1

Fuzzy Relations & Operations on Fuzzy Relations

•Fuzzy RelationConsider two universes:A crisp set consisting of a subset of ordered points

is a crisp relation in the Cartesian product

1 1 2 2{ } and { }X x X x= =

1 2( , )x x

1 2X X×1X

2X

1 2( , ) 0f x x =

2

The notation of relation in crisp sets is also extendable to fuzzy sets. Let us consider the sets of numbers in x and y that are simultaneously close to 0. This relation could be expressed using the Gaussian membership function:

2 2( )( , ) /( , ) /( , )x yRX Y X Y

R x y x y e x yµ − +

× ×= =∫ ∫

3

Let R be a binary fuzzy relation defined on .Similar to fuzzy sets, R can be defined as

Alternatively, in a discrete Cartesian product, R can be expressed as

In a Continuous one

X Y×

( ){ }( , ), ( , )RR x y x yµ=

( , )

( , ) ( , )i j

R i j i jx y X Y

R x y x yµ∈ ×

= ∑

for ( , ) ( )x y X Y∈ ×

( , ) ( , )R i j i jX YR x y x yµ

×= ∫

4

The membership matrix of an mxn binary fuzzy relation has the general form

1 1 1 2 1

2 1 2 2 2

1 2

( , ) ( , ) ( , )( , ) ( , ) ( , )

( , ) ( , ) ( , )

R R R n

R R R n

R m R m R m n

x y x y x yx y x y x y

R

x y x y x y

µ µ µµ µ µ

µ µ µ

=

LL

M M M ML

5

Example: (Representing a Fuzzy Relation)Let and be two discrete sets. The fuzzy relation R = “x is similar toy” may be represented in five different ways:

1. Linguistically, such as by the statement “x is similar to y”

2. By listing (or taking the union of) all fuzzy singletons

3. As a directed graph4. In a tabular form5. As a membership matrix

1 2 3 4{ , , , }X x x x x= 1 2 3 4{ , , , }Y y y y y=

6

Basic Operations with Fuzzy Relations:

Suppose that we have two fuzzy relations R1 and R2. Their union is a new relation

Where the membership function of is

1 21 2 ( , ) ( , ) ( , )R RX YR R x y x y x yµ µ

× ∪ = ∨ ∫

1 2R R∪

1 2 1 2( , ) ( , ) ( , )R R R Rx y x y x yµ µ µ∪ = ∨

7

The intersection of the two fuzzy relations R1 and R2is a new relation

Where the membership function of is

1 21 2 ( , ) ( , ) ( , )R RX YR R x y x y x yµ µ

× ∩ = ∧ ∫

1 2R R∪

1 2 1 2( , ) ( , ) ( , )R R R Rx y x y x yµ µ µ∩ = ∧

8

Example: (Union and Intersection of Fuzzy Relations)Consider the following two fuzzy relations

1

0.0 0.0 0.1 0.8

0.0 0.8 0.0 0.00.1 0.8 1.0 0.8

R =

R1 = “x is larger than y”

2

0.4 0.4 0.2 0.1

0.5 0.0 1.0 1.00.5 0.1 0.2 0.6

R =

R2 = “y is much bigger than x”

9

1 2

0.4 0.4 0.2 0.8

0.5 0.8 1.0 1.00.5 0.8 1.0 0.8

R R ∪ =

The union of the two relations is formed by taking the maximum (t-norm) of the two grades of membership for the corresponding elements of the two matrices.The new relation is defined by

10

For the intersection, we take the minimum (s-norm) of the two grades of membership for the corresponding elements of the two matrices.The new relation is defined by

1 2

0.0 0.0 0.1 0.1

0.0 0.0 0.0 0.00.1 0.1 0.2 0.6

R R ∩ =

11

•Projection

The first projection is a fuzzy set that results by eliminating the second set Y of X×Y by projecting the relation on X.

( , )( , )R

X Y

x yR

x yµ

×

= ∫

1

11

maximize over all

( )( ) [ ( , )]R

R RyX y

xR x x y

x

µµ µ= = ∨∫

12

2

22

( )( ) [ ( , )]R

R RxY

yR y x y

y

µµ µ= = ∨∫

The second projection is a fuzzy set that results by eliminating the first set X of X×Y by projecting the relation on Y.

13

14

15

•Total ProjectionThis is a combined projection over the space X and Y. It is represented by the following:

( , )( )

( , )T

RR x y

x yy

x yµ

µ = ∨ ∨

16

•Example

[ ]

1

2

1

1 2 3

2

1 2 3 4 5 6

2

0.1 0.2 0.4 0.8 1 0.60.2 0.4 0.8 0.9 0.8 0.60.5 0.9 1 0.8 0.4 0.2

1( ) 1 0.9 1

0.91

( ) 0.5 0.9 1 0.9 1 0.6

0.5 0.9 1 0.9 1 0.6

1

iR

i

iR

i

T

y

R x

xR

x x x x

yR

y y y y y y y

R

R

µ

µ

=

= = + + =

= = + + + + +

=

=

∑

∑

17

•Cylindrical Extension

1 2

1 2

1 2, , ,

( , , , )( , , , )

n

R nR

nx x x

x x xCE

x x xµ

= ∫…

……

2

0.5 0.9 1 0.9 1 0.60.5 0.9 1 0.9 1 0.60.5 0.9 1 0.9 1 0.6

RCE =

18

•Fuzzy GraphA fuzzy graph describes a functional mapping between a set of linguistic variables and an output variable.

