On a Theory of Fuzzy Numbers and Fuzzy Arithmetic
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Advances in Applied and Pure Mathematics
On a Theory of Fuzzy Numbers and Fuzzy Arithmetic
Yingxu Wang
International Institute of Cognitive Informatics and Cognitive Computing (ICIC)
Dept. of Electrical and Computer Engineering, Schulich School of Engineering
University of Calgary
2500 University Drive NW, Calgary, Alberta T2N 1N4
CANADA
Tel: (403) 220 6141, Fax: (403) 282 6855
yingxu@ucalgary.ca http://www.ucalgary.ca/icic/
The domains of number theories in mathematics have
been continuously expanding from binary numbers (),
natural numbers (), integers (), and real numbers () to
fuzzy numbers () and hyperstructures () [Kline, 1972;
Zadeh, 1975a; Artin, 1991; Smith, 2001; Timothy, 2008;
Sibley, 2009; Wang, 2002, 2008b, 2011a, 2013, 2014c]. It
demonstrates an interesting course of advances in human
ability of abstraction and quantification in order to deal with
the real-world entities and their perceptive representations in
the brain. The difference between and is that the members
of the former are divisible by one, while those of the latter
may not. The characteristic of the domain of fuzzy numbers
is a 2-demential hyperstructure, = ´ = {(, )} , with
a crisp set of member elements in [-∞, +∞] and an associate
crisp set of degrees of membership in [0, 1] for each of the
members.
Abstract
Fuzzy numbers in number theory are a foundation of
fuzzy sets and fuzzy mathematics that extend the domain of
numbers from those of real numbers to fuzzy numbers. Fuzzy
arithmetic is a system of fuzzy operations on fuzzy numbers.
A theory of fuzzy arithmetic is presented towards a fuzzy
mathematical structure for fuzzy inference and cognitive
computation. The mathematical models of fuzzy numbers and
their algebraic properties enable rigorous modelling of fuzzy
entities in fuzzy systems and efficient manipulation of fuzzy
variables in fuzzy analysis, fuzzy inference, and fuzzy
computing. The denotational mathematical structure of fuzzy
arithmetic not only explains the fuzzy nature of human
perceptions and language semantic representation, but also
enables cognitive machines and fuzzy systems to mimic
human fuzzy inference mechanisms in cognitive informatics,
cognitive computing, soft computing, cognitive linguistics,
measurement theory, and computational intelligence.
Sets are the fundamental mathematical means for
abstraction, which is the essence of mathematics. Therefore,
set theory becomes the foundation of almost all branches of
mathematics. Lotfi A. Zadeh extends the classic set theory to
fuzzy sets [Zadeh, 1965] and fuzzy logic [Zadeh, 1975b,
2008]. It is found that, although logical inferences may be
carried out on the basis of crisp sets and predicate logic, more
inference mechanisms and rules such as those of intuitive,
empirical, heuristic, perceptive, and semantic inferences, are
fuzzy and uncertain [Zadeh, 1971, 1975b; Wang, 2012b,
2014a, 2014b].
Keywords: Fuzzy number theory, fuzzy functions, fuzzy
arithmetic, fuzzy algebra, fuzzy mathematics, fuzzy systems,
fuzzy semantics, fuzzy inference, denotational mathematics,
cognitive computing, computational intelligence.
1. Introduction
Fuzzy numbers and their fuzzy operations [Zadeh, 1965,
1975a] are foundations of fuzzy number theory, fuzzy sets,
and fuzzy arithmetic for rigorously modelling fuzzy entities,
phenomena, semantics, measurement, knowledge, intelligence,
systems, and cognitive computational models [Zadeh, 2004;
Ross, 1995; Bender, 1996; Eugene, 1996; Pullman, 1997;
Wang, 2003, 2007, 2008a, 2009, 2010a, 2010b, 2011b, 2012a,
2013, 2014b; Wang & Berwick, 2012; Wang & Farelleo,
2012; BISC, 2014]. Fuzzy numbers are initially formulated by
Lotfi A. Zadeh [Zadeh, 1975a] as instances of fuzzy sets
[Zadeh, 1965]. It is recognized that a rigorous study on fuzzy
number theory may lead to the establishment of a framework
of fuzzy mathematics based on the fuzzy set theory [Zadeh,
1975b, 1999].
