1

FUZZY SETS

Precision vs. Relevancy

31

is approachingyour head

at 45.3 m/sec.

A 1500 Kg massis approaching

your headat 45.3 m/s. OUT!!

LOOKOUT!

2

Introduction

How to simplify very complex systems?Allow some degree of uncertainty in theirdescription!

How to deal mathematically with uncertainty?Using probabilistic theory (stochastic).Using the theory of fuzzy sets (non-stochastic).

Proposed in 1965 by Lotfi Zadeh (Fuzzy Sets,Information Control, 8, pp. 338-353).Imprecision or vagueness in natural language doesnot imply a loss of accuracy or meaningfulness!

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Examples

Give travel directions in terms of city blocks OR inmeters?The day is sunny OR the sky is covered by 5% of clouds?

If the sky is covered by 10% of clouds is still sunny?And 25%?And 50%?Where to draw the line from sunny to not sunny?Member and not member or membership degree?

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3

Probability vs. Possibility

Event u: Hans ate X eggs for breakfast.Probability distribution: PX(u)Possibility distribution: X(u)

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u 1 2 3 4 5 6 7 8

PX(u) 0.1 0.8 0.1 0 0 0 0 0

X(u) 1 1 1 1 0.8 0.6 0.4 0.2

You’re lost in the Outback; Dying of Thirst

You Come Upon Two Bottles Containing Liquid

Which One Will You Choose?

How Will You Process the Information?

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Probability vs. Fuzzy Set Membership

4

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Potable?Probability

= 0.91

OR

Potable?Membership

= 0.91

Might Taste Funky,but shouldn’t kill you

Probability vs. Fuzzy Set Membership

Applications of fuzzy sets

Fuzzy mathematics (measures, relations, topology,etc.)Fuzzy logic and AI (approximate reasoning, expertsystems, etc.)Fuzzy systems

Fuzzy modelingFuzzy control, etc.

Fuzzy decision makingMulti-criteria optimizationOptimization techniques

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5

Classical set theory

Set: collection of objects with a common property.Examples:

Set of basic colors:A = {red, green, blue}

Set of positive integers:

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{ | 0}A x x

A line in 3:A = {(x,y,z) | ax + by + cz +d = 0}

Representation of sets

Enumeration of elements: A = {x1, x2,..., xn}Definition by property P: A = {x X | P(x)}Characteristic function A(x): X {0,1}

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1, if is member of( )

0, if is not member ofA

x Ax

x A

( ) mod 2A x xExample:

Set of odd numbers:

6

Set operations

Intersection: C = A BC contains elements that belong to A and BCharacteristic function: C = min( A , B) = A · B

Union: C = A BC contains elements that belong to A or to BCharacteristic function: C = max( A , B)

Complement: C =C contains elements that do not belong to ACharacteristic function: C = 1 – A

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Fuzzy sets

Represent uncertain (vague, ambiguous, etc.)knowledge in the form of propositions, rules, etc.

Propositions:

expensive cars,cloudy sky,...

Rules (decisions):Want to buy a big and new house for a low price.If the temperature is low, then increase the heating.…

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7

Classical set

Example: set of old people A = {age | age 70}

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A

50 600

70 80 90 100

0.5

1

age [years]

Logic propositions

“Nick is old” ... true or falseNick’s age:

ageNick = 70, A(70) = 1 (true)ageNick = 69.9, A(69.9) = 0 (false)

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A

50 600

70 80 90 100

0.5

1

age [years]

8

Fuzzy set

Graded membership, element belongs to a set to acertain degree.

