.
Anti symmetric if R(x,y) >0 and R(y,x)>0 implies x=z.
Strongly fuzzy transitive if
for all (x,y)R
It is clear there exist a fuzzy transitive relations R* that R* is strongly
transitive and R*(x,y)≥R(x,y)(for example R*(x,y)=1).
The fuzzy transitive closer of R
Let R* is strongly transitive relations and R*(x,y)≥R(x,y) and for any
strongly transitive transitive relation S,S(x,y)≥R(x,y) S(x,y)≥R*(x,y), then R* is.
If U is reflexive, transitive and has n elements, then R n 1  R  R  ...  R
n 1
is fuzzy transitive closer of R transitive closer of R.
45
Proof: Is evident. We leave it to reader.
Example: Let
 1 0.2 0.5 .7 


 0.3 1 0.5 0.7 
R
0.2 0.5 1 0.7 


 0.6 0.2 0.4 1 


 1 0.2 0.5 .7   1 0.2 0.5 .7 

 

1 0.5 0.7   0.3 1 0.5 0.7 
2  0.3
R 


0.2 0.5 1 0.7   0.2 0.5 1 0.7 

 

 0.6 0.2 0.4 1   0.6 0.2 0.4 1 

 

 max 1,.2,.2,.6 max .2,.2,.5,.2 max .5,.2,.5,.4 max .7,.2,.5,.7


 max .3,.3,.2,.6 max .2,1,.5,.2 max .3,.5,.5,.4 max .3,.7,.5,.7


max .2,.3,.2,.6 max .2,.5,.5,.4 max .2,.5,1.4 max .2,.5,.7,.7


 max .6,.2,.2,.6 max .2.2,.4,.2 max .5,.2,.4,.4 max .6,.2,.4,1 


 1 0.5 0.5 0.7 


 0.6 1 0.5 0.7 

0.6 0.5 1 0.7 


 0.6 0.4 0.5 1 


 1 0.5 0.5 0.7   1 0.2 0.5 .7 

 

0
.
6
1
0
.
5
0
.
7
0
.
3
1
0
.
5
0
.
7




R3  R 2  R  



0.6 0.5 1 0.7
0.2 0.5 1 0.7 

 

 0.6 0.4 0.5 1   0.6 0.2 0.4 1 

 

 max 1,.3,.2,.6 max .2,.5,.5,.2 max .5,.5,.5,.4 max .7,.5,.5,.7


 max .6,.3,.2,.6 max .2,1,.5,.2 max .5,.5,.5,.4 max .6,.7,.5,.7


max .6,.3,.2,.6 max .2,.5,.5,.2 max .5,.5,1.4 max .6,.5,.7,.7


 max .6,.3,.2,.6 max .2.4,.5,.2 max .5,.4,.5,.4 max .6,.4,.5,1 


 1 0.5 0.5 0.7 


 0.6 1 0.5 0.7 

0.6 0.5 1 0.7 


 0.6 0.5 0.5 1 


46
Let R* is reflexive, symmetric relation then R* is fuzzy similarity
relation.
 1 0.5 0.7 


Example: The relation R   0.5 1
0  is reflexive(R(x,x)=1 for all x) and
 0.7 0
1 

symmetric(R(1,2)=R(2,1)=0.5, R(1,3)=R(3,1)=0.7, R(2,3)=R(3,2)=0) and so is is
fuzzy similarity reletion.
The converse fuzzy relation is usually denoted as Rc is defined as
Rc (x,y)=R(y,x)
For all x,yU
Identity relation
I(x,x)=1 for all xU
I(x,y)=0 for all xyU
Zero relation
o(x,y)=0 for all x,yU
Universe relation
u(x,y)=1 for all x,yU
Example: The following are examples of these relations
 1 0.5 0.7 
 1 0.2 0.1



c 
R   0.2 1
0   R   0.5 1
0 
 0.1 0
 0.7 0
1 
1 


 1 0.5 0.7 


R   0.5 1
0 
 0.7 0
1 

 0 0 0
1 1 1




O   0 0 0  U  1 1 1
 0 0 0
1 1 1




47
The Fuzzy equivalence relation.
Let R* is reflexive, symmetric and is strongly fuzzy transitive relation
then R* is fuzzy similarity relation often called fuzzy equivalence relation.
Theorem: R is fuzzy equivalence relation if and only if its -cut R is
relation equivalence for all 0,1.
Proof: Let R is fuzzy relation equivalence. Then R is fuzzy reflexive (R(x,y)=1)
and so R(x,y)=1 and R is reflexive. R is symmetric(R(x,y)=R(y,x)). It implies
R(x,y)=R(y,x) and R is symmetric. R is transitive and so R is transitive too
and R is relation equivalence.
Let R is relation equivalence for all 0,1. Then R is fuzzy reflexive,
symmetric and transitive. It implies R is fuzzy relation equivalence.
Example: Let fuzzy relation is defined by its *-cuts
1

1
R0.4  
1

1

1 1 1

1 1 1
1 1 1

1 1 1
1

0
R0.9  
0

0

0 0 0

1 0 0
0 1 0

0 0 1 
1

1
R0.5  
0

1

1 0 1
1 1


1 0 1
1 1
R

0
.
8
0 0
0 1 0


0 0
1 0 1 

0 0

0 0
1 0

0 1 
All -cuts are relations equivalence and so R is fuzzy relation equivalence.
The basic properties of fuzzy relations
We wil now try to give some basic properties of compositions of fuzzy
relations which plays a major role in areas such as fuzzy control, fuzzy
diagnosis and fuzzy expert systems.
48
1. R  I  I  R  R
2. R  O  O  R  O
3. In general R  S  S  R
4. R m 1  R m  R  R
5. R m  R n  R n  m
 n  R mn
6. R m
7. ( R  S )  T  R  ( S  T )
8. R  (S  T )  R  S   R  T 
9. R  (S  T )  R  S   R  T 
10. S  T  R  S   R  T 
Fort inverse relarions
11. R  S c  R c  S c
R  S c  R c  S c
R  S c  R c  S c
 c  R
12. R c
13. R  S  R c  S c
Minimum fuzzy equivalence closer of R.
Let R* I fuzzy equivalence relation and R*(x,y)≥R(x,y) and for any fuzzy
equivalence relation S, S(x,y)≥R*(x,y), then R* is minimum fuzzy equivalence
closer of R.
Example: Let
 0.9 0.2