ISBN: 978-960-474-380-3
It is recognized that fuzzy numbers as a foundation of
fuzzy sets and fuzzy mathematics are yet to be rigorously
modeled and intensively studied [Zadeh, 1975a; Yager, 1986;
Wang & Klir, 1992; Ross, 1995; Wang, 2014b, 2014c]. Fuzzy
numbers were used to be treated as a fuzzy set with a convex
triangle membership function for arbitrary number of elements
with a uniformed incremental among values of the elements.
Arithmetical operations on fuzzy numbers, such as those of
addition, subtraction, multiplication, and division, were
initially formulated by Lotfi A. Zadeh known as the extension
principle of fuzzy sets [Zadeh, 1975a].
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Advances in Applied and Pure Mathematics
Definition 1. Zadeh’s extension principle for fuzzy set
composition is a fuzzy union of fuzzy intersections of two
and Y
, which derives a composite fuzzy set
fuzzy sets X
*Y
, i.e.:
X
it is almost impossible to manually or mentally manipulate
arbitrary fuzzy arithmetical operations as for those of real
numbers.
This problem leads to a number of proposed solutions and
algorithms for fuzzy arithmetic such as fuzzy interval analysis
and fuzzy vectors [Dubois & Prade, 1980; Ross, 1995; BISC,
2014]. Because of the intensive complexity involved in fuzzy
number representations and operations, various approximate
algorithms were proposed for fuzzy set extensions such as
those of the vertex method [Dong & Shah, 1987] and the DSW
algorithm [Dong et al., 1985]. However, they are not
mathematically intuitive enough and are over complicated for
arbitrary fuzzy variable operations.
X *Y ( x * y | x
X y Y )
||
X Y ||
(1)
{ X ( xk ) Y (
yk )}
k 1
||
X *Y ||
max{min k [ X ( xk ), Y (
yk )]}
k 1
where * denotes a fuzzy arithmetic operation,
*{ , , , } ,
or the number of elements
|| the cardinal size of X
and || X
Theoretically, both the mathematical models and physical
meanings of conventional fuzzy numbers and their fuzzy
arithmetical operations were not rigorously defined in
literature. The uncertainty error of fuzzy numbers, , is used to
be modeled by integers as e ³ 1 with a certain membership
value in conventional fuzzy set representation. However, it
would be otherwise in general engineering practice where a
e where e < 1 as a decimal
fuzzy number is always n
.
in X
Example 1. Given two fuzzy variables described by the
following convex triangle fuzzy sets:
= {(0, 0.2),(1,1.0),(2, 0.2)} and Y
= {(2, 0.3),(3,1.0),(4, 0.3)}
X
Applying the extension principle, a fuzzy addition between the
fuzzy variables
X Y can be derived as follows:
number.
In order to address the aforementioned challenging
problems, an improved theory of fuzzy numbers and fuzzy
arithmetic is presented for applications in fuzzy inference
systems, cognitive robots, cognitive informatics, cognitive
computing, and computational intelligence. In the remainder
of this paper, a mathematical model of fuzzy numbers in the
fuzzy variable discourse is introduced in Section 2, which
extends the domain of number theories from real numbers
to fuzzy numbers . An arithmetic structure on fuzzy
numbers is developed in Section 3 with the abstract model of
fuzzy arithmetic and formal operators of fuzzy addition,
subtraction, multiplication, and division. A set of algebraic
properties of fuzzy numbers and fuzzy arithmetic are explored
in Section 4. The denotational mathematical structure of fuzzy
numbers and fuzzy arithmetic not only explains the fuzzy
nature of human abstract perceptions and language semantic
representation, but also enables cognitive machines and fuzzy
systems to mimic the human fuzzy inference mechanisms in
cognitive
informatics,
cognitive
computing,
fuzzy
measurement theory, fuzzy engineering mathematics, and
computational intelligence.