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A

50 600

70 80 90 100

0.5

1

mem

bers

hip

grad

e

age [years]

Mem

bers

hip

grad

e

Fuzzy proposition

“Nick is old”... degree of truthageNick = 70, A(70) = 0.5ageNick = 69.9, A(69.9) = 0.49ageNick = 90, A(90) = 1

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A

50 600

70 80 90 100

0.5

1

mem

bers

hip

grad

e

age [years]

9

Context dependent

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h [cm]

A

0170 180 190

0.5

1

mem

bers

hip

grad

e

tall in China tall in USA tall in NBA

Typical linguistic values

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20 400

60 80 100

1

mem

bers

hip

grad

e

young middle age old

age [years]

10

Support of a fuzzy set

supp(A) = {x X | A(x) > 0}

48

xsupp( )A

A

Core (nucleous, kernel)

core(A) = {x X | A(x) = 1}

49

x

A

core( )A

11

-cut of a fuzzy set

Crisp set: A = { x X | A(x) }Strong -cut: A = { x X | A(x) > }

50

xA

A

Resolution principle

Every fuzzy set A can be uniquely represented as acollection of -level sets according to

Resolution principle implies that fuzzy set theory is ageneralization of classical set theory, and that itsresults can be represented in terms of classical settheory.

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[0,1]( ) sup [ ( )]A Ax x

12

Resolution principle

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x

1

0

A

Other properties

Height of a fuzzy set: hgt(A) = sup A(x), x XFuzzy set is normal(ized) when hgt(A) = 1.A fuzzy set A is convex iff x,y X and [0,1]:

A( x + (1 – ) y) min( A(x), A(y))

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x

A B

13

Other properties (2)

Fuzzy singleton: single point x X where A(x) = 1.Fuzzy number: fuzzy set in that is normal andconvex.Two fuzzy sets are equal (A = B) iff:

x X, A(x) = B(x)A is a subset of B iff:

x X, A(x) B(x)

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Non-convex fuzzy sets

Example: car insurance risk

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lifetimeage [years]

14

Representation of fuzzy sets

Discrete Universe of Discourse:Point-wise as a list of membership/element pairs:

A = A(x1)/x1+...+ A(xn)/xn = i A(xi)/xi

A = { A(x1)/x1,..., A(xn)/xn} = { A(xi)/xi | xi X}

As a list of -level/ -cut pairs:A = { 1/A 1

, ..., n/A n} = { i/A i

| i [0,1]}

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Representation of fuzzy sets

Continuous Universe of Discourse:

A = X A(x)/x

Analytical formula:

Various possible notations:

A(x), A(x), A, a, etc.

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2

1( ) ,1A x x

x

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Examples

Discrete universeFuzzy set A =“sensible number of children”.

number of children: X = {0, 1, 2, 3, 4, 5, 6}A = 0.1/0 + 0.3/1 + 0.7/2 + 1/3 + 0.6/4 + 0.2/5 + 0.1/6

Fuzzy set C = “desirable city to live in”X = {SF, Boston, LA} (discrete and non-ordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}

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Examples

Continuous universeFuzzy set B = “about 50 years old”

X = + (set of positive real numbers)

B = {(x, B(x)) | x X}

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41( )

50110

B xx

Mem

bers

hip

Gra

des

Age

16

Complement of a fuzzy set

c: [0,1] [0,1]; A(x c( A(x))

Fundamental axioms1. Boundary conditions - c behaves as the ordinary

complementc(0) = 1; c(1) = 0

2. Monotonic non-increasinga,b [0,1], if a < b, then c(a) c(b)

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Complement of a fuzzy set

Other axioms:

c is a continuous function.c is involutive, which means that

c(c(a)) = a, a [0,1]

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17

Complement of a fuzzy set

Equilibrium pointc(a) = a = ec, a [0,1]

Each complement has at most oneequilibrium.If c is a continuous fuzzy complement, it hasone equilibrium point.

