CLASSICAL RELATIONS AND FUZZY RELATIONS

報告流程 卡氏積 (Cartesian Product) 明確關係 (Crisp Relations)

Cardinality Operations Properties 合成 (Composition)

模糊關係 (Fuzzy Relations) Cardinality Operations Properties Fuzzy Cartesian Product and Compositon Noninteractive Fuzzy Sets

Crisp Tolerance and Equivalence Relations Fuzzy Tolerance and Equivalence Relations Value Assignments

Cosine Amplitude Max-min Method Other Similarity Methods

Cartesian Product

Producing ordered relationships among sets

X × Y = {(x,y)│x X, y Y}∈ ∈ All the Ar = A

A1 × A2 × ……. × Ar = Ar

Cartesian Product

Example 3.1 Set A = { 0,1 } Set B = { a, b, c }

A × B = {(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)}

B × A = {(a,0),(a,1),(b,0),(b,1),(c,0),(c,1)}

A × A = A2 = {(0,0),(0,1),(1,0),(1,1)}

B × B = B2 ={(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)}

Crisp Relations

Measure by characteristic function ： χ X × Y = {(x,y)│x X, y Y}∈ ∈

Binary relation χX×Y(x,y)= 1, (x,y) X × Y ∈ 0, (x,y) X × Y χR(x,y)= 1, (x,y) X × Y ∈ 0, (x,y) X × Y

Crisp Relations

a b c

1

R = 2

3

111

111

111

　　

　　

　　

Sagittal diagram Relation Matrix

EX： X={1,2,3} Y={a,b,c}

Crisp Relations

Example 3.2 ( 一 )

X={1,2} Y={a,b} 1 a Locations of zero 2 b

R={(1,a),(2,b)} R X × Y ( 二 )

A={0,1,2} UA ： universal relation IA ： identity relation 以 A2 為例

UA = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)} IA = {(0,0),(1,1),(2,2)}

Crisp Relations

Example 3.3 Continous universes R={(x,y) | y ≥ 2x, x X, y Y}∈ ∈ χR(x,y)= 1, y ≥ 2x

0, Y < 2x

Cardinality of Crisp Relations

X ： n elements Y ： m elements

n X ： the cardinality of X

n Y ： the cardinality of Y Cardinality of the relation

n X × Y = nX * nY

power set The cardinality ： P(X × Y) n P(X × Y) = 2(nXnY)

Operations on Crisp Relations

Union

Intersection

Complement

Containment

Identity(Ø → O and X → E)

)],(),,(max[),(:),( yxyxyxyxSR SRSRSR

)],(),,(min[),(:),( yxyxyxyxSR SRSRSR

),(1),(:),( yxyxyxR RRR

),(),(:),( yxyxyxSR SRR

()

Properties of Crisp Relations 交換律 (Commutative law)

結合律 (Associative law)

分配律 (Distributive law)

乘方 (Involution) 冪等律 (Idempotence)

狄摩根定律 (De Morgan’s law)

排中律 (Low of Excluded Middle)UAA

ABBAABBA ,

CBACBACBA )()(

)()()( CABACBA

AAAAAA ,

,, BABABABA

Composition

R={(X1,Y1),(X1,Y3),(X2,Y4)}

S={(Y1,Z2),(Y3,Z2)}

Composition oeration Max-min composition

T=R 。 S

Max-product comositon T=R 。 S

)),(),((),( zyyxzx SRYy

T

)),(),((),( zyyxzx SRYy

T

Composition Example 3.4

Max-min composition y1 y2 y3 y4 z1 z2

R= x1 S= y1

x2 y2

x3 y3 y4 z1 z2

T= x1

x2 x3

0000

0001

1010

00

01

00

01

00

00

01

Fuzzy Relations

Membership function Interval [0,1] Cartesian space X × Y =>

Cardinality of Fuzzy Relations Universe is infinity

Operations on Fuzzy Relations

Union

Intersection

Complement

Containment

Properties of Fuzy Relations

排中律 (Low of Excluded Middle) 在 Fuzzy 集合中並不成立 !