+Y
= {(0, 0.2),(1,1.0),(2, 0.2)} + {(2, 0.3),(3,1.0),(4, 0.3)}
X
6
= {R (i, maxi [mini (mX (xi ), mY (yi ))])}
i =0
= {(0, max0[min0 (mX (x0 ), mY (y0 ))]) =
(0, max0[min0 (0.2, -)]) = (0, -)
(1, max [min (m (x ), m (y ))]) =
1
1
X
1
Y
1
(1, max1[min1(1.0, -)]) = (1, -)
(2, max2[min2 (mX (x2 ), mY (y2 ))]) =
(2, max2[min0+2 (0.2, 0.3)]) = (2, 0.2)
(3, max [min (m (x ), m (y ))]) =
3
3
X
3
Y
3
(3, max3 [min0+3 (0.2,1.0), min1+2 (1.0, 0.3), ]) = (3, 0.3)
(4, max4 [min4 (mX (x4 ), mY (y4 ))]) =
(4, max4 [min0+4 (0.2, 0.3), min1+3 (1.0,1.0),
min2+2 (0.2, 0.3)]) = (4,1.0)
(5, max5[min5 (m (x5 ), mY (y5 ))]) =
X
(5, max5 [min1+4 (1.0, 0.3), min2+3 (0.2,1.0)]) = (5, 0.3)
(6, max6[min6 (mX (x6 ), mY (y6 ))]) =
(6, max6[min2+4 (0.2, 0.3)]) = (6, 0.2)
}
= {(2, 0.2),(3, 0.3),(4,1.0),(5, 0.3),(6, 0.2)}
2. Mathematical Model of Fuzzy Numbers
It is recognized that the discourse of classic mathematics
is
built on real numbers n , arithmetic operators ,
U
It is noteworthy via Example 1 that the extension method
for fuzzy arithmetic is extremely complicated for such a
simple operation on two small fuzzy numbers. It would be
much more difficult when the fuzzy numbers are distributed in
a wide range of points in the fuzzy sets. Therefore, in general,
ISBN: 978-960-474-380-3
and functions f , i.e., U (, , f ) . As that of U , the
discourse of fuzzy mathematics can be formally described as
follows.
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Advances in Applied and Pure Mathematics
is a
Definition 2. The discourse of fuzzy mathematics U
triple:
( , ,
(2)
U
f)
where is a universal set of fuzzy numbers n ,
determined by a membership function
to the base value n and n , respectively, i.e.:
a set of
{(n \ , n (n \ )) | n n, n (n \ ) } (5)
{(n , n (n )),(n, n (n) 1.0),(n , n (n ))}
where n \ n represents the most accurate value of n
with n (n) 1.0 , and the upper and lower limits of the
2.1 The Domain of Fuzzy Numbers
The concepts of fuzzy numbers and fuzzy variables are a
natural extension of classic numbers from the real number
domain to that of fuzzy numbers .
fuzzy errors of n , respectively.
According to Definition 4, a fuzzy number is a fuzzy
quantification of a fuzzy variable on both its accuracy and
confidentiality. A fuzzy variable can be illustrated by a threepoint convex triangle fuzzy membership function in .
Definition 3. The domain of fuzzy numbers is a fuzzy
set of the range of values of all fuzzy variables N and
) in the
associate degrees of memberships ( N
hyperstructure of the fuzzy mathematical discourse U , i.e.:
Example 2. Given a 2, a 0.2, and (a \ a ) {0.5,1.0,0.7}
a
(3)
corresponding to
a \ , a fuzzy number a , a , can be
formally expressed according to Definition 4 as follows:
where denotes the dimension inclusion or that set i is an
elemental dimension of a hyperstructure S with n multiple
dimensions, i.e.:
a {(a \ a , a (a \ a )) | a 2 a 0.2 a (a \ a ) {0.5,1.0,0.7}}
n
S R i , i S
n {(n .v, n . ) | n .v n \ n . n (n \ )}
Details of each element of the discourse of fuzzy
mathematics will be formally described in the following
subsections.
{( , ) | [ , ] [0,1] }
with respect
fuzzy arithmetic operators, and f a set of fuzzy functions.
U
n (n \ )
{(2 0.2, a (2 0.2)), (2, a (2)), (2 0.2, a (2 0.2))}
(4)
i =1
{(1.8, 0.5), (2,1.0), (2.2, 0.7)}
n
where R i is the big-R notation [Wang, 2002] that denotes an
where the degrees of membership a ( a a ) are case specific.
iterative operation on or a recurring structure of i [Wang,
Example 3. A fuzzy number b , b , can be similarly
expressed according to Definition 4 as follows:
i =1
2002].
