Fuzzy Relations
Review
Fuzzy Relations
Crisp Relation
Definition (Product set):
Let A and B be two nonempty sets, the
product set or Cartesian product A  B is
defined as follows,
A  B  {(a, b) | a  A, b  B }
Extension to n sets
A1A2...An =
{(a1, ... , an) | a1  A1, a2  A2, ... , an  An }
Crisp Relation
Example: A  {a1, a2, a3}, B  {b1, b2}
A  B  {(a1, b1), (a1, b2), (a2, b1), (a2, b2),
(a3, b1), (a3, b2)}
b2
b1
a1
(a1 , b2)
(a2 , b2)
(a3 , b2)
(a1 , b1)
(a2 , b1)
(a3 , b1)
a2
Product set A  B
a3
Crisp Relation
A  A  {(a1, a1), (a1, a2), (a1, a3), (a2, a1), (a2,
a2), (a2, a3), (a3, a1), (a3, a2), (a3, a3)}
a3
(a1 , a3)
(a2 , a3)
(a3 , a3)
(a1 , a2)
(a2 , a2)
(a3 , a2)
(a1 , a1)
(a2 , a1)
(a3 , a1)
a2
a1
a1
a2
a3
Cartesian product A  A
Crisp Relation
Definition

Binary Relation
R = { (x,y) | x  A, y  B }

n-ary Relation
(x1, x2, x3, … , xn)  R ,
R  A1  A2  A3  …  An

AxB
Crisp Relation
Domain and Range
dom(R) = { x | x  A, (x, y)  R for some y  B }
ran(R) = { y | y  B, (x, y)  R for some x  A }
A
B
A
B
x1
dom (R )
R
ran(R )
dom(R) , ran(R)
f
y1
x2
y2
x3
y3
Mapping y  f(x)
Crisp Relation
Characteristics of relation
(1) One-to-many
 x  A, y1, y2  B (x, y1)  R, (x, y2)  R
(2) Surjection (many-to-one)
f(A)  B or ran(R)  B. y  B,  x  A, y  f(x)
Thus, even if x1  x2, f(x1)  f(x2) can hold.
A
B
A
y
x
x1
B
f
y
1
y
x2
2
One-to-many relation
(not a function)
Surjection
Crisp Relation
(3) Injection (into, one-to-one)
for all x1, x2  A, x1  x2 , f(x1)  f(x2).
if R is an injection, (x1, y)  R and (x2, y)  R then x1 
x2.
(4) Bijection (one-to-one correspondence)
both a surjection and an injection.
A
x1
x2
x3
A
B
f
Injection
y1
y2
y3
y4
x1
B
f
y1
x2
x3
y2
y3
x4
y4
Bijection
Crisp Relation
Representation of Relations
(1)Bipartigraph
representing the relation by drawing arcs or edges
(2)Coordinate diagram
plotting members of A on x axis and that of B on
y axis
y
A
B
4
a1
b1
a2
b2
a3
a4
x
-4
4
b3
-4
Binary relation from A to B
Relation of x2 + y2  4
Crisp Relation
(3) Matrix
MR  (mij)
 1, (ai , b j )  R
mij  
 0, (ai , b j )  R
i  1, 2, 3, …, m
j  1, 2, 3, …, n
(4) Digraph
R
a1
a2
a3
a4
b1 b2 b3
1
0
0
0
0
1
1
0
Matrix
0
0
0
1
1
2
3
4
Directed graph
Crisp Relation
Operations on relations R, S  A  B
(1) Union T  R  S
If (x, y)  R or (x, y)  S, then (x, y)  T
(2) Intersection T  R  S
If (x, y)  R and (x, y)  S, then (x, y)  T.
(3) Complement
If (x, y)  R, then (x, y)  RC
(4) Inverse
R-1  {(y, x)  B  A | (xR
, y)  R, x  A, y  B}
(5) Composition T
R  A  B, S  B  C , T  S  R  A  C
T  {(x, z) | x  A, y  B, z  C, (x, y)  R, (y, z)  S}
Types of Relation on a set
Reflexive relation
x  A  (x, x)  R or R(x, x) = 1,  x  A