Fuzzy Cartesian Product

Cartesian product space

Fuzzy relation has membership function

Example 3.5

Fuzzy Composition

Fuzzy max-min composition

Fuzzy max-product composition

不論 crisp 或 fuzzy 的 composition

Fuzzy Composition Example 3.6

X={x1,x2} Y={y1,y2} Z={z1,z2,z3}

Max-min composition

Max-product compositon

Noninteractive Fuzzy Sets Fuzzy set on the Cartesian space X =X1 × X2

noninteractive

interactive

Noninteractive Fuzzy Sets

Example 3.7

Noninteractive Fuzzy Sets Example 3.7( 續 )

Cartesian product

120

2.0

100

0.1

60

7.0

30

3.0)(% seseR

120

1.0

100

0.1

80

8.0

60

6.0

40

4.0

20

2.0)(% aaI

1800

15.0

1500

0.1

1000

67.0

500

33.0)( rpmN

Noninteractive Fuzzy Sets Example 3.7(續 )

Max-min composition

Example 3.8

Max-min composition

Noninteractive Fuzzy Sets Example 3.9

Tolerance and Equivalence relations

自返性 (reflexivity)

對稱性 (symmetry)

傳遞性 (transitivity)

Crisp Eqivalence Relation

自返性 (reflexivity) (xi,xi) R or

對稱性 (symmetry) (xi,xj) R (xj,xi) R

or 傳遞性 (transitivity)

(xi,xj) R and (xj,xk) R (xi,xk) R

or

1),( iiR xx

),(),( ijRjiR xxxx

1),(1),(),( kiRkjRjiR xxxxandxx

Crisp Tolerance Relation

Also called proximity relation Only the reflexivity and symmetry Can be reformed into an equivalence

relation By at most (n-1) compositions with itself

RRRRR n 111

11 ....

Crisp Tolerance Relation

Example 3.10 X={x1,x2,x3,x4,x5}={Omaha, Chicago, Rome,

London, Detroit}

R1 does not properties of transitivity e.g. (x1,x2) R1 (x2,x5) R1 but (x1,x5) R1

Crisp Tolerance Relation

Example 3.10(續 ) R1 can become an equivalence relation through two

compositions

Fuzzy tolerance and equivalence relations

自返性 (reflexivity)

對稱性 (symmetry)

傳遞性 (transitivity)

Fuzzy tolerance and equivalence relations

Equivalence relations Fuzzy tolerance relation Can be

reformed into an equivalence relation By at most (n-1) compositions with itself

Fuzzy tolerance and equivalence relations

Example 3.11

It is not transitive

One composition

Reflexive and symmetric

Fuzzy tolerance and equivalence relations

Example 3.11(續 )

Value assignments Cartesian product Closed-from expression

Simple observation of a physical process No variation

model the process crisp relation Y= f(X)

Lookup table Variability exist

Membership values on the interval [0,1] Develop a fuzzy relation

Linguistic rules of knowledge If-then rules

Classification Similarity methods in data manipulation

Cosine Amplitude

X={x1,x2,….,xn}

xi={ }

miii xxx ,....,, 21

))((

||

1 1

22

1

m

k

m

k jkik

m

k jkikij

xx

xxr

Cosine Amplitude

Example 3.12

r12=0.836

Regions x1 x2 x3 x4 x5

Xi1—Ratio with no damage 0.3 0.2 0.1 0.7 0.4

Xi2—Ratio with medium damage 0.6 0.4 0.6 0.2 0.6

Xi3—Ratio with serious damage 0.1 0.4 0.3 0.1 0

))((

||3

1

3

1

22

3

1

k k jkik

k jkikij

xx

xxr

Cosine Amplitude Example 3.12(續 )

Tolerance relation

Equivalence relation

Max-min Method

rij= where i, j =1,2,…n

Example 3.13 Reconsider Example 3.12

Tolerance relation

m

kjkik

m

kjkik

xx

xx

1

1

),max(

),min(

3

1

3

112

))4.0,1.0max(),4.0,6.0max(),2.0,3.0(max(

))4.0,1.0min(),4.0,6.0min(),2.0,3.0(min(

k

kr

Summary

Q & A