1-1
Chapter 1 From Crisp to Fuzzy sets
1.1. Introduction
○ Uncertainty
-- Arising from imprecision, vagueness
non-specificity, inconsistency
○ Traditional view:
-- Uncertainty is undesirable and should be
avoided.
Modern view:
-- Uncertainty is not only an unavoidable
plague, but it has a great utility.
○ Traditionally, three key characteristics are to
be minimized in order to maximize system
performance
 complexity

incredibility
 uncertainty

Although undesirable when considered alone,
uncertainty becomes valuable when considered
together with the other characteristics
1-2
e.g., allowing more uncertainty tends to
(a) reduce complexity
(b) increase credibility
○ Transition of views
Newtonian mechanics－ precise laws,
analytic methods
Statistical mechanics－ probability theory
Fuzzy mechanics－ theories of uncertainty
○ Analytic methods－
(a) Base on calculus
(b) Involve a small number of variables
that are related to one another in a
predictable way.
Statistical methods－
(a) Base on probability theory
(b) Specific manifestations of microscopic
entities are replaces with their statistical
averages, which are connected with
appropriate macroscopic variables.
1-3
(c) Involve a large number of variables and a
high degree of randomness.
1.2. Crisp Sets
○ Methods for describing sets
1. Enumeration A  {a1 , a2 ,....., an }
2. Description A  {x | p ( x)}
3. Characteristic function
1 x  A
mA ( x )  
0 x  A
1-4
○ Power sets of A︰P(A)
Second order power set of A︰ P2 ( A)  P( P( A))
3
4
Higher order power set of A︰ P ( A), P ( A)
○ Relative complement (difference) ︰A－B
Absolute complement U  B  B
○ General principle of duality
 , , 
U , , 
○ Fundamental properties of set operations
1-5
Example:
Show DeMorgan’s laws A  B  A  B

show 1) A  B  A  B
2) A  B  A  B
(1) Show A  B  A  B
Let S  A  B and T  A  B
Show A  B  A  B  show S  T
 show x  S  x T
Given x  S => x  A  B
=> x  A  B => x  A  x  B
=> x  A  x  B => x  A  B  T
S  T
Similarly, T  S (Assignment 1)
so
S＝T
(2) Show A  B  A  B (Assignment 2)
○ Partial order
Let X  {a, b, c}
Let P( X ) be the power set of X ignoring the
empty set
.
1-6
P( X )  {{a},{b},{c},{a, b},{a, c},{b, c},{a, b, c}}
( P( X ),  ) forms a partial order, i.e., transitive ,
anti-symmetric (no loop) , reflective
Arrows (  ) represent inclusion (  )
○ Lattice
– is formed by a partial ordering and for
every pair A, B  P ( X ) , exists an LUB
(supremum , join , A  B ) and a GLB
(infimum , meet , A  B ).
○
 P ( X )
, ＝  P( X ), ,  
Boolean Lattice
Boolean algebra
e.g., ( A  B iff A  B  B or A  B  A )
1-7
○ Partition:  ( A)  {Ai | i  I , Ai  A}
i)
Ai  
ii)
Ai  Aj  
iii)
Ai  A
iI
○ Refinement relation  :
Let  1 ,  2 be 2 partitions of A.
1
2
1
2
If Ai  one and only one Aj s.t. Ai  Aj
=> 1 is a refinement of  2
2
1
○ Let  ( A) be the set of all partitions of A
  ( A),  forms a lattice (partition lattice).
○ Nested family:
A = {A1 , A2 ,.............., An }
if
Ai  Ai 1
A1 : innermost set, An : outermost set
1-8
○ Convex set A:
r , s  A , and   [0,1]
t   r  (1   ) s  A
Any set defined by a single interval of real
number is convex
Any set defined by more than one separate
interval can not be convex
○ For a partial ordering  on A
i) An upper bound r of A :
x  A, x  r
ii) A lower bound s of A :
x  A
sx
iii) r is a lowest upper bound (LUB) of A :
iff (a) r is an upper bound of A
(b) no a  r is an upper bound of A
s is a greatest lower bound (GLB) of A
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iff
(a)’ s is a lower bound of A
(b)’ no b  s is a lower bound of A
iv) Supremum r: sup A
iii) (a), (b) and r  A
Infimum s:
inf A
iii) (a)’, (b)’ and s  A
1.3 Fuzzy Sets: Basic Types
○ Fuzzy sets
－Sets with vague boundaries
－Membership of x in A is a matter of degree
to which x is in A
○ Utilization of fuzzy sets
(1) Representation of uncertainty
(2) Representation of conceptual entities
e.g., expensive, close, greater, sunny, tall
○ Fuzzy Sets
membership

