fuzzy relations -...

Fuzzy Relations

Review

Fuzzy Relations

Crisp RelationCrisp RelationCrisp RelationCrisp Relation

DefinitionDefinitionDefinitionDefinition (Product(Product(Product(Product set)set)set)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 }


Example:Example:Example:Example: A = {a1, a2, a3}, B = {b1, b2}

A × B = {(a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2)}

(a1, b2) (a2, b2)

(a1, b1)

(a3, b2)

(a2, b1) (a3, b1)

b2

a1

b1

a3a2Product set A × B


A × A = {(a1, a1), (a1, a2), (a1, a3), (a2, a1), (a2, a2), (a2, a3), (a3, a1), (a3, a2), (a3, a3)}

(a1, a2) (a2, a2)

(a1, a1)

(a3, a2)

(a2, a1) (a3, a1)

a2

a1

a1

a3a2

(a1, a3) (a2, a3) (a3, a3) a3

Cartesian product A × A


Definition� Binary RelationBinary RelationBinary RelationBinary Relation

RRRR ==== {{{{ ((((xxxx,,,,yyyy)))) |||| xxxx ∈∈∈∈ AAAA,,,, yyyy ∈∈∈∈ BBBB }}}} ⊆⊆⊆⊆ AAAA xxxx BBBB

� nnnn----aryaryaryary RelationRelationRelationRelation

((((xxxx1111,,,, xxxx2222,,,, xxxx3333,,,, � ,,,, xxxxnnnn)))) ∈∈∈∈ RRRR ,,,,

RRRR ⊆⊆⊆⊆ AAAA1111 ×××× AAAA2222 ×××× AAAA3333 ×××× � ×××× AAAAnnnn


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 }

Rdom(R) ran(R )

A B

fx1

x2

x3

A B

y1

y2

y3

Mapping y = f(x)dom(R) , ran(R)

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

y

1x

y

2

One-to-many relation(not a function)

fx1

x2

A B

y

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.

fx1

A B

y1x2 y2x3 y3

y4

Injection

fx1

x2

A B

y1

y2

x3 y3

x4 y4

Bijection


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

Relation of x2 + y2 = 4

A B

a1 b1

a2b2

a3

b3a4

Binary relation from A to B

4

-4

-4 4

y

x

Crisp Relation

(3) Matrix

(4) Digraph

R b1 b2 b3

a1 1 0 0

a2 0 1 0

a3 0 1 0

a4 0 0 1

MR = (mij)

∉

∈=

Rba

Rbam

ji

jiij

),(,0

),(,1 i = 1, 2, 3, …, m

j = 1, 2, 3, …, n

2 3

4

1

Matrix 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 | (x, 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}

R

Types of Relation on a set

Reflexive relationx ∈ A → (x, x) ∈ R or µR(x, x) = 1, ∀ x ∈ A

� irreflexive

if it is not satisfied for some x ∈ A

� antireflexiveif it is not satisfied for all x ∈ A

Symmetric relation (x, y) ∈ R → (y, x) ∈ R or µR(x, y) = µR(y, x), ∀ x, y ∈ A

� asymmetric or nonsymmetricwhen 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 SetTransitive relationFor all x, y, z ∈ A

(x, y) ∈ R, (y, z) ∈ R→ (x, z) ∈ R

1 3

4

2

(a) R

1

4

2

(b) R∞

Transitive Closure

3

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


Equivalence classes

a partition of A into n disjoint subsets A1, A2, ... , An

(a) Expression by set

b

ad

c

e

(b) Expression by undirected graph

Partition by equivalence relationπ(A/R) = {A1, A2} = {{a, b, c}, {d, e}}

b

a

c

d e

A1 A2

A


Compatibility relation (tolerance relation) (1) Reflexive relation (2) Symmetric relation

x ∈ A → (x, x) ∈ R (x, y) ∈ R → (y, x) ∈ R

(a) Expression by set

b

ad

c

e

(b) Expression by undirected graph

Partition by compatibility relation

b

a

c

d e

A1 A2

A


PrePrePrePre----order relationorder relationorder relationorder relation(1) Reflexive relation

x ∈ A → (x, x) ∈ R


(x, y) ∈ R, (y, z) ∈ R → (x, z) ∈ R

h g

fd

b

c

e

a

A

(a) Pre-order relation

a

c

b, d

e

f, h g

(b) Pre-order

Pre-order relation


Order relation(1) Reflexive relation

x ∈ A → (x, x) ∈ R

(2) Antisymmetric relation

(x, y) ∈ R → (y, x) ∉ R


(x, y) ∈ R, (y, z) ∈ R → (x, z) ∈ R

� strict order relation(1’) Antireflexive relationAntireflexive relationAntireflexive relationAntireflexive 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

