Extension principle method
( A * B)( Z )  sup min[ A( X ), B(Y )] --(4.9)
Z  X Y
e.g. *=+
 ( A  B)( Z )  sup min[ A( X ), B(Y )]
Z  X Y
Theorem 4.2 : A , B : continuous fuzzy numbers
 A*B : continuous fuzzy number
( i.e. (a) continuous (b) normal
Proof :
(B) A*B : normal
∵A , B : normal
∴ x0 , y0 s.t. A( x0 )  1, B( y0 )  1
Let Z0  X 0 * Y0
( A * B ) (Z0 )
(c) convex
s u p mAi nX[ (B Y
),
Z0  X* Y
(d) bounded support )
( ) ] 0A mx i n 0[B (y ) ,
( ) ]
m i n [1, 1]
∴ A*B : normal
(C) A*B : convex
 (  ,  ( A * B) : closed interval )
(if  ( A * B)   A *  B) , =>  A,  B : closed =>  A *  B : closed (pp.103) )
(i) 
Z   A*  B
x0   A, y0   B, s.t. Z  X 0 * Y0
 ( A * B)(Z )  sup min[ A( X ), B(Y )]  min[ A( x0 ), B( y0 )]  
Z  X *Y
 Z   ( A * B)  A *  B  ( A * B)
(ii) 
Z   ( A * B)
( A * B ) (Z )
s u p mAi nX[ (B Y

) , ( ) ]
Z  X* Y
We want to show there exists [ X i ],[Yi ] and ☆ x0   A  [ X i ]   , y0   B  [Yi ]  
s.t. z  x0 * y0   A *  B
1
Heine-Borel Theorem :
-- A subset K of R is compact iff it is closed and bounded
Closed : sets with end points , boundaries
( neighbored → open → close )
Bounded : sets contained in finite spaces
(r , s .t . A [ r , r ] )
Monotone Subsequence Theorem :
-- if X  ( X n ) is a sequence of real numbers , then there is a subsequence of X that is monotone
Bolzano-Weierstrass Theorem :
-- A bounded sequence of real numbers has a convergent subsequence
Theorem :
A subset K of R is compact iff every sequence in K has a subsequence that converges to a point in K

Let A  R A point c  R is a cluster point of A if every   neighborhood V (c)  (c   , c   ) of c

contains at least one point of A distinct from c
A neighborhood of a point x  R is any set V that contains an   neighborhood
V ( c) ( x  , x  of
) x for some   0
Theorem :
-- A number c  R is a cluster point of a subset A of R iff there exists a sequence ( an ) in A with an  c for all
n  N s.t. lim(an )  c
 1
1
Given  , n     1, X n , Yn , s.t.Z  X n * Yn and X n  n1 A ,
 
Consider { X n },{Yn } : sequences
1
n
   
 1
 n1 A 
{ X n } 
 1
n
 1
n
 1
n
1
n 1
 1
A, n1 B 
A,{Yn } 
 1
n
B
 1
n
B
 1
A,
n
B : closed intervals
{ X n },{Yn } : bounded prove (D)


convergent
{ X n } of { X n } , s.t. X n i  X 0
s u b s e q u e n c ie
s u b s e q u e{nYncij }e of {Yni } ,
s.t. Yn ij  Y0
Corresponding subsequence { X nij } of { X ni } , s.t. X n ij  X 0
Where X n ij *Yn ij  Z
Yn 
 1
n 1
B ---- (1)
4-26
 ：continuous operator


 Z  lim X nij  Ynij  (lim X nij )  (lim Ynij )  X 0  Y0
j 
A( X nij )   

1
nij
j 
j 
B(Ynij )   
,

1
nij
 A( X 0 )  A lim X nij  lim A( X nij )  lim( 

j 

j 
j 
 B(Y0 )  B lim Ynij  lim B(Ynij )  lim( 
j 
j 
j 
1
) 
nij
1
) 
nij
X 0   A, Y0   B, s.t. Z  X 0  Y0   A   B
i,e., Z   A   B
(A)
A  B : continuous
Assume A  B : not continuous at Z 0 and
l i m A(  B Z) ( )A  (B Z 0 )( s u) p
Z  Z0
Z0  X *Y
 AmXi nB Y ( 
),
( )
4-27
 X 0 , Y0 , s.t, Z0  X 0  Y0 ,
lim ( A  B)( Z )  min  A( X 0 ), B(Y0 )  --- (A)
Z  Z0
 ：monotonic
Xn  X0
as n  
 X n  , Yn  , s.t. 
Yn  Y0
Where X n  Yn  Z n  n
Let Z n  X n  Yn
 Z n  Z0 as n  
lim ( A  B)( Z )  lim( A  B)( Z n )  lim sup min  A( X ), B(Y ) 