Crisp Sets

characteristic
function
 A : X  [0,1]
function

mA : X  { 0 , 1 }
1-10
e.g.,
1
i) “close to 0” :  A ( x)  1  10 x 2
1



(
x
)


2 
ii) “very close to 0” : A
 1  10 x 
2
1
iii) “close to a” :  A ( x)  1  10( x  a)2
○ Difference between crisp, random, and fuzzy
variables:
Crisp variable: a uniform probability
distribution
Random variable: a probability
distribution
Fuzzy variable: a membership function
is associated with its domain.
1-11
○ Generalization
i) Ordinary fuzzy sets:  A : X  [0,1]
Abbreviated as A : X  [0,1] .
i.e., Each element of X is assigned a
particular real number (i.e., precise
membership grades).
ii) L-fuzzy sets: A : X  L , where L is a partial
order set.
iii) Interval–valued fuzzy sets: A : X   ([0,1]) ,
where  ([0,1]) is the family of all closed
interval in [0,1].
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iv) Fuzzy sets of type-K
-- Interval–valued fuzzy sets possess
fuzzy Intervals
(a) Type-2: A : X  ([0,1]) , where
([0,1]) : fuzzy power set of [0,1], the
set of all ordinary fuzzy sets
defined on [0,1].
(b) Type-3
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v) Level-K fuzzy sets
-- Elements in a universal set are
themselves fuzzy sets.
(a) Level-2: A : ( X )  [0,1]
e.g., fuzzy set “x is close to r”
x : a fuzzy variable
r : a particular number , e.g., 5.
(b) Level 3:
vi) Combinations of interval-valued, L,
type-K, level-K fuzzy sets.
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1.4 Fuzzy Sets: Basic Concept
○ Example: 3 fuzzy sets defined on age.
A1 : “young”, A2 :”middle-aged”, A3 : ”old”
Membership functions:
1


A1 ( x)  (35  x) /15

0

0

( x  20) /15

A2 ( x)  
(60  x) /15

1
0


A3 ( x)  ( x  45) /15

1

x  20
20  x  35
x  35
x  20 or
x  60
20  x  35
45  x  60
35  x  45
x  45
45  x  60
x  60
1-15

○  -cut
A:

A  {x | A( x)  }
If 1   2 
1
A  2 A
Strong  -cut

A:

A  {x | A( x)  }
e.g.,

A1  [0,35  15 ]



A2  [15  20, 60  15 ]   (0,1]

A3  [15  45,80] 

A1  (0,35  15 )



A2  (15  20, 60  15 )    [0,1)

A3  (15  45,80) 
○ Level set ( A) ︰
( A)  { | x  X , s.t


{

|
A  }
or
A( x)   }
1-16
e.g.,
Continuous case --( A1 )  ( A2 )  ( A3 )  [0,1]
Discrete case --( D1 )  {0 , 0.13 , 0.27 , 0.4 , 0.5 , 0.67 , 0.8 , 0.93 , 1}
○ Support︰
S ( A) = [x  X | A( x)  0]
S ( A) 
0
A , e.g., S ( D2 )  {22,24, ,58}
○ Core︰ 1 A (i,e, 1 - cut)
1-17
○ Hight h(A)︰the largest membership grade
h( A)  sup A( x)
xX
○ Normal︰h(A) = 1
Subnormal︰h(A) < 1
○ Convex fuzzy set︰
  (0,1] ,  -cut is convex
1-18
○ Theorem 1.1: A convex fuzzy set on R
iff x1 , x2  R ,  [0,1] ,
A( x1  (1   ) x2 )  min [ A( x1 ), A( x2 )]
Proof︰
i, (  ) Given A : convex ,
x1, x2 , Let a = min[A(x1 ), A(x2 )]
 x1, x2  a A
A : convex  a A convex
  [0,1], x   x1  (1   ) x2  a A
(definition of convex set)
 A( x)  a  min[ A( x1 ), A( x2 )]
ii, (  )
x1, x2 , Given A( x1  (1   ) x2 )
 min[ A( x1 ), A( x2 )]
(Show that   (0,1] ,  A : convex )
x1, x2 ,  , s,t.
A(x1 )   , A(x2 )   (i,e., x1,x2   A)  (1)
  [0,1]
A( x1  (1   ) x2 )  min[ A( x1), A( x2 )]
 min( ,  )   ,
i.e.,  x1  (1   ) x2   A
 (2)
(1), (2)   A : convex  A : convex
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◎ Fuzzy Set Operations
․Standard complement︰ A( x)  1  A( x)
․Equilibrium points︰ A( x)  A( x)
A( x)  A( x)  1  A( x)  2 A( x)  1  A( x)  A( x)  0.5
․Standard intersection︰ ( A  B)( x)  min[ A( x), B( x)]
․Standard union: ( A  B)( x)  max[ A( x), B( x)]
․Difference
A  B  A  B  min( A( x),1  B( x))
Symmetric difference
A B  ( A  B)  ( B  A)
1-20
○ Example:
1-21
○ Example: A1  A3 ?
A1  A3 ︰not young and not old
A2 ︰middle age
1-22
○ Any fuzzy power set P(X) with  form a
lattice, referred to as a De Morgan lattice
(De Morgan algebra)
In such a lattice ,
A, B  P( X ) ,