irreflexive
if it is not satisfied for some x  A
antireflexive
if it is not satisfied for all x  A
Symmetric relation
(x, y)  R  (y, x)  R or R(x, y) = R(y, x),  x, y  A


asymmetric or nonsymmetric
when for some x, y  A, (x, y)  R and (y, x)  R.
antisymmetric
if for all x, y  A, (x, y)  R and (y, x)  R
Types of Relation on a Set
Transitive relation
For all x, y, z  A
(x, y)  R, (y, z)  R (x, z)  R
2
2
1
3
3
1
4
4
(b) R
(a) R
Transitive Closure
Types of Relation on a Set
Equivalence relation
(1) Reflexive
x  A  (x, x)  R
(2) Symmetric
(x, y)  R  (y, x)  R
(3) Transitive relation
(x, y)  R, (y, z)  R  (x, z)  R
Types of Relation on a Set
Equivalence classes
a partition of A into n disjoint subsets A1, A2, ... , An
A
A1
A2
b
a
b
d
d
e
a
e
c
c
(a) Expression by set
(b) Expression by undirected graph
Partition by equivalence relation
(A/R)  {A1, A2}  {{a, b, c}, {d, e}}
Types of Relation on a Set
Compatibility relation (tolerance relation)
(1) Reflexive relation
(2) Symmetric relation
x  A  (x, x)  R
(x, y)  R  (y, x)  R
A
A1
A2
b
b
a
d
e
d
a
e
c
c
(a) Expression by set
(b) Expression by undirected graph
Partition by compatibility relation
Types of Relation on a Set
Pre-order relation
(1) Reflexive relation
x  A  (x , x )  R
(2) Transitive relation
(x, y)  R, (y, z)  R  (x, z)  R
A
e
e
b
a
f
d
a
b, d
c
h
g
f, h
c
(a) Pre-order relation
Pre-order relation
(b) Pre-order
g
Types of Relation on a Set
Order relation
(1) Reflexive relation
x  A  (x, x)  R
(2) Antisymmetric relation
(x, y)  R  (y, x)  R
(3) Transitive relation
(x, y)  R, (y, z)  R  (x, z)  R

strict order relation
(1’) Antireflexive relation
x  A  (x, x)  R

total order or linear order relation
(4)  x, y  A, (x, y)  R or (y, x)  R
Types of Relations on a Set
Comparison of relations
Property
Relation
Equivalence
Compatibility
Pre-order
Order
Strict order
Reflexive
Anti
reflexive




Symmetric
Anti
symmetric



Transitive






Fuzzy Relation
Definition of fuzzy relation

Crisp relation
membership function R(x, y)
R : A  B  {0, 1}
R (x, y) =

1 iff (x, y)  R
0 iff (x, y)  R
Fuzzy relation
R : A  B  [0, 1]
R = {((x, y), R(x, y))| R(x, y)  0 , x  A, y  B}
Fuzzy Relation
R
R  A B
1
0.5
(a1, b1 )
(a 1, b 2 )
( a 2,b1)
...
Fuzzy relation as a fuzzy set
A B
Fuzzy Relation
Example
Crisp relation R
R(a, c)  1, R(b, a)  1, R(c, b)  1 and R(c, d)  1.
Fuzzy relation R
R(a, c)  0.8, R(b, a)  1.0, R(c, b)  0.9, R(c, d)  1.0
a
a
A
0.8
c
1.0
b
b
d
(a) Crisp relation
a
b
c
d
a
0.0
0.0
0.8
0.0
b
1.0
0.0
0.0
0.0
c
0.0
0.9
0.0
1.0
d
0.0
0.0
0.0
0.0
A
c
0.9 1.0
d
(b) Fuzzy relation
crisp and fuzzy relations
corresponding matrix
Fuzzy Relation
Operation of Fuzzy Relation
1) Union relation
 (x, y)  A  B
R  S (x, y)  Max [R (x, y), S (x, y)]  R (x, y)  S (x, y)
2) Intersection relation
R  S (x) = Min [R (x, y), S (x, y)] = R (x, y)  S (x, y)
3) Complement relation
 (x, y)  A  B
R (x, y)  1 - R (x, y)
4) Inverse relation
For all (x, y)  A  B,
R-1 (y, x)  R (x, y)
Fuzzy Relation
Examples
Fuzzy Relation
(Standard) Composition