The domain of fuzzy numbers enlarges the discourse of
mathematics from the domains of binary, integer, natural, and
real numbers to a two-dimensional domain of fuzzy numbers
= ´ .
{(4.7,0.8),(5,1.0),(5.3,0.4)}
In the hyperstructural domain , a fuzzy number carries
2.2 Properties of Fuzzy Numbers
two dimensions of uncertainties for its base, lower, and upper
values in the accuracy dimension of quantities as well as the
degrees of memberships of its values in the confidentiality
dimension. This notion leads to the formal definition of fuzzy
numbers as follows.
Corollary 1. The domains of , , , and are subsets
or special cases of , i.e.:
ÌÌÌ
= ´ = {(, )} U
is a set of
, in U
Definition 4. A fuzzy number, n
(6)
Definition 5. The fuzziness of a fuzzy number, ( n ) , in
is the rate of changes of its degree of membership
U
pairs n {( n .v, n . )} where its values n .v is constrained by
a fuzzy errors and the associate uncertainty, n . is
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b {(b \ b , b (b \ b )) | b 5 b 0.3 b (b \ b ) {0.8,1.0,0.4}}
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Advances in Applied and Pure Mathematics
0 , is a pair
Corollary 4. The special fuzzy number 0,
of the real number 0 and the minimum degree of membership
at , i.e.:
over unit distance of its base value n in both the left and right
sides, respectively, i.e.:
( n )
d n ( n )
d
n ( n ) n ( n ) 1 n ( n ) (7)
l
( n )
r ( n ) n ( n ) n ( n ) 1 n ( n )
0 {(0, ( )), (0, 1.0), (0, ( ))
{(0, min( ( ), ( ))}
1
, is a pair
Corollary 5. The special fuzzy number 1,
of the real number 1 and the minimum degree of membership
at 1 , i.e.:
It is obvious that the shaper the triangle membership
function for a given fuzzy number, the higher the degree of its
fuzziness. When the triangle membership function of the fuzzy
number is symmetric, the left and right fuzziness are the same:
(8)
l ( n ) r ( n ), iff ( n ) ( n )
n
1 {(1, (1 )), (1, 1.0), (1, (1 ))
{(1, min( (1 ), (1 ))}
n
Example 4. The fuzziness the fuzzy number b given in
(12)
Both Corollaries 4 and 5, as well as Definition 4, indicate
the approach to real number fuzzification.
Example 3, b , can be determined according to Definition
5 as follows:
l 1 b (b )
(b)
(b )
1
(b )
r (b )
b
1
0.8
0.3 0.7
1 0.4 2.0
0.3
3. Arithmetic of Fuzzy Numbers
Classic fuzzy arithmetic is based on Zadeh’s extension
principle [Zadeh, 1975a] as described in Definition 1.
However, it is over complicated and does not work on
disintersected fuzzy numbers (sets) in general. On the basis of
the mathematical models of fuzzy numbers and variables as
created in Section 2, a set of fuzzy arithmetic operations can
be formally defined in this section. The typical fuzzy
arithmetic operators are addition, subtraction, multiplication,
and division [Zadeh, 1975a; Ross, 1995; BISC, 2014], which
form an algebraic structure of fuzzy arithmetic.
Corollary 2. The fuzziness of an unfuzzy real number
n is always zero, i.e.:
l (n | n ) r (n | n ) 0
(11)
3.1 The Abstract Algebraic Model of Fuzzy
Arithmetical Operations
(9)
Eq. 9 can be directly proven based on Definition 5. It
indicates that a real number in is a special case of fuzzy
numbers in where its fuzziness is zero.
The generic operator of fuzzy arithmetic on fuzzy
numbers and variables can be introduced as a unique abstract
operation in the following theorem.
as denoted by a fuzzy set
Corollary 3. A fuzzy number n
degrades to a single real number n known as the base value of
Theorem 1. The abstract arithmetic operator on fuzzy
is:
and y in U
numbers x
n when the fuzzy errors of n is zero, i.e.:
n n, iff 0
x · y {([x - ex ] · [y - ey ], min(mX (x - ex ), mY (y - ey ))),
(10)
(x · y, 1.0),
([x + ex ] · [y + ey ], min(mX (x + ex ), mY (y + ey )))
}
(13)
Proof. Corollary 4 can be directly proven according to
Definition 5 as follows:
n {(n \ , n (n \ )) | 0 n (n) 1.0}
, , , } represent the arithmetical operators of
where {
addition, subtraction, multiplications, and division on fuzzy
numbers, respectively. Correspondingly, {, , , }
denote the classic arithmetical operators on real numbers or
the base of the corresponding fuzzy numbers.