Z  Z0
n 
n 
Z n  X *Y
 lim min  A( X n ), B(Yn )  min  A(lim X n ), B(lim Yn )   min  A( X 0 ), B(Y0 ) 
n 
x 
 x

Contradict Eq. (A)
 A  B : continuous
4-28
4.5 Lattice of fuzzy numbers
。partial ordering ：a binary relation R ( x, y ) ,
that is reflexive, antisymmetric, transitive ( e.g.,   )
Lower bound：
X  X ， y  A ， X  Y  X ：lower bound of A on X
Upper bound：
X  X ， y  A ， Y  X  X ：upper bound of A on X
Infimum ：greatest lower bound (GLB)
Supremum：least upper bound
(LUB)
Lattice --- a partial ordering R on a set X
That contains an infimum and a supremum for every pair of elements in X
( X , R)
。Example ( R, ) : a lattice
 can be expressed in terms of lattice operations
x x  y
----- Inf.
min( x, y )  
y y  x
y x  y
----- Sup.
max( x, y )  
x y  x
。Example ( P( X ), ) : a lattice,  can beexpressed
   Inf .
  Sup.
4-29
◎ Fuzzy numbers

Real numbers
Partially orderd

linearly ordered
MIN(meet)
MAX(joint)
min
max
Let A,B ：fuzzy numbers
Define MIN ( A, B)( Z ) 
sup
min  A( x), B(Y ) 
Z  min( x , y )
M A X( A
, B) ( 
Z)
。Examples：
0

A( x)  ( x  2) / 3
(4  x) / 3

0

B( x)   x  1
3  x

m i n A (x ) B
, Y( )
sup
Z  m a xx( y,
x  2 ,x  4
2  x  1
1 x  4
x  1 ,x  3
1 x  2
2 x3
)
4-30
0
( x  2) / 3

i, MIN ( A, B)( x)  
(4  x) / 3
3  x
x  2 ,x  3
2  x  1
1  x  2.5
2.5  x  3
x  1, x  3
0
x 1
1 x  2

MAX ( A, B)( x)  
2  x  2.5
3  x
(4  x) / 3 2.5  x  4
※ MIN、MAX：different from standard
、
、min、 max
4-31
Let R: the set of all fuzzy numbers
<R , MIN , MAX>: a distributive Lattice
A, B, C  R 
b  F a  E

v0  RB(v0 )  1
·Theorem 4.3
 M  min( A( z ), A( z ), B(v0 ))  A( z )
min( x, u )  z  x  max( x, u )
A, B, C  R
A( x1  (1   ) x2 )  min[ A( x1 ), A( x2 )]
A( z )  min( A(min( x, u )), A(max( x, u )))
 MIN ( A, ( MAX ( B, C ))  A
(a) Commutativity : MIN(A,B)=MIN(B,A)
MAX(A,B)=MAX(B,A)
(b)Associativity :
MIN(MIN(A,B),C)=MIN(A,MIN(B,C))
MAX(MAX(A,B),C)=MAX(A,MAX(B,C))
(c)Idenpotence : MIN(A,A)=A
MAX(A,A)=A
(d)Absorption : MIN(A,MAX(A,B))=A
MAX(A,MIN(A,B))=A
(e)Distributivity: MIN(A,MAX(B,C))=MAX(MIN(A,B),MIN(A,C)
MAX(A,MIN(B,C)=MIN(MAX(A,B),MAX(A,C))
Proof: : (a),(b),(c),(d) can be easily seen by ※ prove (b),(d),(e)
4-32
(b)Prove associativity –
z  R ,
i , MIN(A,MIN(B,C)(z) = sup min(A(x),MIN(B,C)(y))
z=min(x,y)
= sup min(A(x),sup min(B(u),C(v))
z=min(x,y)
y=min(u,v)
A( x) is irrelevant
To sup
y=m(u,v)
=
sup
sup min(A(x),B(u),C(v))

 they can
z=min(x,y) y=min(u,v)
be switched
= sup min(A(x),B(u),C(v))
z=min(x,u,v)
= sup
sup min(A(x),B(u),C(v))