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Examples of fuzzy complements

Satisfying only fundamental axioms:

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tata

acif,0if,1

)(

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Fuzzy complement of threshold type: t=0.3

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Examples of fuzzy complements

Satisfying fundamental axioms and continuity:

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aac cos121)(

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Continuous fuzzy complement (non involutive)

Examples of fuzzy complements

Sugeno complement:

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1( ) , ] 1, ]1

ac aa

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Sugeno fuzzy complement, lambda = -0.9, -0.5, 0, 2, 10

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Yager fuzzy complement, w = 0.2, 0.5, 1, 2, 10

Examples of fuzzy complement

Yager complement:

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],0],1)(1

waacww

w

Representation of complement

(x) = 1 – A(x)

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1

x

A

20

0 20 40 60 80 1000

0.5

1

x

0 20 40 60 80 1000

0.5

1

x

0 20 40 60 80 1000

0.5

1

x

A(x)

1- A(x)

A(x)

=-0.9=10

A(x)

w=0.2w=5

Representation of complement

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Intersection of fuzzy sets

i: [0,1] [0,1] [0,1];A B(x i( A(x), B(x))

Fundamental axioms: triangular norm or t-norm1. Boundary conditions - i behaves as the classical

intersectioni(1,1) = 1;i(0,1) = i(1,0) = i(0,0) = 0

2. Commutativityi(a,b) = i(b,a)

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21

Intersection of fuzzy sets

3. MonotonicityIf a a’ and b b’, then i(a,b) i(a’,b’)

4. Associativityi(i(a,b),c) = i(a,i(b,c))

Other axioms:i is a continuous function.i(a,a) = a (idempotent).

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Examples of fuzzy conjunctions

Zadeh

A B(x) = min( A(x), B(x))Probabilistic

A B(x) = A(x) B(x)Lukaziewicz

A B(x) = max(0, A(x) B(x) 1)

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22

Intersection of fuzzy sets

A B(x) = min( A(x), B(x))

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x

BA

Intersection of fuzzy sets

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0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

x

ABmin(A,B)A*Bmax(0,A+B-1)

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Yager t-norm

Example of week and strong intersections:

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1( , ) 1 min 1, 1 1 , ]0, ]

ww wwi a b a b w

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Yager fuzzy intersection, w = 1.5, 2, 5

Union of fuzzy sets

u: [0,1] [0,1] [0,1];A B(x u( A(x), B(x))

Fundamental axioms: triangular co-norm or s-norm1. Boundary conditions - u behaves as the classical

unionu(0,0) = 0;u(0,1) = u(1,0) = u(1,1) = 1

2. Commutativityu(a,b) = u(b,a)

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Union of fuzzy sets

3. MonotonicityIf a a’ and b b’, then u(a,b) u(a’,b’)

4. Associativityu(u(a,b),c) = u(a,u(b,c))

Other axioms:u is a continuous function.u(a,a) = a (idempotent).

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Examples of fuzzy disjunctions

Zadeh

A B(x) = max( A(x), B(x))Probabilistic

A B(x) = A(x) B(x) A(x) B(x)Lukasiewicz

A B(x) = min(1, A(x) B(x))

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25

Union of fuzzy sets

A B(x) = max( A(x), B(x))

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x

BA

Union of fuzzy sets

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0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

x

ABmax(A,B)A+B-A*Bmin(1,A+B)

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Yager s-norm

Example of week and strong disjunctions:

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1( , ) min 1, , ]0, ]

ww wwu a b a b w

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Yager fuzzy union, w = 2.5, 5, 10

General aggregation operations

h: [0,1]n [0,1];A(x h( A1

(x),..., An(x))

Axioms1. Boundary conditions

h(0,...,0) = 0h(1,...,1) = 1

2. Monotonic non-decreasingFor any pair ai, bi [0,1], iIf ai bi then h(ai) h(bi)

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27

General aggregation operations

Other axioms:

h is a continuous function.h is a symmetric function in all its arguments:

h(ai) = h(ap(i))for any permutation p on

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Averaging operations

When all the four axioms hold:

Operator covering this range: Generalized mean

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1 1 1min( , , ) ( , , ) max( , , )n n na a h a a a a

1

11( , , ) n

n

a ah a a

n

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Generalized mean

Typical cases:Lower bound:Geometric mean:

Harmonic mean:

Arithmetic mean:Upper bound:

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1

1

1 1n

nh

a a

1max( , , )nh a a

1min( , , )nh a a

0 1 2( )nnh a a a

11

na ahn

Fuzzy aggregation operations

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min max umax

Generalized mean

Parametric t-norms Parametric s-norms

Intersection operators Averaging operators Union operators

imin

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Membership functions (MF)

Triangular MF:

Trapezoidal MF:

Gaussian MF:

Generalized bell MF:

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( ; , , ) max min , ,0x a c xTr x a b cb a c b

( ; , , , ) max min ,1, ,0x a d xTp x a b c db a d c

212( ; , , )

x c

Gs x a b c e

2

1( ; , , )1

aBell x a b cx c

b

Membership functions

87

0 20 40 60 80 1000

0.5

1

(a) Triangular MF

0 20 40 60 80 1000

0.5

1

(b) Trapezoidal MF

0 20 40 60 80 1000

0.5

1

(c) Gaussian MF

0 20 40 60 80 1000

0.5

1

(d) Generalized Bell MF

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Left-right MF

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,( ; , , )

,

L

R

c xF x cLR x c

x cF x c

c = 65= 60= 10

c = 25= 10= 40

0 50 1000

0.2

0.4

0.6

0.8

1

Mem

bers

hip

Gra

des

0 50 1000

0.2

0.4

0.6

0.8

1

Mem

bers

hip

Gra

des

Two-dimensional fuzzy sets

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( , ) ( , ) ( , )A AX YA x y x y x y X Y

x y

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2-D membership functions

90

-100

10

-100

100

0.5

1

X

(a) z = m in(trap(x ), trap(y))

Y -100

10

-100

100

0.5

1

X

(b) z = m ax (trap(x), trap(y ))

Y

-100

10

-100

100

0.5

1

X

(c) z = m in(bell(x ), bell(y))

Y -100

10

-100

100

0.5

1

X

(d) z = m ax (bell(x), bell(y ))

Y

Compound fuzzy propositions

Small = Short and Light (conjunction)

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( , ) ( ) ( )Small Short Lighth w h w

LONG

LIG

HT

HEA

VY

Weight(w)

Height(h)

Thin

BigCompact

Small

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Cylindrical extension

Cylindrical extension of fuzzy set A in X into Y resultsin a two-dimensional fuzzy set in X Y, given by

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0

0.5

1

X

(a) Base Fuzzy Set A

0

0.5

1

X

(b) Cylindrical Extension of A

y

cext ( ) ( ) /( , ) ( ) ( , ) ( , )y A AX YA x x y x x y x y X Y

Projection

93

0

0.5

1

X

(a) A Two-dimensional MF

Y

0

0.5

1

X

(b) Projection onto X

Y

0

0.5

1

X

(c) Projection onto Y

Y

( , )R x y ( ) max ( , )A Ryx x y ( ) max ( , )B Rx

y x y

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Cartesian product

Cartesian product of fuzzy sets A and B is a fuzzy set inthe product space X Y with membership

Cartesian co-product of fuzzy sets A and B is a fuzzyset in the product space X Y with membership

94

( , ) min( ( ), ( ))A B A Bx y x y

( , ) max( ( ), ( ))A B A Bx y x y

Cartesian product

95

-10 0 100

0.5

1

X

(X)

(a) trap(x)

-10 0 100

0.5

1

Y

(Y)

(b) trap(y)

-100

10-10

010

0

0.5

1

X

(c) z = min(trap(x), trap(y))

Y -100

10-10

010

0

0.5

1

X

(d) z = max(trap(x), trap(y))

Y

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Classical relations

Classical relation R(X1, X2,..., Xn) is a subset of theCartesian product:

Characteristic function:

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1 2 1 2, , , n nX X X X X XR

1 21 2

1, iff , , ,, , ,

0, otherwisen

n

x x xx x xR

R

Example

X = {English, French}Y = {dollar, pound, euro}Z = {USA, France, Canada, Britain, Germany}

R(X, Y, Z) = {(English, dollar, USA),(French, euro, France), (English, dollar,Canada),