P ropertyP ropertyP ropertyP roperty

RelationRelationRelationRelationReflexiveReflexiveReflexiveReflexive

AntiAntiAntiAnti

reflexivereflexivereflexivereflexiveSymmetricSymmetricSymmetricSymmetric

AntiAntiAntiAnti

symmetricsymmetricsymmetricsymmetricTransitiveTransitiveTransitiveTransitive

EquivalenceEquivalenceEquivalenceEquivalence � � �

CompatibilityCompatibilityCompatibilityCompatibility � �

P reP reP reP re----orderorderorderorder � �

OrderO rderO rderO rder � � �

S trict orderS trict orderS trict orderS trict order � � �

Fuzzy Relation

Definition of fuzzy relation

� Crisp relation

membership function µR(x, y)

µR : A × B → {0, 1}

� Fuzzy relationFuzzy relationFuzzy relationFuzzy relation

µR : A × B → [0, 1]

R = {((x, y), µR(x, y))| µR(x, y) ≥ 0 , x ∈ A, y ∈ B}

µR (x, y) =1 iff (x, y) ∈ R

0 iff (x, y) ∉ R

Fuzzy Relation

(a1, b1) ...(a1, b2 ) ( a2, b1)

0.5

1

Rµ

BAR ×⊆

BA×

Fuzzy relation as a fuzzy set

Fuzzy Relation

ExampleExampleExampleExampleCrisp relation R

µR(a, c) = 1, µR(b, a) = 1, µR(c, b) = 1 and µR(c, d) = 1.

Fuzzy relation R

µR(a, c) = 0.8, µR(b, a) = 1.0, µR(c, b) = 0.9, µR(c, d) = 1.0

a

b

c

d

a

b

c

d

0.8

1.0

0.9 1.0

(a) Crisp relation (b) Fuzzy relation

crisp and fuzzy relations

A

A 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

corresponding matrix

Fuzzy Relation

Operation of Fuzzy Relation 1) Union relation

∀ (x, y) ∈ A × B

µR ∪ S (x, y) = Max [µR (x, y), µS (x, y)] = µR (x, y) ∨ µS (x, y)

2) Intersection relation

µR ∩ S (x) = Min [µR (x, y), µS (x, y)] = µR (x, y) ∧ µS (x, y)

3) Complement relation

∀ (x, y) ∈ A × B

µR (x, y) = 1 - µR (x, y)

4) Inverse relation

For all (x, y) ⊆ A × B, µR-1 (y, x) = µR (x, y)

Fuzzy Relation

Examples

Fuzzy Relation

(Standard) Composition � For (x, y) ∈ A × B, (y, z) ∈ B × C,

µR•S (x, z) = Max [Min (µR (x, y), µS (y, z))]y

= ∨ [µR (x, y) ∧ µS (y, z)] y

MR • S = MR • MS

� Example

=>

Fuzzy Relation

=>

Composition of fuzzy relation

Note: Matrix Multiplication

Fuzzy Relation

α-cut of fuzzy relationRα = {(x, y) | µR(x, y) ≥ α, x ∈ A, y ∈ B} : a crisp relation.

Example

Fuzzy Relation

Decomposition of Fuzzy Relation

� Example

( ) ( )( )]1,0[

R ,),(

x )(for ,,

∈

=

∈•=

αα

αα

α

α

µµ

µαµ

yxyx

BAx,yyxyx

R

RR

U

Fuzzy Relation

Projection

� Example

( ) ( ) A toProjection : , yxMaxx Ry

RAµµ =

( ) ( ) B toProjection : ,

and allFor

yxMaxy

ByAx

Rx

RBµµ =

∈∈

Fuzzy Relation

� Projection in n dimension

� Cylindrical extension

µC(R) (a, b, c) = µR (a, b)

a ∈ A, b ∈ B, c ∈ C

� Example

( ) ( )nR

XXX

ikiiR xxxxxx Maxjmjj

ikXiXiX,,,,,, 21

,,,

21

21

21

KKK

Kµµ =

×××

Types of Fuzzy Relations

Reflexive

� Irreflexive

� Antireflexive

� Epsilon Reflexive

Symmetric

� Asymmetric

� Antisymmetric

XxxxR ∈= allfor 1),(

XxxxR ∈≠ somefor 1),(

XxxxR ∈≠ allfor 1),(

XxxxR ∈≥ allfor ),( ε

XxxyRyxR ∈= allfor ),(),(

XxxyRyxR ∈≠ somefor ),(),(

XyxyxxyRyxR ∈=→>> , allfor 0 ),( and 0),(


Transitive (max-min transitive)

� Non-transitive: For some (x,z), the above do not satisfy.

� Antitransitive:

Example: X = Set of cities, R=“very far”Reflexive, symmetric, non-transitive

X allfor )],(),,(min[max),( ∈≥∈

x,zzyRyxRzxRYy

X allfor )],(),,(min[max),( ∈<∈

x,zzyRyxRzxRYy


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:

TRR

RRRR

RRRR

=

=≠

∪=

'

''

'

:Stop 3.