Assume that a function is approximated by the following if-then rules.

f: if x is small, then y is smallif x is medium, then y is largeif x is large, then y is small

This forms a fuzzy graph

: , ,f U V x U y V→ ∈ ∈

*f* (small small, medium large, large small)f =∨ ∧ ∧ ∧

19

µA(x)lms

l

s

y

x

f*C2

C1

C3

20

•Composition of Fuzzy Relations§ Fuzzy logic (if-then rules) → relation§ Fuzzy inference system (multi if-then rules) → set of

relations

Notation: ο composed with⊗ Cartesian product

Suppose we have

1

2

( , )

( , )

R x y X Y

R y z Y Z

→ ⊗→ ⊗

21

§ Max-Min Composition

The max-min composition of is1 2R Ro

1 2

1 2

( , ) ( , )

( , )

R Ry

X Z

x y y zR R

x z

µ µ

⊗

∨ ∧ = ∫o

Similar to product of two matrices

22

§ Max-Star Operation (Composition)

§ Max-Product Composition (algebraic product)

1 2

1 2

( , ) ( , )

( , )

R Ry

X Z

x y y zR R

x z

µ µ

⊗

∨ ∗ = ∫o

1 2

1 2

( , ). ( , )

( , )

R Ry

X Z

x y y zR R

x z

µ µ

⊗

=

∑∫o

23

•Example Max-Min compositionSuppose we have the two following relations:

1 1

2 21 2

3 3

4 4

1 2 3 4 1 2 31 0.3 0.9 0 1 0.3 0.9

0.3 1 0.3 0 1 1 0.50.9 0.8 1 0.8 0.3 0.1 00 1 0.8 1 0.3 0.3 0.1

X Y Y Z

y y y y z z zx yx y

R Rx yx y

→ →

= =

24

[ ]

1 2

1 1

1 2

1 2

1 2 ( , )

(max-min operation)

( , ) ( , )

11

( ) 1 0.3 0.9 00.3

0.3

[1 1] [0.3 1] [0.9 0.3] [0 0.3]1 0.3 0.3 0 1

R Ry

x z

R R

R R y x y y y z

R R

= ∨ ∧

=

= ∧ ∨ ∧ ∨ ∧ ∨ ∧= ∨ ∨ ∨ =

oo

o o

25

1 2

1 2

Max-Min is

1 1 0.91 0.3 0.5

0.9 0.9 0.91 0.3 0.5

Max-Product would be

1 1 0.91 0.3 0.5

0.9 0.9 0.811 0.3 0.5

R R

R R

⇒ =

=

o

o

Notice the difference

26

[ ]

[ ]

1 2 (3,3)

0.90.5

( ) 0.9 0.8 1 0.80

0.1

0.81 0.4 0 0.08

0.81

R R

= ⋅

= ∨=

o

As an example for computing terms (3,3)

27

Fuzzy Intersection and T-norm

The intersection of two fuzzy sets A and B is given by an operation T which maps two membership functions to:

T is known as the T-norm operator

( ( ), ( ))A B A BT x xµ µ µ∩ =

28

Min ( , ) min( , )Algebraic product ( , )Bounded product ( , ) 0 ( 1)

if 1Basic product ( , ) if 1

0 if , 1

T a b a b a bT a b a b

T a b a b

a bT a b b a

a b

→ = = ∧→ = ⋅

→ = ∨ + −

=→ = = <

Four of the well known T-norm operators are:

29

Minimum:Tm(a, b)

Algebraicproduct:Ta(a, b)

Boundedproduct:Tb(a, b)

Drasticproduct:Td(a, b)

T-norm Operator

30

•Fuzzy Union and T-conorm(S-norm)

The union of two fuzzy sets A and B is given by an operation S which maps two membership functions to:

S is known as the T-conorm or S-norm operator

T-Conorm Operator

( ( ), ( ))A B A BS x xµ µ µ∪ =

31

T-conorm or S-norm

Maximum:Sm(a, b)

Algebraicsum:

Sa(a, b)

Boundedsum:

Sb(a, b)

Drasticsum:

Sd(a, b)

32

•ExampleRd for an automobile relates the system set S to the fault set F.