 0.3 1
R
0.2 0.5

 0.6 0.2

0.5 .7 

0.5 0.7 
0.4 0.7 

0.4 0.8 
What is minimum fuzzy equivalence closer of R?
The minimum fuzzy equivalence closer of R is fuzzy reflexive relation.
The fuzzy relation is reflexive if for all xU R(x,x)=1. The minimum reflexive
49
relation R*R is relation R*(x,x)=1 and R*(x,y) =R(x,y) for all xy. Hence
 1 0.2 0.5 .7 


1 0.5 0.7 
*  0.3
R 
0.2 0.5 1 0.7 


 0.6 0.2 0.4 1 


The fuzzy relation is symmetric if for all x,yU R(x,y)=R(y,x). The
minimum symmetric relation R*R is relation R*(x,y)=max {R(x,y),R(z,x)} for all
xy. Hence
1
max 0.2,0.3 max 0.2,0.5 max 0.6,0.7



1
max 0.2,0.5 max 0.2,0.7
*  max 0.2,0.3
R 

max 0.2,0.5 max 0.5,0.5
1
max 0.4,0.7


 max 0.6,0.7 max 0.2,0.7 max 0.4,0.7

1


 1 0.3 0.5 0.7 


 0.3 1 0.5 0.7 

0.5 0.5 1 0.7 


 0.7 0.7 0.7



The minimum fuzzy transitive relation fuzzy closer of R and if U is finite
then R*=Rn-1. Hence
 1 0.3 0.5 0.7   1 0.3 0.5 0.7 

 

0
.
3
1
0
.
5
0
.
7
0
.
3
1
0
.
5
0
.
7




R2  



0.5 0.5 1 0.7
0.5 0.5 1 0.7 

 

 0.7 0.7 0.7 1   0.7 0.7 0.7 1 

 

 max 1,.3,.5,.7 max .3,.3,.5,.7 max .5,.3,.5,.7 max .7,.3,.5,.7


 max .3,.3,.5,.7 max .3,1,.5,.7 max .3,.5,.5,.7 max .3,.7,.5,.7


max .5,.3,.5,.7 max .3,.5,.5,.7 max .5,.5,1.7 max .5,.5,.7,.7


 max .7,.3,.5,.7 max .3,.7,.5,.7 max .5,.5,.7,.7 max .7,.7,.7,1 


 1 0.7 0.7 0.7 


 0.7 1 0.7 0.7 

0.7 0.7 1 0.7 


 0.7 0.7 0.7 1 


50
 1 0.7 0.7 0.7   1 0.3 0.5 0.7 

 

0
.
7
1
0
.
7
0
.
7
0
.
3
1
0
.
5
0
.
7




R3  R 2  R  



0.7 0.7 1 0.7
0.5 0.5 1 0.7 

 

 0.7 0.7 0.7 1   0.7 0.7 0.7 1 

 

 max 1,.3,.5,.7 max .3,.3,.5,.7 max .5,.3,.5,.7 max .7,.7,.7,.7


 max .3,.3,.5,.7 max .3,1,.5,.7 max .5,.5,.7,.7 max .7,.7,.7,.7


max .5,.3,.5,.7 max .5,.5,.7,.7 max .5,.5,1.7 max .7,.7,.7,.7


 max .7,.7,.7,.7 max .7,.7,.7,.7 max .7,.7,.7,.7 max .7,.7,.7,1 


 1 0.7 0.7 0.7 


 0.7 1 0.7 0.7 

0.7 0.7 1 0.7 


 0.7 0.7 0.7 1 


If fuzzy relations is not symmetric then for symmetric closer of R pay
R*(x,y)≥R(x,y) and R*(x,y)≥R(y,x). At first we take R*(x,y)=max{ R(y,x), R(x,y) }.
It can be interesting to take R*(x,y)=min{ R(y,x), R(x,y) }.
Example: Let
 1 0.2 0.5 .7 


 0.3 1 0.5 0.7 
R
0.2 0.5 1 0.7 


 0.6 0.2 0.4 1 


Then the first estimation of R* is
 1 0.2 0.2 0.6 


 0.2 1 0.5 0.2 
R´ 
0.2 0.5 1 0.4 


 0.6 0.2 0.4 1 


The minimum fuzzy transitive relation fuzzy closer of R´, f U is finite, is
R*=Rn-1. Hence
 1 0.2 0.2 0.6   1 0.2 0.2 0.6 

 

1 0.5 0.2   0.2 1 0.5 0.2 
2  0.2
R 


0.2 0.5 1 0.4   0.2 0.5 1 0.4 

 

 0.6 0.2 0.4 1   0.6 0.2 0.4 1 

 

51
 max 1,.2,.2,.6 max .2,.2,.2,.2 max .2,.2,.2,.4 max .6,.2,.2,.6


 max .2,.2,.2,.2 max .2,1,.5,.2 max .2,.5,.5,.2 max .2,.2,.4,.4


max .2,.2,.2,.4 max .2,.5,.5,.2 max .2,.5,1.4 max .2,.2,.4,.4


 max .6,.2,.2,.6 max .2,.2,.4,.4 max .2,.2,.4,.4 max .6,.2,.4,1 


 1 0.2 0.4 0.6 


 0.2 1 0.5 0.4 

0.4 0.5 1 0.4 


 0.6 0.4 0.4 1 


 1 0.2 0.4 0.6   1 0.2 0.2 0.6 

 

1 0.5 0.4   0.2 1 0.5 0.2 
3  0.2
R 


0.4 0.5 1 0.4   0.2 0.5 1 0.4 

 

 0.6 0.4 0.4 1   0.6 0.2 0.4 1 

 