For (x, y)  A  B, (y, z)  B  C,
RS (x, z) = Max [Min (R (x, y), S (y, z))]
y
=  [R (x, y)  S (y, z)]
y
MR  S  MR  M S

Example
=>
Fuzzy Relation
=>
Composition of fuzzy relation
Note: Matrix Multiplication
Fuzzy Relation
-cut of fuzzy relation
R = {(x, y) | R(x, y)  , x  A, y  B} : a crisp relation.
Example
Fuzzy Relation
Decomposition of Fuzzy Relation
R x, y      R x, y  for ( x,y)  AxB
 R ( x, y )   R x, y 
 [ 0 ,1]

Example
Fuzzy Relation
Projection
For all x  A and y  B
 RB  y   Max  R x, y  : Projectionto B
x
 R x   Max  R x, y  : Projection to A
A

y
Example
Fuzzy Relation

Projection in n dimension
R
X i1 X i 2 X ik
xi1, xi 2 ,, xik   Max  R x1, x2 ,, xn 
X , X ,, X
j1

Cylindrical extension
C(R) (a, b, c)  R (a, b)
a  A, b  B, c  C

Example
j2
jm
Types of Fuzzy Relations
Reflexive



R( x, x)  1 for all x  X
Irreflexive
Antireflexive
Epsilon Reflexive
Symmetric


Asymmetric
Antisymmetric
R( x, x)  1 for some x  X
R( x, x)  1 for all x  X
R( x, x)   for all x  X
R( x, y)  R( y, x) for all x  X
R( x, y)  R( y, x) for some x  X
R( x, y)  0 and R( y, x)  0  x  y for all x, y  X
Types of Fuzzy Relations
Transitive (max-min transitive)
R( x, z )  max min[ R( x, y ), R( y, z )] for all x,z  X
yY


Non-transitive:
For some (x,z), the above do not satisfy.
Antitransitive:
R( x, z )  max min[ R( x, y ), R( y, z )] for all x,z  X
yY
Example: X = Set of cities, R=“very far”
Reflexive, symmetric, non-transitive
Types of Fuzzy Relations
Transitive Closure



Crisp: Transitive relation that contains
R(X,X) with fewest possible members
Fuzzy: Transitive relation that contains
R(X,X) with smallest possible membership
Algorithm:
1. R '  R  ( R  R).
2. If R '  R, make R  R ' and go to step1
3.Stop : R '  RT
Types of Fuzzy Relations
Fuzzy Equivalence or Similarity Relation


Reflexive, symmetric, and transitive
Decomposition:
R

  R
 [ 0,1]

R is a crisp equivalence relation.
Set of partitions:
 (R)  { ( R ) |  [0,1]}

Partition Tree
Types of Fuzzy Relations
Fuzzy Compatibility or Tolerance Relation


Reflexive and symmetric
Maximal compatibility class and complete cover
Compatibility class Subset A of X such that x, y  R
Maximal compatibility class: largest compatibility class
Complete cover: Set of maximal compatibility classes



Maximal alpha-compatibility class
Complete alpha-covers
Note:
Relation from distance metrics forms tolerance
relation in clustering.
Fuzzy Morphism
Homomorphism


Preserve relations by a function
Example:
Log function preserves the order of real
data.
Let R( X , X )  X  X and Q(Y , Y )  Y  Y .
h : X  Y is said to be homomorhis
m if
 x1 , x 2  R  h( x1 ), h(x2 )  Q
Let R( X , X )  X  X and Q(Y , Y )  Y  Y .
h : X  Y is said to be homomorhis
m if
R( x1 , x 2 )  Q(h( x1 ), h(x2 ))
Other Compositions
Sup-I composition
i
[ P  Q](x, z )  sup yY i[ P( x, y), Q( y, z )]
INF-omega i composition