1 step togo and make , If .2

).( .1 o


Fuzzy Equivalence or Similarity Relation

� Reflexive, symmetric, and transitive

� Decomposition:

� Partition Tree

[0,1]}|) ({(R)

:partitions ofSet

relation. eequivalenc crisp a is

]1,0[

∈=

⋅=

∏

∈

απ

α

α

α

α

α

R

R

RR U


Fuzzy Compatibility or Tolerance Relation� Reflexive and symmetric

� Maximal compatibility class and complete coverCompatibility class

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.

Ryx,such that of Subset >∈<XA

Fuzzy Morphism

Homomorphism

� Preserve relations by a function

� Example:

Log function preserves the order of real data.

QhhR

YXh

YYYYQXXXXR

>∈→<>∈<

→

×⊆×⊆

)(x),x(x,x

if mhomomorhis be tosaid is :

.),( and ),(Let

2121

))(x),x(()x,x(

if mhomomorhis be tosaid is :

.),( and ),(Let

2121 hhQR

YXh

YYYYQXXXXR

>

→

×⊆×⊆

Other Compositions

Sup-I composition

INF-omega i composition

� Degree of Implication

� i=min: a < b then 1, otherwise b.

� INF-omega i composition

)],(),,([sup),]([ zyQyxPizxQP Yy

i

∈=o

{ }bxaixbai ≤∈= ),(|]1,0[sup),(ω

)],(),,([),)(( inf zyQyxPzxQP iYy

i

ωω

∈

=o

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 if f -1(y) ≠∅

Example: A = {(a1, 0.4), (a2, 0.5), (a3, 0.9), (a4, 0.6)}, B = {b1, b2, b3}

f -1(b1) = {(a1, 0.4), (a3, 0.9)}, Max [0.4, 0.9] = 0.9

⇒ µB' (b1) = 0.9

f -1(b2) = {(a2, 0.5), (a4, 0.6)}, Max [0.5, 0.6] = 0.6

⇒ µB' (b2) = 0.6

f -1(b3) = {(a4, 0.6)}

⇒ µB' (b3) = 0.6

BBBB '''' ==== {({({({(bbbb1111,,,, 0000....9999),),),), ((((bbbb2222,,,, 0000....6666),),),), ((((bbbb3333,,,, 0000....6666)})})})}

( )( )

( )[ ]xy Ayfx

B Max µµ1−∈

′ =

Extension principle

� Extension principleµA1 × A2 × ... × Ar ( x1 × x2 × ... × xr )

= Min [ µA1 (x1), ... , µAr(xr) ]

f(x1, x2, ... , xr) : X → Y

( )( )

( )( ) ( )( )[ ]

∅=

=

=

−

otherwise , ,,

if , 0

1,,,

1

1

21

rAAxxxfy

B

xxMinMax

yf

y

r

r

µµ

µ

KK

Extension of Fuzzy Set


Example:

.3/10 ... .5/21/0

)5/(3.)4/(5.)3/(5.)2/(5.),(

),( :

5/3.4/5.3/12/5.)(

2/3.1/5.0/1)1/(5.)(

21

++++

−+−+−+−=

⋅=→×

+++=

+++−=

BAf

xxxxfXXXf

xB

xA

7/3.6/3.5/5.4/5.3/12/5.1/5.),(

),( :21

++++++=

+=→×

BAf

xxxxfXXXf

Extension by fuzzy relation For x ∈ A, y ∈ B, and B′ ⊆ B

µB' (y) = Max [Min (µA (x), µR (x, y))] x ∈ f -1(y)

� Example For b1 Min [µA (a1), µR (a1, b1)] =Min [0.4, 0.8] = 0.4

Min [µA (a3), µR (a3, b1)] =Min [0.9, 0.3] = 0.3

Max [0.4, 0.3] = 0.4 � µB ' (b1) = 0.4

For b2, Min [µA (a2), µR (a2, b2)] =Min [0.5, 0.2] = 0.2

Min [µA (a4), µR (a4, b2)] =Min [0.6, 0.7] = 0.6

Max [0.2, 0.6] = 0.6 �µB ' (b2) = 0.6

For b3, Max Min [µA (a4), µR (a4, b3)] =Max Min [0.6, 0.4] = 0.4

� µB ' (b3) = 0.4

B' ==== {(b1, 0.4), (b2, 0.6), (b3, 0.4)}


� ExampleA = {(a1, 0.8), (a2, 0.3)}

B = {b1, b2, b3}

C = {c1, c2, c3}

B' = {(b1, 0.3), (b2, 0.8), (b3, 0)}

C' = {(c1, 0.3), (c2, 0.3), (c3, 0.8)}


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

∀ δ ∈ ℜ +, µd(A, B)(δ) =Max [Min (µA(a), µB(b))]

δ = d(a, b)


Example

A = {(1, 0.5), (2, 1), (3, 0.3)} B = {(2, 0.4), (3, 0.4), (4, 1)}


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