join︰ A  B (LUB, supremum)
meet︰ A  B (GLB,
infimum)
This lattice possesses all the properties (Table
1.1) of the Boolean lattice (or Boolean algebra)
except the laws of contradiction ( A  A   )
and exclusive middle ( A  A  X )
․Verify A  A   (law of contradiction) is
violated for fuzzy sets,
i.e., Show x min{ A( x),1  A( x)}  0
e.g.,
A( x)  0.3  1  A( x)  0.7
min{0.3, 0.7}  0.3  0
1-23
․Verify A  ( A  B)  A (law of absorption)
i.e., Show
x max{ A( x),min{ A( x), B( x)}}  A( x)
x
i, if A(x)  B(x) ,
 min[A(x) , B(x)] = A(x) and
max[A(x ), B (x )] = A(x )
ii, if A(x) > B(x) ,
 min[A(x) , B(x)] = B(x) and
max[A(x ) , B (x )] = A(x )
◎ Fuzzy set inclusion (subset) 
A  B iff x , A( x)  B( x)
 A  B  A, A  B  B
1-24
○ Description of fuzzy sets with finite supports
i, Finite universersal set X (discrete case)
A
or A 
a
a1 a2
  ....  n
x1 x2
xn
ai
, ai  A( xi )

xi Supp ( X ) xi
ii, X is an interval of real numbers (continuous
case)
A 
X
A( x)
x
◎Scalar cardinality (or sigma count) | A |
| A |  A( x)
xX
1-25
◎ Fuzzy cardinality
․Fuzzy number: convex, normalized fuzzy set
․Fuzzy cardinality | A |
--- a fuzzy number define on N whose
membership function is
   A
| A | (|  A |)  


or | A | 
 A | A |

|  A | ︰ the degree to which fuzzy set A
contain the number of members ,
|  A | , is 
1-26
○ Example
X︰crisp universal set
X = {5 , 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80}
Fuzzy sets labeled as
“infant” , “adult” , “young” , “old”
1-27
Consider Fuzzy set labeled “old”
 Scalar cardinaliity:
|old|  0  0  0.1  0.2  0.4
 0.6  0.8  1  1  4.1
 Fuzzy cardinality:
  {0,0.1,0.2,0.4,0.6,0.8,1}
old
when
 = 0.1,
old  {20,30, 40,50, 60, 70,80}
0.1
| 0.1 old|  7
 = 0.2,
0.2
old  {30, 40,50, 60, 70,80}
| 0.2 old|  6
 = 0.4,
0.4
old  {40,50, 60, 70,80}
| 0.4 old|  5
 = 0.6,
0.6
old  {,50, 60, 70,80}
| 0.6 old|  4
 = 0.8,
0.8
old  {60, 70,80}
| 0.8 old|  3
 = 1,
old  {70,80}
1
| 1 old|  2
| old |
0.1 0.2 0.4 0.6 0.8 1





7
6
5
4
3 2
1-28
◎ Degree of subsethood , S(A,B) , of A in B
S ( A, B) 
| A B |
| A|
1
(| A |   max{0, A( x)  B ( x)})
| A|
x X
1
=
(| B |   max{0, B ( x)  A( x )})
| A|
x X
1
=
(  min{ A( x), B ( x )})
| A | xX
S ( A, B ) 
1-29
◎ Distances between fuzzy sets
X︰universal set containing n elements
A, B︰fuzzy sets defined on X
an
a1
a1
A

  
x1
x2
xn
bn
b1
b2
B

  
x1
x2
xn
0  ai , bi  1
From A  PA  (a1 , a2 , , an )
From B  PB  (b1 , b2 , , bn )
In an n-D space ,
d ( A, B)   | A( x)  B( x) |
xX
 d ( A, B)  d ( B, A)
The n-cube represents the fuzzy power set  (X)
The vertices represents the crisp power set P(X)
1-30
※ Scalar cardinality |A| = d(A,Φ):
Probability distributions are represented by sets
whose cardinality is 1 (
 P  1)
i
the set of
all probability distributions is represented by a
(n-1)-D simplex of the n-cube (
 P  1)
i
◎ Assignment︰
1.8 , 1.10 , 1.11 (1996),
1.4 , 1.6 , 1.13 (1997)
1.12 , 1.14
1.7 , 1.9 , 1.15 (2000)
(1998),