Degree of Implication
i (a, b)  supx [0,1] | i(a, x)  b


i=min: a < b then 1, otherwise b.
INF-omega i composition
i
( P  Q)(x, z )  inf i [ P( x, y), Q( y, z )]
yY
Extension of fuzzy set
Extension by relation

Extension of fuzzy set
x  A, y  B y  f(x) or x  f -1(y)
for y  B
 B  y   Max  A x 
x f
1
y
if f -1(y) 
Example: A  {(a1, 0.4), (a2, 0.5), (a3, 0.9), (a4, 0.6)}, B  {b1, b2, b3}
f
-1(b
1)
 {(a1, 0.4), (a3, 0.9)}, Max [0.4, 0.9]  0.9
 B' (b1)  0.9
f
-1(b
2)
 {(a2, 0.5), (a4, 0.6)}, Max [0.5, 0.6]  0.6
 B' (b2)  0.6
f
-1(b
3)
 {(a4, 0.6)}
 B' (b3)  0.6
B'  {(b1, 0.9), (b2, 0.6), (b3, 0.6)}
Extension of Fuzzy Set
Extension principle

Extension principle
A1  A2  ...  Ar ( x1  x2  ...  xr )
 Min [ A1 (x1), ... , Ar(xr) ]
f(x1, x2, ... , xr) : X  Y

1
0 , if f  y   

B  y  

Max
Min  A  x1 ,,  A  xr  , otherwise

 y  f  x , x ,, x 
 
1
2
r
1
r

Extension of Fuzzy Set
Example:
A( x)  .5 /(1)  1 / 0  .5 / 1  .3 / 2
B( x)  .5 / 2  1 / 3  .5 / 4  .3 / 5
f :XX  X
f ( x, x)  x1  x2
f ( A, B)  .5 /(2)  .5 /(3)  .5 /(4)  .3 /(5)
 1/0  .5/2  ...  .3/10
f :XX  X
f ( x, x)  x1  x2
f ( A, B)  .5 / 1  .5 / 2  1 / 3  .5 / 4  .5 / 5  .3 / 6  .3 / 7
Extension of fuzzy set
Extension by fuzzy relation
For x  A, y  B, and B  B
B' (y)  Max [Min (A (x), R (x, y))]
x  f -1(y)

Example
For b1 Min [A (a1), R (a1, b1)]  Min [0.4, 0.8]  0.4
Min [A (a3), R (a3, b1)]  Min [0.9, 0.3]  0.3
Max [0.4, 0.3]  0.4  B' (b1)  0.4
For b2, Min [A (a2), R (a2, b2)]  Min [0.5, 0.2]  0.2
Min [A (a4), R (a4, b2)]  Min [0.6, 0.7]  0.6
Max [0.2, 0.6]  0.6 B' (b2)  0.6
For b3, Max Min [A (a4), R (a4, b3)]
 Max Min [0.6, 0.4]  0.4
 B' (b3)  0.4
B'  {(b1, 0.4), (b2, 0.6), (b3, 0.4)}
Extension of Fuzzy Set

Example
A  {(a1, 0.8), (a2, 0.3)}
B  {b1, b2, b3}
C  { c 1, c 2, c 3}
B'  {(b1, 0.3), (b2, 0.8), (b3, 0)}
C'  {(c1, 0.3), (c2, 0.3), (c3, 0.8)}
Extension of fuzzy set
Fuzzy distance between fuzzy sets

Pseudo-metric distance
(1) d(x, x)  0,  x  X
(2) d(x1, x2)  d(x2, x1),  x1, x2  X
(3) d(x1, x3)  d(x1, x2)  d(x2, x3),  x1, x2, x3  X
+ (4) if d(x1, x2)=0, then x1 = x2  metric distance

Distance between fuzzy sets
    , d(A, B)() Max
  d(a, b)
[Min (A(a), B(b))]
Extension of Fuzzy Set
Example
A  {(1, 0.5), (2, 1), (3, 0.3)} B  {(2, 0.4), (3, 0.4), (4, 1)}