 1 0.2 0.4 0.6 


 0.2 1 0.5 0.4 

0.4 0.5 1 0.4 


 0.6 0.4 0.4 1 


As it is well known, within a classical context, an equivalence relation in a
set defines a partition or a classification in it, and viceversa. There have been
several attempts to extend these concepts to the fuzzy framework, so that in the
existing literature on this subject, two different trends have been followed. The
first one puts its emphasis on the definition of fuzzy partition and then, studies
the properties of the associated relation, if it exists. The papers by (Bezdek &
Harris, 1978) and (Ovchinnikov & Riera, 1982) are representative of this
research line. We follow an opposite path where the stress lies on the
conditions that must fulfil a fuzzy partition in order to have an indistinguishability
relation as its associated relation. In this trend we find the papers by (Ruspini,
1982), (Valverde, 1985) and (Jacas, Trillas & Valverde, 1987). In the first one by
Ruspini, the concept of R-cluster and fuzzy R-cluster coverage is introduced as
a convenient definition of classification associated to a likeness relation, such
definition of R-cluster links up the relation R defined in the basic set X, with a
“metric in the unit interval” that, in this case, is the restriction to this interval of
the ymmetric metric.
52
In (Valverde, 1985) these previous ideas have been collected and used
in order to obtain a metric characterization of fuzzy cluster coverages
associated to T-transitive relations. In the last section of this work we present
results which go one step further and characterize the classifications associated
to any given S-metric m in the unit interval, showing that the m-cluster
coverages defined in this way are intrinsically linked to T-transitive relations. In
fact, the definition of m-cluster we introduce is a generalization of the one given
for classical clusters in terms of its characteristic functions, where the
underlying metric is the discrete distance in the two point set 0,1 .
The applicability of the above mentioned results drew, essentially, on the
representation theorem for fuzzy T-transitive relations. As it is stressed in the
next section, this theorem is a powerful tool to build fuzzy transitive relations
starting from arbitrary fuzzy subsets of the given set. Such fuzzy subsets are
called generators of the fuzzy relation. The fact that they appear as the closed
sets associated with the fuzzy topology generated by the given relation may be
considered as the most significative result concerned with the generators of a
relation. An algorithm to compute a suitable family of such generators for
similarity relations is given.
T-indistinguishability relation
Definition. T-indistinguishability relation E is a reflexive and symmetric fuzzy
relation such that
T(E(x,y),E(y,z))≤E(x,z)
for all x,y,zU.
Definition. A S-pseudometric m is a mapping m:UU<0,1> such that
-m(x,x)=0
-m(x,y)=m(y,x)
S(m(x,y),m(y,z))≥m(x,z)
for all x,y,zU.
There is a close relation between T-indistinguishability relations and Spseudometrics as is shown in the following theorem:
Theorem. Let E be a T- indistinguishability relation and let  be a continuous
order-reversing bijection from <0,1> to <0,1>. Then
mE(x,y)=(E(x,y))
is a S-pseudometric.
53
For a long time, the only available methods to build up fuzzy transitive
relations have been the transitive closure and related methods. As it has been
pointed out repeatedly, these methods carry on a number of major problems,
like the requirements of both storage and computer-time and, in spite of this, no
one is satisfied with the results they yield, because there is no way to control
the distortion that its application produces on the data sample, so that the
transitive closure methods do not fit the desiderata of having a method to
specify a similarity measure which matches with the data.
To be more concrete, in order to apply the transitive closure method to
construct a similarity relation and, in general, a fuzzy T-transitive relation, a
reflexive and symmetric fuzzy relation has to be used as a starting point. In
others words, an index of similarity relating each couple of elements in the
sample set has to be given: each two elements should be matched, in some
way, and then the method is applied to obtain either a similarity or dissimilarity
measure. At this point, the first arising question is the following: Does it mean
that, for instance, from a single criterion, or from the matching of all elements to
one given, no similarity measure can be given? The obvious negative answer
can be stated by assuming that as a result of the single criterion evaluation or
the matching-to-one process, a function
h:U<0,1>
is given, h(x) representing the degree to which x fits the given conditions. In this
assumption it is easy to check that
m(x,y)=h(x)-h(y)
is a pseudo-distance on U. It is also quite obvious, that
E(x,y)=1-m(x,y)
is a likeness relation on U . It is the measure of similarity between the element
y , and any perfect prototype.
Such a construction can be extended in order to get T-transitive fuzzy
relations for any t-norm. If T* stands for the quasi-inverse of the t-norm T , i.e.
then it is also easy to check that
E( x, y)  T *max hx , h y min hx , h y 
is a T-fuzzy transitive relation, such that
h( x)  E ( x, x0 )
for any x0  h 11 . Thus, for instance,
min hx , h y , hx   h y 
E ( x, y )  
1, hx   h y 

54
is a the similarity relation induced by h , i.e. E is min-transitive. On its own part,
E ( x, y ) 
min hx , h y 
max hx , h y 
is a probabilistic relation, i.e. transitive with respect to the t-norm T(a,b)=a.b and
m(x,y)=1-E(x,y)
is a generalized pseudo-metric with respect to the t-conorm s(a,b)=a+b-a.b.
Summing up, the above considerations show what to do in order to
obtain a similarity (or symmetrically) measure which matches to the data from a
single symmetrical evaluation of the degrees of similarity in the sample set.
Next, suppose that several criteria or prototypes are given in the form of a
family of functions
h j : U  0,1
in this case the most natural procedure seems, first, to get the similarity
measure –in the form of a fuzzy transitive relation for a fixed t-norm T –
associated with each hj , Ej , and then to take as the degree of the similarity of
two elements, E(x,y), the minimum of all the degrees E j(x,y), which, as it is easy
to check, is also a T-transitive relation. Obviously, there are other ways to
combine fuzzy transitive relations which also preserve the transitive character of
the relation. , any reflexive, symmetric and T –transitive fuzzy relation on a set X
is generated by a family of fuzzy subsets of the given set through the procedure
described in this section. In (Valverde and Ovchinnikov, 1986) it has been
shown that the above representation also holds for left-continuous T –norms,
this fact is specially interesting when the minimal T –norm Z is considered. As it
is known, this T –norm is defined by
min x, y, if max x, y  1
Q ( x, y )  
1, if max x, y  1