1

2

3

4

5

1

2

3

4

(low gas mileage)x (excessive vibration)x (loud noise)x (high coolant temperature)x (steering instability)

[ (bad spark plugs) (wheel imbalance) (bad muffler) (thermostat stuck close

S x

F yyyy

=

=

d)

33

Linguistic representation of the diagnosis

is strongly related to and weakly related tois strongly related to and weakly related tois strongly related to and weakly related tois strongly related to and weakly related tois strongly related to and weakly related to

1x 1y 2 4y y…2x 2y 1 4y y…3x 3y 1 4y y…4x

4y 1 4y y…5x 5y 1 4y y…

34

[ ][ ] [ ]

1

2

3

4

1 2 3 4 11 0.3 0.9 0 1

00.3 1 0.3 0 0

0.050.9 0.8 1 0.8 0

0.10 1 0.8 1 0

0

1 0 0 0 0 5 4 1 0.1 0.05 0

y y y y Fxx

Rxx

S

= =

× =

35

•ExampleThree relations involved in max-min composition

When two of the components in the above equation are given and the other is unknown, we have a set of equations known as fuzzy equations:

.P Q R=o

[ ]

[ ]1 2 3

1 2 3

1 2 3

1 2 3

0.9 0.6 10.8 0.3 0.5 0.6 0.6 0.50.6 0.4 0.6

( 0.9) ( 0.8) ( 0.6) 0.6

( 0.6) ( 0.3) ( 0.4) 0.6( 1) ( 0.5) ( 0.6) 0.5

P

P p p p

p p p

p p pp p p

=

=

∧ ∨ ∧ ∨ ∧ =

∧ ∨ ∧ ∨ ∧ =∧ ∨ ∧ ∨ ∧ =

o

36

Generalized DeMorgan’s Law

T-norm and T-conorms are dual operators. They satisfy DeMorgan’s laws.

( )( )( )( )

( , ) ( ), ( ) ( , )

( , ) ( ), ( )

( , ) ( , )

( , ) ( , )

( , ) ( , )

( , ) ( , )

m m

a a

b b

d d

T a b N S N a N b S a b

S a b N T N a N b

T a b S a b

T a b S a b

T a b S a b

T a b S a b

= =

=

↔↔

↔↔

37

Fuzzy Reasoning

Definition: A linguistic variable is characterized by a quintuplet

is the name of the variableis the term set of ≡ set of linguistic values

is the universe of discourseis a syntactic rule which generates the terms in is a semantic rule, it associates with each fuzzy

set A, its meaning

( ), ( ), , ,x T x X G Mx

x( )T xXG ( )T x

M( )M A

38

Example

If age is interpreted as a linguistic variable; then its term set: •T(age)={young, not young, middle age, old, not old, very old}•T(x) is a term set over X=[0,100]•The syntactic refers to the way the linguistic values are represented in T(x)•The semantic rules define the membership function of each linguistic value of the term set T(x)

39

40

Fuzzy If-Then Rules

A fuzzy if-then rule (fuzzy rule ≡ fuzzy implication) assumes the form of: if x is A then y is B.•A and B represent fuzzy sets•x and y are two fuzzy linguistic variables•“x is A” is called antecedent•“y is B” is called consequent

41

Example:

•If pressure is high, then the volume is small.•If tomato is red, then it is ripe.

A fuzzy if-then rule is a binary relation R .

IF A an B are two fuzzy sets over X and Y respectively, thentwo ways for representing the implication

§ A is coupled with B§ A entails B

:R X Y→

42

A coupled with B

AA

B B

A entails B

Two ways to interpret “If x is A then y is B”:

y

xx

y

43

A is coupled with B

( ) ( ):

( , )A B

X Y

x yR X Y A B

x yµ µ

×

∗→ = × = ∫

44

A entails B

1 (1 ( ) ( )): :

( , )A B

aX Y

x yR A B

x yµ µ

×

∧ − +∪ ∫

“Zadeh” arithmetic operator:(1 ( )) ( ( ) ( ))

: ( ):( , )

A A Bmn

X Y

x x yR A A B

x yµ µ µ

×

− ∨ ∧∪ ∩ ∫

45

Presume that

f is called the fuzzy implication. It transforms elements of A to and elements of B in

For A entails B we have the two following Oprts:•Material implication•Extended proposition calculus

For A Coupled with B

Minimum operator:

Mamdani operator:

:R X Y A B→ = ∪: ( )R X Y A B B→ = ∩ ∪

( )( , ) ( ), ( ) ( , )R A Bx y f x y f a bµ µ µ= =A B→

( ) ( )( , )

A Bm

X Y

x yR A B

x yµ µ

×

∧= × = ∫

( ) ( )( , )

A Bp

X Y

x yR A B

x yµ µ

×

= × = ∫o

46

Derivation of y = b from x = a and y = f(x):

a and b are points

y = f(x) is a curve

a

b

y

xx

y

a and b are intervals

y = f(x) is an interval-valued

function

a

b

y = f(x) y = f(x)