Download Introduction to Fuzzy Set Theory

II. BASICS – Lecture 2
OBJECTIVES
1. To define the basic ideas and entities in fuzzy set
theory
2. To introduce the operations and relations on
fuzzy sets
3. To learn how to compute with fuzzy sets and
numbers - arithmetic, unions, intersections,
complements
August 12, 2003
II. BASICS: Math Clinic Fall 2003
1
OUTLINE
II. BASICS
A. Definitions and examples
1. Sets
2. Fuzzy numbers
B. Operations on fuzzy sets – union, intersection, complement
C. Operations on fuzzy numbers – arithmetic, equations, functions
and the extension principle
August 12, 2003
II. BASICS: Math Clinic Fall 2003
2
DEFINITIONS
A. Definitions
1. Sets
a. Classical sets – either an element belongs to the set
or it does not. For example, for the set of integers, either
an integer is even or it is not (it is odd). However, either
you are in the USA or you are not. What about flying into
USA, what happens as you are crossing? Another
example is for black and white photographs, one cannot
say either a pixel is white or it is black. However, when
you digitize a b/w figure, you turn all the b/w and gray
scales into 256 discrete tones.
August 12, 2003
II. BASICS: Math Clinic Fall 2003
3
Classical sets
Classical sets are also called crisp (sets).
Lists: A = {apples, oranges, cherries, mangoes}
A = {a1,a2,a3 }
A = {2, 4, 6, 8, …}
Formulas: A = {x | x is an even natural number}
A = {x | x = 2n, n is a natural number}
Membership or characteristic function

1 if x A
 ( x)  

A
0 if x A
August 12, 2003
II. BASICS: Math Clinic Fall 2003
4
Definitions – fuzzy sets
b. Fuzzy sets – admits gradation such as all tones between
black and white. A fuzzy set has a graphical description
that expresses how the transition from one to another
takes place. This graphical description is called a
membership function.
August 12, 2003
II. BASICS: Math Clinic Fall 2003
5
Definitions – fuzzy sets (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
6
Definitions: Fuzzy Sets (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
7
Membership functions (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
8
Fuzzy set (figure from Earl Cox)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
9
Fuzzy Set (figure from Earl Cox)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
10
The Geometry of Fuzzy Sets (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
11
Alpha levels, core, support, normal
z
z
August 12, 2003
z z
z
z
II. BASICS: Math Clinic Fall 2003
z
12
Definitions: Rough Sets
A rough set is basically an approximation of a crisp set A in
terms of two subsets of a crisp partition, X/R, defined on
the universal set X.
Definition: A rough set, R(A), is a given representation of
a classical (crisp) set A by two subsets of X/R, R(A) and
R(A) that approach A as closely as possible from the
inside and outside (respectively) and
R( A)  R( A), R( A)
where R(A) and R(A) are called the lower and upper
approximation of A.
August 12, 2003
II. BASICS: Math Clinic Fall 2003
13
Definitions: Rough sets (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
14
Definitions: Interval Fuzzy Sets (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
15
Definitions: Type-2 Fuzzy Sets (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
16
2. Fuzzy Number
A fuzzy number A must possess the following three
properties:
1. A must must be a normal fuzzy set,
2. The alpha levels A( ) must be closed for every  (0,1],
3. The support of A, A(0 ) , must be bounded.
August 12, 2003
II. BASICS: Math Clinic Fall 2003
17
Membership function
Fuzzy Number (from Jorge dos Santos)
 z, z 
is the suport of z
1
z1 is the modal value
’
[ z]   z , z  is an -level of

 < '  [z]'  [z]
z
August 12, 2003
z ,   (0,1]
z z1
z
z
II. BASICS: Math Clinic Fall 2003
18
Fuzzy numbers defined by its -levels (from Jorge dos Santos)
1
.7
.5
A fuzzy number can be given by a
set of nested intervals, the -levels:
z
[z]1  [z]0.7 [ z]0.5 [ z]0.2 [ z]0
.2
0
August 12, 2003
 z z

1 z z
z0.2
z  z0.2
0.5 0.7 z
0.7 0.5
z
II. BASICS: Math Clinic Fall 2003
19
Triangular fuzzy numbers
[z]0  [z, z]; [z]1  [z1, z1]
1
z  (z / z1 / z)
z
August 12, 2003
z1
z
II. BASICS: Math Clinic Fall 2003
20
Fuzzy Number (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
21
B. Operations on Fuzzy Sets: Union and Intersection
(figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
22
Operations on Fuzzy Sets: Intersection (figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
23
Operations on Fuzzy Sets: Union and Complement (figure
from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
24
C. Operations on Fuzzy Numbers: Addition and Subtraction (figure
from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
25
Operations on Fuzzy Numbers: Multiplication and Division
(figure from Klir&Yuan)
August 12, 2003
II. BASICS: Math Clinic Fall 2003
26
Fuzzy Equations
~ ~ ~
A* X  B
One interpretation is :
A( ) X ( )  B( ).
Let A( ) [a( ),a( )], B( ) [b( ),b( )] and X ( ) [ x( ), x( )].
Then the solution t o the fuzzy equation exists iff :
i) b( ) / a( )  b( ) / a( )  (0,1]
ii)    b( ) / a( )  b( ) / a( )  b( ) / a( )  b( ) / a( ).
Another interpretation is :
{x |  (ax  b)  0.8 where a A,b B}
August 12, 2003
II. BASICS: Math Clinic Fall 2003
27
Example of a Fuzzy Equation (figure from Klir&Yuan)
0 for x  3, x  5
A( x)  x  3 for 3  x  4
5  x for 4  x  5


















0 for x 12, x  32
B( x)  ( x 12) / 8 for 12  x  20
(32  x) /12 for 20  x  32
A( )  [  3, 5  ]
B( )  [8 12, 32 12 ]
Verify (i) that :
8 12  32 12 so that
 3
5 


 8 12 32 12 
X ( )  
,



5 
  3

Verify (ii) that :
    X ( )  X ( )
August 12, 2003
II. BASICS: Math Clinic Fall 2003
28
The Extension Principle of Zadeh
Given a formula f(x) and a fuzzy set A defined by,  A(x)
how do we compute the membership function of f(A) ?
How this is done is what is called the extension principle (of
professor Zadeh). What the extension principle says is that
f (A) =f(A(  )). The formal definition is:
[f(A)](y)=supx|y=f(x){ A(x)}
August 12, 2003
II. BASICS: Math Clinic Fall 2003
29
Extension Principle - Example
Let f(x) = ax+b,
a A 1/ 2 / 3, bB  2 / 3/ 5, and x  6. Then
f(x)  6 A  B  8/15/ 23
August 12, 2003
II. BASICS: Math Clinic Fall 2003
30