It is worth noting that reflexive, symmetric and Z-transitive fuzzy relations
are simply those reflexive and symmetric relations for which the 1-level set is a
classical equivalence elation. From this standpoint, the Z-transitive relation, S ,
obtained by applying the procedure implied by the representation theorem
starting from a strict reflexive fuzzy relation, R , is simply the greatest symmetric
relation contained in R , i.e.
s( x, y)  min R( x, y), R( y, x)
On the other hand, when the representation theorem is applied to build
the T – indistinguishability relation generated by a reflexive and symmetric fuzzy
relation R , i.e. when the functions hj are the rows of R , then E(x,y) is either
R(x,y) or the greatest number among those which satisfy both
55
for all z in U . The point is that from the representation theorem both the
existence of such a fuzzy relation and the method to compute it follow.
Moreover, the use of the representation theorem no longuer requires a
complete fuzzy binary relation; neither reflexivity nor symmetry are required. As
it has been shown the initial data may be just one function from the set X into
<0,1>.
Theorem. Let U be nonempty universal set, S a continuous t-conorm and m a
mapping UU into <0,1>. Then m is pseudometric if, and only if there exist a
 
family h j n , such that
j 1
 

m( x, y)  sup ms j  h j x , h j  y 
j
For some continuous and order reversing bijection  on the unit interval.
In other words, any S-pseudometric on a given set U comes from a
family of fuzzy subsets of the given set. So that, in the case of ordinary
(bounded) metrics, the corresponding S- metric is m( x, y)  x  y . That is, once
a “distance” on the unit interval is fixed, this distance is carried to the given set
U through the fuzzy subsets of U. Let it be noticed that such procedure is
implicitely used in order to associate a likenes relation to a fuzzy partition. As it
is known, at the very ®, a fuzzy partition of a set U was defined as a finite family
of fuzzy subsets ui  of U such that
 ui ( x)  1, for any I and  ui ( x)0 , for any xU.
xU
i
Definition: A function h from U to <0,1> is termed a generator of given T
indistinguisability relation E , if Eh≥E,, HE will denote the set of all generators of
E.
The next definition will play as important role in order to give a more
convenient characterization of the generators of a T-indistinguishability relation
E. It follows immediately from the representation theorem that, given a Tindistin-guishability relation E on U, the set E ( x, y)yU of fuzzy subsets of X is


a generating family of E and will be denoted by h y x 
. The next definition
yU
will play as important role in order to give a more convenient characterization of
the generators of a T-indistinguishability relation E.
Definition. If E be T-indistinguishability relation then E is a map from <0,1>U
into <0,1>U defined by  E h  x   sup T E x, y , h y  for any xU.
yU
If U is a finite set then E is represented by a matrix and  E h may be
understood as the max-T product of E by the column vector representing the
fuzzy set h.
56
Theorem. A fuzzy set h<0,1>U is generator of a T-indistinguishability relation
E if and only if  E h  h .
Taking into account definition, the next proposition follows immediately
 
-
If h is a singleton, then  E h x   E x,   .
-
If E is T-indistinguishability relation, then
-
HE=E(<0,1>U),
-
If hk<0,1>U is a constant fuzzy set, then hkHE
-
If h1,h2HE then min{h1,h2}HE
If U is a finite set, the columns of the matrix associated to E are the
images of the singletons and the image of a crispl set is obtained as the
supremum of elements of the set {E(x,y)}yU.
For similarity relations the existence of an injective column is a
symmetrical condition for unidimensionality, therefore:
On the other hand, the characterization of unidimensional relations,
shows that these relations define a “betweeness” in the underlying set X,
structured as a chain, giving a “geometrical” interpretation of the
unidimensionality. Finally, the existing duality between T-indistinguishability
relations and a type of generalized metrics, leads to the interesting problem of
the study of the topological structures induced by these metrics.
Next, the problem of determining a minimal generating family for a similarity
relation over a finite set X, is completely solved. This result may be an important
tool in all algorithms dealing with similarity relations because all the information
contained in the matrix representing the mentioned relation can be ‘packed’ in a
few (even one!) fuzzy sets. As an application, two versions of an algorithm for
the automatic search of one of such families are presented.
Let E be a similarity relation and kU, will denote the fuzzy set defined by
and E the similarity generated by h i.e.
Algorithm 1.
1. Compute the columns’ orders and select the maximal ones.
57
2. Compute the similarity order and the upper bound of the dimension (
+1).
3. Select a maximal column
whose order is the similarity order.
4. Build ( +1). Column vectors hj of dimension n and initialize them with
1’s.
5. If hk(x) is a unrepeated value in hk, do hj(x) hk(x) for any j. Do this
ymmetrica for all unrepeated values in hk(x).
6. Select a repeated value in hk(x) and find its associated set U.
7. For any pair of elements of U. xj,xk, such that E(xj,xk)=´ , do h1(xi)=
h1(xj)   h1(xi)= h1(xj)   h2(xi)= 1h2(xj)  ´.
8. For any element x of U not selected in the preceding step, do hj(xr)  
for some j, such that any pair of elements xr,xs included in this step there
exists a j such that hj(xr) hj(xs).
9. If there exists in hj another repeated value, select it and go to step 6.
10. If for two elements , with different repeated values , , hj(x´)= hj(x´´)
for all j, select p such that hp(x´)= hp(x´´) and do hp(x´) E(x´,x´´)
.
11. Delete all constant columns.
12. End.
Preference relations
A fundamental component of preference modeling is how alternatives are
related to one another. A relation R from a set U to a set V is, in essence, a rule
that assigns certain objects in the set X to certain other objects in the set Y. If
both sets U and V are identical, we say that R is a binary relation on U. The
expression “x is in relation R with y” (or also “x is R- related to y”) is usually
denoted as xRy or (x,y).R. A matrix is usually the most convenient way to
represent relations on relatively small sets.
We usually refer to the R-afterset of an element x in X as the set of those
elements of Y that are R-related to x in X, denoted as xR and given by {y ; xRy}.
Similarly, the R-foreset Ry of an element y in Y is the set of those elements of
X that are R-related to y. In preference modeling, the objects we are looking at
are decision alternatives (or in short, alternatives), and the rules express a
user’s preference (or lack thereof, either an indifference or an incomparability)
among all possible pairs of alternatives. Denoting the set of alternatives as A,
then the relations expressing preferences (or indifferences or incomparabilities)
all are relations in A. More specifically, we define three important relations in A:
58

A couple of alternatives (a,b) belongs to the strict preference relation P if
and only if the user prefers a to b;

A couple of alternatives (a,b) belongs to the indifference relation I if and
only if the user is indifferent between alternatives a and b;

A couple of alternatives (a,b) belongs to the indifference relation I if and
only if the user is indifferent between alternatives a and b;