(French, dollar, Canada), (English, pound,Britain)}

97

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Matrix representation

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USA Fra Can Brit GerDollar 1 0 1 0 0Pound 0 0 0 1 0

Euro 0 0 0 0 0 English

USA Fra Can Brit GerDollar 0 0 1 0 0Pound 0 0 0 0 0

Euro 0 1 0 0 0 French

Fuzzy relation

Fuzzy relation:

R: X1 X2 ... Xn [0,1]

Each tuple (x1, x2,..., xn) has a degree of membership.Fuzzy relation can be represented by an n-dimensionalmembership function (continuous space) or a matrix(discrete space).Examples:

x is close to yx and y are similarx and y are related (dependent)

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36

Discrete examples

Relation R “very far” between X = {New York, Lisbon}and Y = {New York, Beijing, London}:R(x,y) = 0/(NY,NY) + 1/(NY,Beijing) +

0.6/(NY,London) + 0.5/(Lisbon,NY) +0.8/(Lisbon,Beijing) + 0.1/(Lisbon,London)

Relation: “is an important trade partner of”

100

Holland Germany USA JapanHolland 1 0,9 0,5 0,2

Germany 0,3 1 0,4 0,2USA 0,3 0,4 1 0,7

Japan 0,6 0,8 0,9 1

Continuous example

R: x y (“x is approximately equal to y”)

101

2( )( , ) x yx y e

-1-0.5

00.5

1

-1

-0.5

00.5

10

0.5

1

xy

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Composition of relations

R(X,Z) = P(X,Y) Q(Y,Z)

Conditions:(x,z) R iff exists y Y such that(x,y) P and (y,z) Q.

Max-min composition

102

( , ) max min ( , ), ( , )y Y

x z x y y zP Q P Q

Properties

Associativity:

Distributivity over union:

Weak distributivity over intersection:

Monotonicity:

103

( ) ( )R S T R S T

( ) ( ) ( )R S T R S R T

( ) ( ) ( )R S T R S R T

( ) ( )S T R S R T

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Other compositions

Max-prod composition

Max-t composition

104

( , ) max ( , ) ( , )y Y

x z x y y zP Q P Q

( , ) max ( , ), ( , )y Y

x z t x y y zP Q P Q

Example

105

YZa

b

c

d

1

2

Q

P

0.70.5

1

1

0.40.3

0.6

0.8

1

0.9

39

Example

Composition R = P Q

Composition R = P Q?

106

x z R(x,z)a 1 0.6a 2 0.7b 1 0.6b 2 0.8c 2 1d 2 0.4

Matrix notation examples

107

0.3 0.5 0.8 0.9 0.5 0.7 0.70 0.7 1 0.3 0.2 0 0.9

0.4 0.6 0.5 1 0 0.5 0.5

0.3 0.5 0.8 0.9 0.5 0.7 0.70 0.7 1 0.3 0.2 0 0.9

0.4 0.6 0.5 1 0 0.5 0.5

0.8 0.3 0.5 0.51 0.2 0.5 0.7

0.5 0.4 0.5 0.6

0.8 0.15 0.4 0.451 0.14 0.5 0.63

0.5 0.2 0.28 0.54

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Relations on the same universe

Let R be a relation defined on U U, then it is called:Reflexive, if u U, the pair (u,u) RAnti-reflexive, if u U, (u,u) RSymmetric, if u,v U, if (u,v) R, then (v,u) RtooAnti-symmetric, if u,v U, if (u,v) and (v,u) R,then u = vTransitive, if u,v,w U , if (u,v) and (v,w) R, then(u,w) R too.

108

Examples

R is an equivalence relation if it is reflexive, symmetricand transitive.R is a partial order relation if it is reflexive, anti-symmetric and transitive.R is a total order relation if R is a partial orderrelation, and u, v U, either (u,v) or (v,u) R.Examples:

The subset relation on sets ( ) is a partial orderrelation.The relation on is a total order relation.

109