fuzzy relations and functions

8/13/2019 Fuzzy Relations and Functions

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Fuzzy Relationsand Functions

By

P. D. Olivier, Ph.D., P.E.From

Driankov, Hellendoorn, Reinfrank


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Classical to Fuzzy Relations

A classical relation is a set of tuples Binary relation (x,y)

Ternary relation (x,y,z)

N-ary relation (x1,x

n)

Connection with Cross product

Married couples

Nuclear family

Points on the circumference of a circle

Sides of a right triangle that are all integers


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Characteristic Function

Any set has a characteristic function.

A relation is a set of points

Review definition of characteristic function Apply this definition to a set defined by a

relation


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Properties of some binary relations Reflexive

Anti-reflexive

Symmetric

Anti-symmetric

Transitive

Equivalence

Partial order Total order

Assignment: Classify: =,,=


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Fuzzy Relations

Let U and V be universes and let the function

: [0,1]R U V

Continuous relations ( , ) /( , )RU V

R u v u v

Discrete relations ( , ) /( , )RU V

R u v u v


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Example 2.50{1, 2,3}U

" " 1/(1,1) 1(2, 2) 1/(3,3)

.8 /(1, 2) .8 /(2,3) .8 /(2,1) .8 /(3, 2)

.3/(1,3) .3(3,1)

Universe of Discourse

Approximately Equals

1

( , ) .8 | | 1

.3 | | 2

R

when x y

x y when x y

when x y

Tabular


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Example 2.52: IF-Then Rule

Programming If-Then

,If e is PB and e is PS then u is NM

Convert to integral form using two versions of

AND


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Operations on Fuzzy Relations

R = x considerably larger than y

S = y very close to x

Intersection of R and S (T-norms) Union of R and S (S-norms)

Projection

Cylindrical extension


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ProjectionSimple case 1:

sup ( , ) /x RY

R X Y

proj R on Y x y y

Case 2:,sup ( , , ) /x y R

Y

R X Y Z

proj R on Z x y z y

proj R on X

proj R on Y

proj R on X Y

proj R on X Z

proj R on Y Z

General

case


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Example 2.60


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Example 2.58

Each x is assigned the highest

membership degree from all tuples with

that x

Projections reduce the number of

variables

Extensions increase the number of

variables


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Cylindrical Extension

Extension from 1 D to 2 D( ) ( ) /( , )F

X Y

ce F y x y

Extension form 2D to 3 D( ) ( , ) /( , , )F

X Y Z

ce F y z x y z

proj ce(S) on V = S

ce(proj R on V) R


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Composition

Combines fuzzy sets and fuzzy relations

with the aid of cylindrical extension and

projection. Denoted with a small circle.

Draw picture of composition of functions

Intersection can be accomplished with any

T norm

Projection can be accomplished with any

S norm


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Extension Principle Allows for the combination of fuzzy and non-

fuzzy concepts Very important

Allows mathematical operations on fuzzy sets

The extension of function f, operating on A1, ,Anresults in the following membership function

for F

1

1

1

1,...( ,... )

( ) sup min( ( ),..., ( ))n

n

n

F A A nu uf u u v

v u u

When f -1exists. Otherwise, 0.

fuzzy relations and functions

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