Fuzzy sets

By
S.Veni, M.Sc.,
Assistant Professor

Sets with fuzzy boundaries
A = Set of tall people
Heights5’10’’
1.0
Crisp set A
Membership
function
Heights5’10’’ 6’2’’
Fuzzy set A
.5
.9
1.0

 Characteristics of MFs:
 Subjective measures
 Not probability functions
MFs
Heights5’10’’
.5
.8
.1
“tall” in Asia
“tall” in the US
“tall” in NBA

 Formal definition:
A fuzzy set A in X is expressed as a set of ordered pairs:
A x x x XA {( , ( ))| }
Universe or
universe of discourse
Fuzzy set
Membership
function
(MF)
A fuzzy set is totally characterized by a
membership function (MF).

Fuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and nonordered)
C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}
Fuzzy set A = “sensible number of children”
X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)
A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}

Fuzzy set B = “about 50 years old”
X = Set of positive real numbers (continuous)
B = {(x, B(x)) | x in X}
B x
x
( ) 







1
1
50
10
2

A fuzzy set A can be alternatively denoted as follows:
A x xA
x X
i i
i


 ( ) /
A x xA
X
  ( ) /
X is discrete
X is continuous
Note that S and integral signs stand for the union of
membership grades; “/” stands for a marker and does not
imply division.

MF
X
.5
1
0
Core
Crossover points
Support
a - cut
a

A fuzzy set A is convex if for any l in [0, 1],
 l l  A A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21  
Alternatively, A is convex is all its a-cuts are
convex.

Fuzzy Singleton: A Fuzzy set whose support is a single
point in X with is called a Fuzzy Singleton .1(x)μA 
Fuzzy numbers: A Fuzzy number A is a fuzzy set in the
real line (R) that satisfies the conditions for normality and
convexity.
Bandwidth: The Bandwidth or width is defined as the
distance between the two unique crossover points:
width(A) = | x2 - x1 | where 5.0)(xμ)(xμ 2A1A 
Symmetry: A Fuzzy set A is symmetric around a certain
point x = c, namely, for all x € Xx)(cμx)(cμ AA 

Subset:
Complement:
Union:
Intersection:
A B A B   
C A B x x x x xc A B A B         ( ) max( ( ), ( )) ( ) ( )
C A B x x x x xc A B A B         ( ) min( ( ), ( )) ( ) ( )
A X A x xA A     ( ) ( )1

Triangular MF: trimf x a b c
x a
b a
c x
c b
( ; , , ) max min , ,















0
Trapezoidal MF: trapmf x a b c d
x a
b a
d x
d c
( ; , , , ) max min , , ,















1 0
Generalized bell MF: gbellmf x a b c
x c
b
b( ; , , ) 


1
1
2
Gaussian MF:
2
2
1
),,;(





 

 
cx
ecbaxgaussmf

Sigmoidal MF: sigmf x a b c
e a x c( ; , , ) ( )
  
1
1
Extensions:
Abs. difference
of two sig. MF
Product
of two sig. MF

L-R MF:
LR x c
F
c x
x c
F
x c
x c
L
R
( ; , , )
,
,
a 
a







 





 







Example: F x xL ( ) max( , ) 0 1 2
F x xR ( ) exp( ) 
3
c=65
a=60
b=10
c=25
a=10
b=40

Base set A Cylindrical Ext. of A

Two-dimensional
MF
Projection
onto X
Projection
onto Y
R x y( , ) 

A
y
R
x
x y
( )
max ( , )
 

B
x
R
y
x y
( )
max ( , )

project.m

Fuzzy sets

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