A couple of alternatives (a,b) belongs to the incomparability relation J if
and only if the user is unable to compare a and b, for instance caused by
conflicting or insufficient information.
A preference structure on a set of alternatives A is the triplet (P,I,J) of a
binary preference, indifference and incomparability relation in A. However, P, I
and J must satisfy some rather basic additional conditions. For instance, any
couple of alternatives belongs to exactly one of the relations P, Pt (the
transpose of P), I or J. More formally, a preference structure is defined as
follows.
Definition: A preference structure on a set of alternatives A is a triplet (P,I,J) of
binary relations in structures since statements over degrees of and incomparability of preferences are natural and satisfy:
i.
I is reflexive and J is irreflexive;
ii.
P is asymmetrical;
iii.
I and J are symmetrical;
iv.
P I =, P  J =  and I  J = ;
v.
P  Pt  I  J = A2
Example. Let A=a,b,c and


a
P
b

c

a b c
a



0 1 1
a 1
I

,
b 0
0 0 0


c 0
0 0 0 

b c

0 0
,
1 0

0 1 


a
J 
b

c

a b c

0 0 0
0 0 1

0 1 0 
Then (P,I,J) is preference structure.
Proof:
(i)
I is reflexive (aIa, bIb, cIc) and J is irreflexive((a,a)J);
(ii)
P is asymmetrical((a,b)P and (b,a)P);
(iii)
I and J are symmetrical(see a matrix);
(iv)
P I =, P  J =  and I  J = (see a matrix);
59
(v)
P  Pt  I  J = A2( is evident true);
Property i means that the user is always indifferent between a and a, and
that a can always be compared to itself, ii is the property that a user
cannot prefer a to b and b to a at the same time; iii means that when a user is
indifferent between a and b, Property iv states that a pair (a,b) cannot belong
to two of relations P and I and J at the same time. Finally, v is property that a
pair (a,b) always belongs one of relations P,P´,I or J.
It is well-known that from any reflexive binary relation R in a set of
alternatives A, a classical preference structure (P,I,J) can be constructed
in the following way
c) P=RcoRt;
ii. R=RRt;
iii. J=co RcoRt
In other words, an alternative a is strictly preferred to b (or (a,b)  P) if
and only if a is at least as good as b, and b is not at least as a. Similarly, a user
is indifferent to two alternatives a and b if and only if a is at least as good as b
and b is at least as good a.
Definition. A triplet (P,I,J) of binary fuzzy relations in A is a fuzzy
preference relation on A if and only if
(i)
I is reflexive or (P and J are irreflexive);
(ii)
I is symmetrical or J is symmetrical
(iii)
( (a,b)A2)( P(a,b) + P(b,a) + I(a,b) + J(a,b) =1)).
Let A be a finite set of objects with at least two elements. We interpret the
elements of A as alternatives among which a choice is to be made taking into
account different points of view, e.g. several criteria or the opinion of several
voters. A common practice in such a situation is to associate with each ordered
pair (a, b) of alternatives a number indicating the strength or the credibility of the
proposition “a is at least as good as b”, e.g. the sum of the weights of the
criteria favoring a or the percentage of voters declaring that a is preferred or
indifferent to b. This leads to a fuzzy (large) preference relation on A. In the
area of ELECTRE III is a typical illustration of such a process. A fuzzy (binary)
relation on a set A is a function R associating with each ordered pair of
alternatives (a, b)  A2 an element of 0, 1. Therefore, we define a choice
procedure for fuzzy preference relations (on a set A) as function associating a
nonempty subset of A, the “choice set”, with each fuzzy reflexive binary relation
on A. In this note, we study “choice procedures” instead of the more classical
notion of “choice functions”, i.e. functions associating a choice set with any
subset of A. If a fuzzy relation R is such that R(a, b)  {0, 1}, for all a, b  A, we
say that R is crisp. In this case, we write a R b instead of R(a, b) = 1.
60
The classical problem of defining “reasonable” choice procedures for
crisp relations is not an easy one. It has generated numerous studies, in
particular in the case of tournaments (i.e., crisp, asymmetric and complete
binary relations). This difficulty is largely due to the fact that when a crisp
preference relation is not complete and/or has cycles in its asymmetric part, the
very notion of a “good” alternative is not easy to define. Such relations are
commonly found in social choice theory. Turning now to fuzzy preference
relations, the situation appears even more complex.
An important class of preference structures consists of those structures for
which there are no couples of incomparable alternatives. A preference structure
of the form (P,I,J=) is called a preference structure without incomparability,
and will be denoted shortly as a preference structure (P,I). The following
theorem provides a useful characterization of a preference structure in terms of
its large preference relation. Recall that a binary relation R in A is called
complete if and only if R Rt = A2.
Theorem (Roubens and Vincke 1985). A preference structure (P,I,J) on A is a
preference structure (P,I) on A if and only if its large preference relation is
complete.
Some properties of choice procedures
Consider a fuzzy relation i.e. a function associating with each ordered
pair of alternatives (a, b)  A2 an element of 0, 1. Suppose that R(a, b) = 0.2
and R(c, d) = 0.8. Should we conclude that the proposition “c is at least as good
as d” is four times more credible than the proposition “a is at least as good as b”
? In some situations, e.g. when the numbers R(a, b) represent a proportion of
voters or of criteria favoring the proposition “a is at least as good as b”, this may
seem reasonable and a choice procedure should take into account such
considerations. In other situations, e.g. when the fuzzy relation has been
obtained on a purely introspective basis or when the weights of the criteria only
reflect an ordinal information about their respective importance, this may well
lead to a somewhat unrealistic preference model. Many attempts have been
made to propose an “ordinal” theory of fuzziness. In this note we shall remain in
the ordinary framework of the theory of fuzzy sets but impose that a choice
procedure should only make use of the underlying ordinal information on
credibility conveyed by a fuzzy preference relation. We say that a choice
procedure C is ordinal if, for all R and all strictly increasing and one-to-one
transformation  from 0, 1 to 0, 1, C® = C( [R]),where [R] is the element of
F(A)(set of all fuzzy relations on A) such that [R](c, d) = (R(c, d)) for all c, d 
61
A.
It is clear that an ordinal choice procedure does not make use of the
cardinal properties of the numbers R(a, b). Many ordinal choice procedures can
be envisaged. Let us mention one of them that has often been discussed in the
literature and may be seen as a direct extension to the fuzzy case of the
classical notion of the “greatest elements” of a crisp preference relation. Let
R F(A) and, for all a  A, define, using the same notation as in Barrett et al.
(1990), the ‘min in Favor’ score of alternative a letting:
mF a, R   min
R(a, c)
c A \ a
A clearly ordinal choice I defined by CmF R   a  A; mF a, R   mF b, R  for all
bA.
Many other ordinal choice procedures can be envisaged. This is the raison d’etre
of the following axiom. We say that a choice procedure C is faithful if, for all
RF(A), [R  U(A) and G® ≠ ]  C®  G®. Faithfulness imposes a constraint
on the result of a choice procedure when applied to (some) crisp relations.
Ordinality imposes that the result of choice procedure is identical when applied to
two “ordinally-equivalent” relations, i.e. to two relations R, S  F(A) such that, for all
a, bA, R(a, b) = φ(S(a, b)) for some strictly increasing and one-to-one
transformation φ on 
. It should be noticed that no relation in F(A)\U(A) can be
“ordinally-equivalent” to a relation in U(A) since only one-to-one transformations are
invoked by ordinality. Thus, these two axioms impose very few constraints on the
desirable behavior of a choice procedure when applied to fuzzy relations outside
U(A). In particular, they leave room for “discontinuities”, which seem rather
paradoxical. Let us illustrate the possibility of discontinuities on a simple example
involving a crisp relation and an “almost crisp” one. Consider the relations
R
a
b
c
a
1
0
0
b
1
1
0
c
1
0
1
R´
a
b
c
a
1
0
0
b
1
1
0
c

