FUZZY SETS
AND
FUZZY LOGIC
PART 4
Fuzzy Arithmetic
Theory and
Applications
1. Fuzzy numbers
2. Linguistic variables
3. Operations on intervals
4. Operations on fuzzy numbers
5. Lattice of fuzzy numbers
6. Fuzzy equations
Fuzzy numbers
•
Three properties
1) A must be a normal fuzzy set;
2) αA must be a closed interval for every
  (0, 1];
3) the support of A, 0+A, must be bounded.
A is a fuzzy set on R.
Fuzzy numbers
Fuzzy numbers
• Theorem 4.1
Let A  F (R ). Then, A is a fuzzy number if and
only if there exists a closed interval
[a, b]   such that
1 for x  [a, b]

A( x)  l ( x) for x  (, a)
r ( x) for x  (b, ),

Fuzzy numbers
• Theorem 4.1 (cont.)
where
l is a function from (, a) to [0, 1] that is
monotonic increasing, continuous from the right,
and such that l ( x)  0 for x  (, 1 ) ; r is a
function from (b, ) to [0, 1] that is monotonic
decreasing, continuous from the left, and such
that r ( x)  0 for x  (2 , )
Fuzzy numbers
Fuzzy numbers
Fuzzy numbers
• Fuzzy cardinality
Given a fuzzy set A defined on a finite universal
~
set X, its fuzzy cardinality, | A | , is a fuzzy number
defined on N by the formula
~ 
| A | (| A |)  
for all
  ( A).
Linguistic variables
• The concept of a fuzzy number plays a
fundamental role in formulating quantitative
fuzzy variables.
• The fuzzy numbers represent linguistic
concepts, such as very small, small, medium,
and so on, as interpreted in a particular context,
the resulting constructs are usually called
linguistic variables.
Linguistic variables
• base variable
Each linguistic variable the states of which are
expressed by linguistic terms interpreted as
specific fuzzy numbers is defined in terms of a
base variable, the values of which are real
numbers within a specific range.
A base variable is a variable in the classical
sense, exemplified by any physical variable
(e.g., temperature, etc.) as well as any other
numerical variable, (e.g., age, probability, etc.).
Linguistic variables
• Each linguistic variable is fully characterized by a
quintuple (v, T, X, g, m).
– v : the name of the variable.
– T : the set of linguistic terms of v that refer to
a base variable whose values range over a
universal set X.
– g : a syntactic rule (a grammar) for generating
linguistic terms.
– m : a semantic rule that assigns to each
linguistic term t T.
Linguistic variables
Operations on intervals
• Let ＊ denote any of the four arithmetic
operations on closed intervals: addition ＋ ,
subtraction —, multiplication • , and division /.
Then,
[a, b]  [d , e]  { f  g | a  f  b, d  g  e},
[a, b]  [d , e]  [a  d , b  e],
[a, b]  [d , e]  [a  e, b  d ],
[a, b]  [d , e]  [min( ad , ae, bd , be), max( ad , ae, bd , be)],
[a, b] /[ d , e]  [a, b]  [1 e , 1 d ]
 [min( a / d , a / e, b / d , b / e),
max( a / d , a / e, b / d , b / e)].
Operations on intervals
• Properties
Let A  [a1, a2 ], B  [b1, b2 ], C  [c1, c2 ], 0  [0, 0], 1  [1, 1].
1. A  B  B  A,
A  B  B  A (commutativity ).
2. ( A  B)  C  A  ( B  C ),
( A  B)  C  A  ( B  C ) (associativ ity ).
3. A  0  A  A  0 ,
A  1 A  A  1 (identity ).
4. A  ( B  C )  A  B  A  C ( subdistrib utivity).
Operations on intervals
5. If b  c  0 for every b  B and c  C , then A  ( B  C )  A  B  A  C
(distributi vity). Furthermor e, if A  [a, a ], then a  ( B  C )  a  B  a  C.
6. 0  A-A and 1  A/A.
7. If A  E and B  F , then :
A  B  E  F,
A  B  E  F,
A B  E  F,
A / B  E / F (inclusion monotonicity).
Operations on fuzzy numbers
• First method
Let A and B denote fuzzy numbers. ＊ denote
any of the four basic arithmetic operations.