z=min(s,v) s=min(x,n)
= sup min( sup min(A(x),B(u)),C(v))
z=min(s,v) s=min(x,u)
= sup min(MIN(A,B)(s),C(u))
Z=min(s,v)
=MIN(MIN(A,B),C)(z)
ii, similarly ,
MAX(A,MAX(B,C))(z)=MAX(MAX(A,B),C)(z)
4-33
(d)Prove absorption
z  R
i , MIN(A,MAX(A,B))(z) = sup
min(A(x),MAX(A,B)(y))
z=min(x,y)
= sup min(A(x),sup min(A(u),B(u))
z=min(x,y)
y=max(u,v)
= sup
min(A(x),A(u),B(v))
z=min(x,max(u,v))
= M (We want to show M=A(z))
B: fuzzy number , v0  R , s.t. B(v0 )  1 (normal)
z=min(z,max(z, v0 )) 恆成立
 M  min( A( z ), A( z ), B(v0 ))  A( z ) -------(i)
z=min(x,max(u,v))
已知
 min( x, u )  z  x  max( x, u )
z is between x and u
fuzzy number : convexity
(see p.4-36)
next page
min(x,u)
max(x,u)
By theorem 1.1 A( x1  (1   ) x2 )  min[ A( x1 ), A( x2 )]
A( z )  min( A(min( x, u )), A(max( x, u )))
=min(A(x),A(u))  min(A(x),A(u),B(v))
x1 , x2
for any x,u,v ---------(ii)
M
(i)(ii)
 M  A( z ) ,  MIN ( A, ( MAX ( B, C ))  A
ii, similarly , MAX(A,MIN(B,C))=A
4-34
z =min (x,max(u,v)) ---------------(A)
 z  x ----------(B)
and min(x,u)  x  max(x,u) -------------(C)
(A)  z =x ,u ,or v
i, if z=z ,
(c), x  min(x,u)
 z  min(x,u)
ii, if z=u ,
(B) , u  x
z = u = min(x,u)
iii, if z=v ,
(A) , max(u,v)=v , u  v
(B) , v  x
 uvx
 min(x,u)=u
 z = v  u = min(x,u)
i,ii,iii,  z  min(x,u) --------------(D)
(B),(C),(D)  min(x,u)  z  x  max(x,u)
i,ii,iii (B) (C)
(D)
4-35
(e) Prove distributivity
i, z  R ,
MIN(A,MAX(B,C)(z) = sup
min(A(x),B(u),C(v)) --------(4.19)
z=min(x,max(u,v))
MAX(MIN(A,B),MIN(A,C))(z)= sup min(A(m),B(n),A(s),C(t))----------(4.20)
Z=max(min(m,n),min(s,t)
Let E= { min(A(x),B(u),C(v)) | min(x,max(u,v))=z }
F= { min(A(m),B(n),A(s),C(t)) | max(min(m,n),min(s,t))=z }
To prove (4.19)=(4.20),
1, show E  F
 a = min(A(x),B(u),C(v)) , s.t.
min(x,max(u,v))=z
i.e. a  E
 m = s = x , n = u , t = v . s.t.
max(min(m,n),min(s,t))=max(min(x,u),min(x,v))
=min(x,max(u,v)) = z
 a = min(A(x),B(u),A(x),C(v))
= min(A(m),B(n),A(s),C(t))  F
 EF
4-36
2, Show b  F , a  E s.t. b  a
b  F ,  m,n,s,t, s.t.
max(min(m,n),min(s,t))=z
b= min(A(m),B(n),A(s),C(t))
 z = min(max(s,m),max(s,n),max(t,m),max(t,n))
Let x= min(max(s,m),max(s,n),max(t,m),max(t,n)) , u=n , v=t
 z = min(x,max(u,v))
min(s,m)  x  max(s,m) , A : convexity
A(x)  min(A(min(s,m),A(max(s,m)))
= min(A(s),A(m))
  a = min(A(x),B(u),C(v)) , s.t.
min(x, max(u,v))=z , i.e. a  F
a = min(A(x),B(u),C(v))
 min(A(s),A(m),B(n),C(t))=b
i.e. b  F  a  F , s.t. b  a
 sup F  sup E  (4.19) = (4.20)
ii similarly , MAX(A,MIN(B,C))=MIN(MAX(A,B),MAX(A,C))
4-37
◎  ：partial ordering
‧Defined as
 MIN ( A, b)  A
A  B iff 
MAX ( A, b)  B
or
 min(  A, b) A
A  B iff 
 

