2-1
Chapter 2: Fuzzy Sets vs Crisp Sets
※ α-cuts bridge fuzzy and crisp sets
2.1 Additional Properties of α-cuts
○ Theorem 2.1: A, B ( X ) ,  ,  [0,1]
(i)

(ii)
(iii)
A A
  

A   A,   A    A

( A  B)   A   B ,

( A  B)   A   B
(iv)   ( A  B)    A    B ,

(v)

( A  B)    A    B
( A)  (1 )  A
Proof:


(ii) (a) Show     A  A
x   A  A( x)  
   ,  A( x)    
 x  A 

A A
2-2
(b) Show    

A

A
x    A  A( x)  
 
 A( x)    
 x  A 
(iii) (a) Show


A   A
( A  B) A B

1) x  ( A  B ) , ( A  B)( x)  
 min{ A( x), B ( x)}  
 A( x)   , B ( x)  
 x   A, x   B  x   A   b
  ( A  B )   A   B - - - - ( A)
2) x   A   B , x   A, x   B
 A( x)   , B( x)    min[ A( x), B( x)]  
 ( A  B)( x)    x   ( A  B)
  A   B   ( A  B)
(A), (B) 
(b) Show


- - - - - - ( B)
( A  B)   A   B
( A  B)   A   B
1) x   ( A  B), ( A  B)( x)  
 max{ A( x), B( x)}  
 A( x)   or B( x)  
 x   A or B( x)    x   A   B
2-3
  ( A  B)   A   B
------- (A)




2) x  A  B, x  A or x  B
 A( x)   or B( x)  
 max[ A( x), B( x)]  
 ( A  B)( x)    x   ( A  B)
  A   B   ( A  B)
------- (B)



(A),(B)  ( A  B)  A  B



(iv) (a) Show ( A  B)  A  B,
x    ( A  B)  ( A  B)( x)  
 min{ A( x), B( x)}  
 A( x)   , B( x)  
 x    A, x    B  x    A    B
   ( A  B)    A    B
------ (A)
x    A    B  x    A, x    B
 A( x)   , B( x)  
 min( A( x), B( x))  
 ( A  B )( x)    x    ( A  B )
   A    B    ( A  B)
------ (B)



(A),(B)  ( A  B)  A  B
2-4

(b) Show
( A  B)    A    B
x    ( A  B)
 ( A  B)( x)    max( A( x), B( x))  
 A( x)   or B( x)  
 x    A or x    B  x    A    B
   ( A  B)    A    B
------ (A)
x    A    B  A( x)   or B( x)  
 max( A( x), B( x))  
 ( A  B)( x)    x    ( A  B)
   A    B    ( A  B)
------ (B)



(A),(B)  A  B  ( A  B)
(v) Show

( A)  (1 )  A

1) x  ( A), A( x )  
 1  A( x)    A( x)  1  
 x  (1 ) A  x  (1 ) A


( A) 
(1 ) 
A
(1 ) 
A, x  (1 ) A
2) x 
 A( x)  (1   )  1  A( x)  
 A( x)    x   ( A)
 (1 )  A   ( A)
2-5
◎

( A): the α-cut of the complement of A

A: the complement of the α-cut of A
○ Theorem 2.2:
Ai  ( X ), i  I infinite index set

(vi)
Ai   (
Ai )
i
i

(vii)
Ai   (
i
Ai   (
i

,
Ai )
i

,
i
Ai )
Ai    (
i
Ai )
i
Proof:

(vi) (a) Show

iI

i
i
x 
A  ( A)
i
i
Ai  i0  I , s.t.
x   Ai0 (i.e. Ai0 ( x)   )
2-6
 sup Ai ( x)    (
iI
 x  (
iI

(b) Show
i

iI
iI
iI
Ai )
Ai   ( Ai )
Ai  i, Ai ( x)  
 inf Ai ( x)    (
iI
 x (
i
Ai   (
i
Ai )( x)  
iI

Ai ) 
iI

(vii) (a) Show
x 
Ai   (
i

x 
Ai ) 
iI
Ai )( x)  
Ai   ( Ai )
i
Ai )
i

Ai , i0  I s.t. x   Ai0
i
(i.e., Ai ( x)   )
0
def .
 supAi ( x)    (
iI
 x   (
Ai ) 

iI
(b) Show

iI
Ai )( x)  
Ai   (
i
i
Ai    ( Ai )
i
i
x    ( Ai )  ( Ai )( x)  
i
i
def .

A ( x)  
inf
meet
iI
i
Ai )
2-7
 i  I , Ai ( x)   (i.e., x    Ai )
 x


Ai

( Ai ) 
i
i

Ai
i
○ Example:
A  F( X )
i
1
Let A ( x)  1  , i  N
i
i
1
x  X , ( A )( x)  supA ( x)  sup(1  )  1
i
i
i
i
i
i
 ( A)  X
1
i
i
1
However, i  N , A   ( x  X , A ( x)  1   1)
i
 A     X  ( A)
1
i
i
1
i
1
i
i
i
i
○ Theorem 2.3: A, B  F ( X ),  [0,1]
(viii) A  B iff A  B , (ix) A  B iff


+
A  B
+
Proof:
(viii) (a) Show A  B iff A  B


i) (  ) Assume  A   B
 x , s.t. x   A, x   B,
0
0
0
i.e. A( x )   , B( x )  
0
0
def.
 A( x )  B( x )  i.e. A  B contradict
0
0

A B
2-8
ii) (  ) Assume A  B  x , s.t.
0
A( x )  B ( x )
0
0
Let  =A( X )  x   A, x   B
0
0
0
i.e.  A   B
(b) Show A  B iff

A  B

1) ( ) Similar to (a)
2) ( )
Assume A  B  x , A( x )  B( x )
0
0
0
Let A( x )    B ( x )  x   A, x   B

0
0
0

0
 A  B

(ix) A  B iff

+
(a) A  B iff

(b) A  B iff

A   +B
A  B
A   B
1,  if A  B, then  A   B
A  B  x, A( x)  B( x)
  , A( x )   iff B ( x )  
  , A( x )   A iff x   B
  ,  A   B
2-9
2.  if

A   B, then A  B
assume A  B  x0 s.t. A( x0 )  B( x0 )
Let  =A( x0 )    B ( x0 )
 x0   A, x0   B
  A  B

(b) A  B iff
A

B
1.  if A  B, then


A
B
A  B  x, A( x)  B ( x)
  , A( x)   iff B( x)  
  , A( x) 
  ,
2.  if

A


A


A iff x    B
B
B, then A  B
assume A  B  x0 s.t. A( x0 )  B( x0 )
Let  =A( x0 )    B ( x0 )
 x0    A, x0    B


A

B