0
1
where 0 < λ < 1.
It is easy to see that R is crisp and that G® = {a}. Let C be a faithful choice
procedure. We have C® = {a}. Even if C is ordinal, it may happen that a ∉
arbitrarily “close” to R. Our final axiom is designed to prevent such situations.
 F(A2) if, for all ε
, there is an integer k such that, for all j ≥ k and all a, b  A, Rj(a,b)-R(a,b) .
A choice procedure C is said to be continuous if, for all RF(A) and all
sequences Rj F(A2) converging to R f(aC(Ri), for all Ri in sequence) aC®
Our definition of continuity implies that an alternative that is always chosen with
fuzzy relations arbitrarily close to a given relation should remain chosen with this
relation. It is not difficult to see that C is continuous.
62
Fuzzy partial ordered relations
The fuzzy relation is fuzzy partial ordered relation if it satisfy following
conditions
a) is reflexive(R(x,x)=1 for all xU)
b) is symmetric(If R(x,y)0 R(y,x)=0 for all xy)
c) is transitive(R(x,z)supminR(x,y),R(y,z) for all x,zU
 1 0,5 0.6 0.8 


 0 1 0.7 0.9 
Example: Fuzzy relation R  
is fuzzy partial ordered relation
0 0
1
1 


0 0

0
1


Note: Fuzzy relation R is fuzzy partial ordered relation if ad only if its -cut is
patial ordered relation for all 0,1.
Proof: We leave to reader.
Transitivity properties for fuzzy relations
We define the following transitivity conditions
1) Strong stochastic transitivity(S-transitivity)
minR(x,y,R(y,z)0.5R(x,z)maxR(x,y),R(y,z)
2) Moderate stochastic transitivity
minR(x,y,R(y,z)0.5R(x,z) minR(x,y,R(y,z)
3) Weak stochastic transitivity
minR(x,y,R(y,z)0.5R(x,z)0.5
4) -transitivity
minR(x,y,R(y,z)0.5
R(x,z)maxR(x,y,R(y,z)+(1-)minR(x,y,R(y,z)
5) G-transitivity
63
R(x,z) R(x,y+R(y,z)-1
The
G-transitivity is often called group transitivity because if n elements
have preference R which are linear ordered then
a) R(x,y)0,1
b) R(x,x)=0
c) R(x,y+R(y,z)=1 for xy
d)
R(x,z) R(x,y+R(y,z)-1
It is well-known that from any reflexive binary relation R in a set of alternatives
A, a classical preference structure (P,I,J) can be constructed in the following
way:
t
(i) P = R ∩ coR ;
t
(ii) I = R ∩ R ;
t
(iii) J = co R ∩ coR .
In other words, an alternative a is strictly preferred to b (or (a,b) P) if and
only if a is at least as good as b, and b is not at least as good as a. Similarly, a
user is indifferent to two alternatives a and b if and only if a is at least as good
as b and b is at least as good as a. Finally, two alternatives a and b are
incomparable if and only if a is not at least as good as b and b is not at least as
good as a. Also, it is important to note the equality R = P  I, which means that
R is the large preference relation of the preference structure (P,I,J) constructed
in this way.
Together, the above construction and re-construction define a full
characterization of a classical preference structure. Note, that users can be
assisted during a preference elicitation process through an interactive conjoint
analysis over A with an agent who has a knowledge base of these preference
axioms. Since preference structures are based on classical set theory and are
therefore restricted to classical relations, they do not allow expressing degrees
of strict preference, indifference or incomparability. This can be seen as a
serious drawback to the practical use of these structures since statements over
degrees of and incomparability of preferences are natural in most real world
decision making.
64
Preference Structures Without Incomparability
An important class of preference structures consists of those structures
for which there are no couples of incomparable alternatives. A preference
structure of the form (P,I,J=) is called a preference structure without
incomparability, and will be denoted shortly as a preference structure (P,I). The
following theorem provides a useful characterization of a preference structure in
terms of its large preference relation. Recall that a binary relation R in U is
t
2
called complete if and only if R  R = U .
Theorem (Roubens and Vincke 1985). A preference structure (P,I,J) on U is a
preference structure (P,I) on U if and only if its large preference relation is
complete.
Two different types of fuzzy preference structures without incomparability
can be distinguished.
Theorem (De Baets and Van de Walle 1995). A fuzzy preference structure
(P,I,J) on U with fuzzy large preference relation R in U is a fuzzy preference
structure (P,I) on A of Type 1 if and only if
(x,y)U2 maxR(x,y),R(y,x)=1
Theorem (De Baets and Van de Walle 1995). A fuzzy preference structure
(P,I,J) on U with large fuzzy preference relation R in A is a fuzzy preference
structure (P,I) on U of Type 2 if and only if
(x,y)U2 R(x,y)+R(y,x)1
In both classes, the following relationship between the fuzzy strict
preference relation P and the fuzzy large preference relation R holds:
(x,y)U2 R(x,y)= 1-R(y,x)
Quasi-order relations and the analysis of preference relations
A binary (fuzzy) relation R in a universe U is called:
(i) reflexive if and only if xU2 R(x,x)= 1
(ii) a (fuzzy) quasi-order relation in U if and only if it is reflexive and transitive.
Theorem 4 (Fodor and Roubens 1994). Consider a binary fuzzy relation R in a
universe U. R is a fuzzy quasi-order relation in U if and only if for all values of α
(with α belonging to the interval <0,1>) it holds that Rα is a (crisp) quasi-order
relation in U.
65
The starting point of the analysis is the realization that every row in the matrix
representation of a preference relation is a profile of the preferences a user has
for an alternative compared to all other alternatives. The i-th row of P contains
all preferences of the form P(xi,xj) the number of alternatives. Recall that we can
denote the i-th row of P as the afterset aiP.
What we can to do is to compare all rows, or all profiles, in a given preference
matrix two by two. We compute how much every row is included in any other
row. The degree to which one row is included in another reflects how much of
one preference profile is contained in another profile or how strongly a user’s
preferences on one alternative are related to his preferences on another
alternative. We summarize that information in a new matrix which represents a
quasi-order relation, and analyze that fuzzy relation at every relevant cut.
From a strict preference relation P, we construct the so-called dependency
relation D, a binary fuzzy relation in the set of alternatives, in the following way:
D(xi,xj)=SH(aiPajP)
with SH(aiPajP) the subsethood of the afterset xiP in the set xjP. The
subsethood measures the degree to which the afterset of xj is included in the
afterset of xj. We for instance use the following definition of subsethood:
SH ( A, B) 
1
 min 1,1  A( x)  B( x)
n xU
We then construct the fuzzy quasi-order closure Q of the dependency
relation D. Obviously, Q is a transitive fuzzy relation in A. The α-cuts of Q have
the following interpretation: (ai,aj)Q if and only if ai is at most as good as aj,
with degree of confidence α.
Each such α-cut is a quasi-order relation in the set of alternatives U. To Qα
corresponds an equivalence relation Eα in A defined by
(x,y)E (x,y)E( y,x)E
The equivalence relation partitions the set of alternatives into classes of
alternatives that are equally good. The equivalence class [a]α of an alternative a
is given by
[x]={y;(x,y)E}
The corresponding quotient set Uα is then given by
U={[x];xU}
The quasi-order relation Qα induces an order relation ≤in the quotient set, in the
following way:
[x] [y](x,y)Q
A relationship of the form [x]α < [y]α means that the alternatives in [y]α are better
than the alternatives in [x]α.
66
A SINGLE EXAMPLE
Consider the following multi-criteria decision problem. A user describes a job
requirement in terms of three criteria C1 (Monthly Salary), C2 (Location) and C3
(Starting date) where each criteria has a particular ideal preference. For
instance ‘5,000 US dollar’, ‘Within a radius of at most 30 kilometers of my
home’, and ‘The earliest at the first week of next month’, respectively. Assume
the user is currently evaluating 5 job offers. Every job description contains
particular values on the selected criteria, and these differ more or less
significantly from the ideal values he has in mind. Clearly, the magnitude of
these value differences will determine the preferences the user has with respect
to the various job offers. Assume that the user has the following preferences,
summarized in the following preference relation P:
 0.1