 max( A, b)  B
‧ Let

A  [a1 , a2 ] ,


B  [b1 , b2 ]

 min( A, b)  [min( a1 , b1 ), min( a2 , b2 )]

 
max( A, b)  [max( a1 , b1 ), max( a2 , b2 )]
eq：
min(  A, b)  [a1 , a2 ]
1 
 
max( A, b)  [b1 , b2 ]
 min(  A, b)  [a1 , b2 ]
2 
 
max( A, b)  [b1 , a2 ]
‧ Define the partial ordering of closed intervals as
[a1 , a2 ]  [b1 , b2 ] iff a1  b1 , a2  b2
4-38
Eample：
Very small ≦ small ≦ median ≦ large ≦ very large
4.6. Fuzzy Equation
var iables
‧Fuzzy 
+ Arithmetic operations → Fuzzy Equations
 numbers
Consider 2 types
A X  B
A X  B
4-39
⊙A+X=B (intervals at some level 2)
‧ X=B-A：not a solution
e.g.：  A  [a1 , a2 ], B  [b1 , b2 ] for some α
 X  B A  [b1  a2 , b2  a1 ],

A X  [a1 , a2 ]  [b1  a2 , b2  a1 ]  [a1  b1  a2 , a2  b2  a1 ]  [b1 , b2 ] B
‧ Let  X  [ x1 , x2 ]
 [a1  x1 , a2  x2 ]  [b1 , b2 ]
a  x  b
x b a
 1 1 1  1 1 1
a2  x2  b2
 x2  b2  a2
 x ：interval x1  x2
 a solution iff b1  a1  b2  a2
 X  [b1  a1 , b2  a2 ],
‧ The solution of a fuzzy equation is obtained by solving a set of interval equations, one for each α in the level
set  A   B
4-40
  (0,1]
Let  A [ a1 , a2 ], B [ b1 , b2 ], X [ x1 , x2 ]
→The equation has a solution iff

(i) b1  a1  b2  a2

(ii )    b  a   b   a   b   a  b  a  X  X
1
1
1
1
2
2
2
2




 (i) X [ b1  a1 , b2  a2 ]



(ii )      X  X , nested
‧ Example：
A
0.2
0.6
0.8
0.9 1 0.5
0.1



 

[0,1) [1,2) [23) [3,4) 4 (4,5] (5,6]
B
0.1
0.2
0.6
0.7
0.8
0.9 1 0.5
0.4
0.2
0.1





 



[0,1) [1,2) [23) [3,4) [4,5) [5,6) 6 (6,7] (7,8] (8,9] (9,10]
 A  {0.1,0.2,0.5,0.6,0.8,0.9,1}
 B  {0.1,0.2,0.4,0.5,0.6,0.7,0.8,0.9,1}
   A   B  {0.1,0.2,0.4,0.5,0.6,0.7,0.8,0.9,1}
4-41

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
X  X 

A B
[4,4] [6,6]
[3,4] [5,6]
[2,4] [4,6]
[2,4] [3,6]
[1,4] [2,6]
[1,5] [2,7]
[1,5] [2,8]
[1,5] [2,8]
[0,5] [1,9]
[0,6] [0,10]

0.1
0.7 1 0.4
0.2

 

[0,1) [1,2) 2 (2,3] (3,4]
◎ A X  B
X  B / A ：not a solution
  (0,1] A X  B
Let  A [ a1 , a2 ], B [ b1 , b2 ], X [ x1 , x2 ]
→The equation has a solution iff

(i) b1 /  a1  b2 /  a2

(ii )    b /  a   b /  a   b /  a  b /  a  X  X
1
1
1
1
2
2
2
2





X
[2,2]
[2,2]
[2,2]
[1,2]
[1,2]
[1,2]
[1,3]
[1,3]
[1,4]
[0,4]
4-42
‧Example：
 0

A( x)  ( x  3)
 5 x

x  3, x  5
3 x4
4 x5
0


B( x)   ( x  12) / 8
(32  x) / 12

x  12, x  32
12  x  20
20  x  32
 A  [2  3,5  2], B  [8  12,32  12 ]
(
8  12 32  12

)
23
52
 X  [
8  12 32  12

] trivial
23
52
 The solution

 0
12  3 x
X   X  

 x 8
 32  5 x
 12  x
x  4, x  32 / 5
4 x5
5  x  32 / 5
Theorems 2.5, 2.6, 2.7
(Decomposition Theorems)