Fuzzy Relations
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Transcript Fuzzy Relations
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
A1A2...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,
RS (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
yY
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
yY
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 yY i[ P( x, y), Q( y, z )]
INF-omega i composition
Degree of Implication
i (a, b) supx [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 )]
yY
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 :XX X
f ( x, x) x1 x2
f ( A, B) .5 /(2) .5 /(3) .5 /(4) .3 /(5)
1/0 .5/2 ... .3/10
f :XX 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)}