 0.2
P   0.7
 0.1

 0.6
0.2 

0.1 
0.3 
0.5 

0.5 0.5 
0.3
0.5
0.8
0.7
where P(i,j) is the degree to which he prefers his/her own value on the j-th
criterion to the value suggested in the description of the i-th job offer on the j-th
criterion. For example, the value of 0.1 for P(1,1) indicates that the user slightly
prefers (to a degree 0.1) her ideal Monthly Salary value of 5,000 dollar to the
Monthly Salary value proposed in the first job’s description. Note that this
preference relation is not a binary relation on U but a relation on U × C, with U
the set of Alternatives (whose elements are the five job offers) and C = {C1, C2,
C3} the set of Criteria.
We should make two important observations here. First, it happens that
in our particular example all values in P are strictly larger than zero. This means
that none of the job offers contains a value that is ‘better than ideal’ for the user,
for instance working a monthly salary of 6,000 dollar, or a job within a range of
30 km. This suggests that perhaps the values she set for a good job are too
high and hard to meet in real life. Secondly, the smaller a degree of preference
in P is, the better the corresponding value of that offer is to the user. This
implies that the interpretation of the α-cuts of Q will be: for any two job offers xi
and yj, (if and only if job offer yi,(yi,yj)Q is at least as good as job offer yj, with
degree of confidence α.
The dependency relation D, measuring the degree of inclusion of the
various rows PI as a binary relation in the set of jobs, and its transitive closure Q
are given by:
1

 0.9
D   0.6

 0.77
 0.67






0.8 0,93 1 0.93 
0.73 0,93 0.83 1 
0.97
1
0.67
1
1
1
1
1
0.97 1
0.77 0.87
1

 0.9
Q   0.6

 0.8
 0.8






0.8 0,93 1 0.93 
0.8 0,93 0.83 1 
0.97
1
0.8
1
1
1
1
1
0.97 1
0.83 0.87
67
Note that we have quite a large difference for some positions in Q as compared
to D – e.g. the most extreme change is for Q(3,1): a change from 0.6 to 0.8.
Indeed, in general, the transitive closure will strengthen the weakest links, i.e.,
increase the degrees of the weakest inclusions.
From Q, it follows that we must consider α-cuts at the following levels:
{0.8,0.83,0.87,0.9,0.93,0.97,1}, leading to the following crisp cut relations:
1

1
Q0.83   0

0
0

1 1 1 1

1 1 1 1
0 1 1 1

0 1 1 1
0 1 1 1 
1

1
Q0.87   0

0
0

1 1 1 1

1 1 1 1
0 1 0 1

0 1 1 1
0 1 0 1 
1

1
Q0.9   0

0
0

1 1 1 1

1 1 1 1
0 1 0 1

0 1 1 1
0 1 0 1 
1

0
Q0.93   0

0
0

1 1 1 1

1 1 1 1
0 1 0 1

0 1 1 1
0 1 0 1 
1

0
Q0.97   0

0
0

1 1 1 1
1 1 1 1
0 1 0 0





0 0 1 0
0 0 0 1 
1

0
Q1   0

0
0






0 0 1 0
0 0 0 1 
0 1 1 1
1 1 0 1
0 1 0 0
Note that Q0.8 is the identity matrix.
Let us now analyze Q at the given cut-levels. At α = 0.8, the user is indifferent to
all job offers. This means that, at this level, no significant differences exist
among the various job offers she is considering. At α = 0.83, we observe a first
separation among the job offers. The user is still indifferent to Offers 1 and 2, as
well
Fuzzy functions
One of the fundamental conceptions of mathematics is the function f:AB
. It is nonempty binary relation fAB satisfying conditions
a) xA yB (x,y)f
b) (x1,y)f(x2,y)fx1=x2.
Let F(U) and F(V) are sets of all fuzzy sets on universes U,V. Then a fuzzy
function U in V denoted by f:UV is a map
f: F(U)F(V)
If two fuzzy functions f a g are given
68
f:UV g:VW
the composition
gf:UW
Examlle: Let U={1,2} and V=R=(-,) and
0, x5