( A  B)  A  B
for any   (0, 1].
A B 
Since


α
α[0, 1]
( A  B).
( A B) is a closed interval for each
  (0, 1]. and A, B are fuzzy numbers, A  B is
also a fuzzy number.
Operations on fuzzy numbers
• Second method
for all z  R
(A  B)( z )  sup min[ A( x), B ( y )],
z  x y
(A  B)( z )  sup min[ A( x), B( y )],
z x y
(A  B)( z )  sup min[ A( x), B( y )],
z  x y
(A  B)( z )  sup min[ A( x), B( y )],
z  x y
(A / B)( z )  sup min[ A( x), B( y )].
zx / y
Operations on fuzzy numbers
Operations on fuzzy numbers
Operations on fuzzy numbers
• Theorem 4.2

Let ＊
{＋, －, •, / }, and let A, B denote
continuous fuzzy numbers. Then, the fuzzy set
A＊B defined by
(A  B)( z )  sup min[ A( x), B( y)]
z  x y
is a continuous fuzzy number.
Lattice of fuzzy numbers
• MIN and MAX
MIN( A, B)( z ) 
MAX( A, B)( z ) 
sup
min[ A( x), B( y )],
sup
min[ A( x), B( y )].
z  min( x , y )
z  max( x , y )
Lattice of fuzzy numbers
Lattice of fuzzy numbers
Lattice of fuzzy numbers
• Theorem 4.3
Let MIN and MAX be binary operations on R.
Then, for any A, B, C  R , the following
properties hold:
Lattice of fuzzy numbers
Lattice of fuzzy numbers
• Lattice R, MIN, MAX
It also can be expressed as the pair R,  ,
where  is a partial ordering defined as:
A  B iff MIN( A, B)  A or, alternativ ely,
A  B iff MAX( A, B)  B
for any A, B  R and all α  (0, 1], we can also define the
partial ordering in terms of the relevant  - cuts :
A  B iff min(  A,  B)  A,
A  B iff max(  A,  B)  B,
where  A,  B are closed intervals.
Lattice of fuzzy numbers
Then,
min ( α A, α B)  [min (a1, b1 ), min (a2 , b2 )],
max ( α A, α B)  [max (a1, b1 ), max (a2 , b2 )].
If we define the partial ordering of closed intervals in the usual
way, that is,
[a1, a2 ]  [b1, b2 ] iff a1  b1 and a2  b2 ,
then for any A, B  R , we have
A  B iff
for all   (0, 1].

A  B
Fuzzy equations
• A+X=B
The difficulty of solving this fuzzy equation is
caused by the fact that X = B－A is not the
solution.
Let A = [a1, a2] and B = [b1, b2] be two closed
intervals, which may be viewed as special fuzzy
numbers. B－A = [b1－ a2 , b2 －a1], then
Fuzzy equations
Let X = [x1, x2].
Then, [a1  x1 , a2  x2 ]  [b1 , b2 ].
a1  x1  b1 , x1  b1  a1.
a2  x2  b2 , x2  b2  a2 .
 X must be an interval, it' s required that x1  x2 .
 the equation has a solution iff b1  a1  b2  a2 .
X  [b1  a1 , b2  a2 ].
Fuzzy equations
Let αA = [αa1, αa2], αB = [αb1, αb2], and
αX = [αx , αx ] for any
  (0,.1]
1
2

A  X  B has a solution iff :
(i)  b1  a1  b2  a2 for every α  (0, 1], and
(ii) α  β implies α b1  α a1  β b1  β a1  β b2  β a2  α b2  α a2 .
the solution X of the fuzzy equation is given by
X
  X.
α( 0 , 1]
Fuzzy equations
• A．X = B
A, B are fuzzy numbers on R+. It’s easy to show
that X = B / A is not a solution of the equation.

A  X  B has a solution iff :
(i)  b1 /  a1  b2 /  a2 for every α  (0, 1], and
(ii) α  β implies α b1 / α a1  β b1 / β a1  β b2 / β a2  α b2 / α a2 .
the solution X of the fuzzy equation is given by
X
  X.
α( 0 , 1]
Exercise 4
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4.1
4.2
4.5
4.6
4.9