 x  5; x  5,6

f(1)  
7  x; x  6,7

0, x 7
0, x3

 x  3; x  3,4

f(2)  
5  x; x  4,5

0, x5
Then f is fuzzy function from U={1,2} in V=R=(-,).
An alternative definition of a fuzzy function is by using notion of fuzzy relation.
Definition: The fuzzy function from U in V, denoted by f:UV, is fuzzy
subset of the product UV.
Remark: It is often considered as the intensity of relation between x and y.
Example: A fuzzy function
approximatively equal 2y.
f ( x, y )  e  x  2 y describes the statement x is
0, x1
0, x 2


 x  1, x  1,2
 x  1, x  2,3
~
~


Example: Let A ={R,  A x  
}, B ={R,  B x  
}
3  x, x  2,3
3  x, x  3,4


0, x 3
0, x 4
 1   ,3   ;   (0,1

R;   0
Z=x+y. Then A = 


 2   ,4   ;   (0,1

R;   0
B = 


If x 1   ,3    and y 2   ,4    then value membership function is more
then ((z) max min{  A (x),  B (y)} and that
x, y 
x  yz
 A (x)   ,  B (y)    x1,  1   , x2,  3   , y1,  2   , y 2,  4  
z1,  x1,  y1,  1    2    3  2 , z 2,  x2,  y 2,  3    4    7  2
69
0, x3

x3
, x  3,5
~

},
C ={R, C z    2
7x

, x  5,7
 2

0, x 7
~
Note: Let A is fuzzy set of U and f is mapping U in V. Then usually projection
~
~
A onto B Is fuzzy set with membership function
 B ( y)  max  A x 
y   A x 
~
Example: Let A ={(-2,0.4), (-1,0.2),(0,1),(1,0,5), (2,0.8)} and y=x2. Then
~
~
projection A is the fuzzy set B ={(4,max{((-2)2,0.4), (22,0.8)}, (1,max{((-1)2,0.2),
(12,0.5)},(0,1)}={(0,1),}1,0.5),(4,0.8)}
0, x1

 x  1, x  1,2
~

Example: Let A ={R,  A x  
} and y=x2
3

x
,
x

2
,
3


0, x 3
Then
 1   ,3   ;   (0,1

R;   0
A = 
 y  x 2  1   2 , 3   2  y1,  1   2 


 1 
y1,  1,  2  3  y 2, ,
If   0  y1  1, y2  9. If   1  y1  4, y2  4. and

 y

 y x   
3 

0, y 1
 1, y  1,4
y , y  4,9
0, y9
Problems:
~
1) Let B
=(0,0,2),(1,0,4), (2,0.5), (3,0,2),(4,0,8),B(x)=0, for x=5,6,7,8,,,,,
Let + and * are arithmetic (functions z=x+y=f(x,y),z=xy=g(x,y)).
What are values membership function ? .
70
~
2) Let B
=(0,0,2),(1,0,4), (2,0.5), (3,0,2), (4,0,8),B(x)=0, for x=5,6,7,8,,,,,
~
Let A
=(0,0,1),(1,0,4), (2,0.3), (3,0,2),(4,0),B(x)=0, for x=5,6,7,8,,,,,
Let + and * are arithmetic (functions z=x+y=f(x,y),z=xy=g(x,y)).
What are values membership function ? .
~
3) Let A
=(-1,1),(0,0,1),(1,0,4), (2,0.3), (3,0,2),(4,0),B(x)=0 else. What
are values membership function if y=x2.
~
~
A 1=(a,0.3),(b,0.8),(c,0.5), A 2=(x,0.5),(y,0.6). What are values
~ ~
membership function of A 1 A 2.
~
5) Let matrix representation fuzzy relation A 1 is
4) Let
a
b
c
1 0.3 0,5 0,3
2 0,2 0.2 0,5
3 0.2 0,7 0,3
~
Let matrix representation fuzzy relation A 2 is
x
y
z
a 0.6 0,2 0,4
b 0,2 0.4 1
c 1 0,3 0,5
~
~
What is the max-product composition of fuzzy relations A 1 and A 2?
6) Prove
a) f  A1   A2  f  A1  f  A2

    
b) f  A1   A2   f  A1   f  A2 .
c) f 1  A1   A2   f 1  A1   f 1  A2 
d) f 1  A1   A2   f 1  A1   f 1  A2  .
0, x 0

sin x; x  0, 

7) Let  A x  
What is the membership function of fuzzy function
0, x


Y=sin3x ?
9) Let R=(0,10,1,(x,y)=xy2). What is the projection R onto axis?
x  y2
10) Let R=(0,10,1,(x,y)=
). What is the projection R onto axis?
2
11) Find of the strongly transitive closer of fuzzy relation
 1 0.4 0.4 0.5 


 0.3 0.3 0.7 0.4 
a)  R  
0.6 0.5 0.8 0.6 


 0.6 0.5 0.5 0.7 


71
 0,7 0.5 0.4 0.5 


 0.2 0.6 0.7 0.4 
b)  R  
0.6 0.5 0.8 1 


 0.4 0.5 0.8 0.7 


 1 0.2 0.4 0.5 


 0.2 0.3 0.7 0.4 
c)  R  
0.6 0.5 0.8 1 


 0.6 0.5 0.2 0.7 


12) Find of the minimum fuzzy relation similarity and equivalence if
 1 0.4 0.4 0.5 


 0.3 0.3 0.7 0.4 
a)  R  
0.6 0.5 0.8 0.6 


 0.6 0.5 0.5 0.7 


0
,
7
0
.
5
0
.
4
0.5 



 0.2 0.6 0.7 0.4 
b)  R  
0.6 0.5 0.8 1 


 0.4 0.5 0.8 0.7 


1
0
.
2
0
.
4
0
.
5




 0.2 0.3 0.7 0.4 
c)  R  
0.6 0.5 0.8 1 


 0.6 0.5 0.2 0.7 


13)