{(n, n (n))}
{(n,1.0)}
n
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Advances in Applied and Pure Mathematics
Proof. Theorem 1 can be proven according to Definition
4 on fuzzy numbers and their properties as follows:
3.2.1 Fuzzy Addition of Fuzzy Numbers
y {(x - e , m (x - e )),(x , m (x )),(x + e , m (x + e ))} ·
x ·
x
x
x
x
X
X
X
{(y - ey , mY (y - ey )),(y, mY (y )),(y + ey , mY (y + ey ))}
= {([x - ex ] · [y - ey ], min(mX (x - ex ), mY (y - ey ))),
([x · y ], min(mX (x ), mY (y ))),
derived from Theorem 1 as follows:
y {([x - e ] + [y - e ], min(m (x - e ), m (y - e ))),
x +
x
y
x
y
X
Y
([x + ex ] · [y + ey ], min(mX (x + ex ), mY (y + ey )))
}
= {([x - ex ] · [y - ey ], min(mX (x - ex ), mY (y - ey ))),
(x · y, 1.0),
(x + y, 1.0),
([x + ex ] + [y + ey ], min(mX (x + ex ), mY (y + ey )))
}
(17)
and b as
Example 5. Applying the fuzzy numbers a
b can be
given in Examples 2 and 3, the fuzzy addition a
([x + ex ] · [y + ey ], min(mX (x + ex ), mY (y + ey )))
}
(14)
carried out according to Definition 6 as follows:
Corollary 6. The hybrid arithmetic operator on a fuzzy
b = {([a - e ] + [b - e ], min(m (a - e ), m (b - e ))),
a +
a
b
a
b
A
B
and a real number k is:
number n
(a + b, 1.0),
([a + ea ] + [b + eb ], min(mA (a + ea ), mB (b + eb )))
}
= {(1.8 + 4.7, min(0.5, 0.8)),
n
{(k · (n - e ), m (n - e )),
k·
n
n
N
(k · n, 1.0),
(15)
(k · (n + en ), mN (n + en )))
}
Proof. Corollary 6 can be proven based on Theorem 1
and Definition 4 given ek = 0, and mk (k ) = 1 as follows:
(2.0 + 5.0, 1.0),
(2.2 + 5.3, min(0.7, 0.4))
}
= {(6.5, 0.5),(7.0, 1.0),(7.5, 0.4)}
n
{(k - e , m (k - e )),(k, m (k )),(k + e , m (k + e ))} ·
k·
k
k
k
k
K
K
K
{(n - en , mN (n - en )),(n, mN (n )),(n + en , mN (n + en ))}
Example 6. Redo Example 1 according to Definition 6 for
the complex solution obtained by applying the extension
y is
principle, the following simplified solution on x
derived:
= {(k · (n - en ), min[mK (k - ek ), mN (n - en )]),
(k · n, 1.0),
(k · (n + en ), min[mK (k + ek ), mN (n + en )])
}, ek = 0, and mK (k ek ) = 1
y = {(0, 0.2),(1,1.0),(2, 0.2)}+
{(2, 0.3),(3,1.0),(4, 0.3)}
x +
= {((1 - 1) + (3 - 1), min(0.2, 0.3)),
= {(k · (n - en ), mN (n - en )),
(k · n, 1.0),
(k · (n + en ), mN (n + en )))
}
(1 + 3, 1.0),
((1 + 1) + (3 + 1), min(0.2, 0.3))
}
= {(2, 0.2),(4, 1.0),(6, 0.2)}
(16)
The above solution to Example 1 as demonstrated in
Example 6 explains the efficiency of the formal fuzzy
arithmetical operations and the nature of fuzzy numbers.
Therefore, a hybrid fuzzy arithmetical operation on mixed
fuzzy and real numbers is a special case as described in
Theorem 1, which will be illustrated in Section 3.2 by specific
instances.
,
, on a
Definition 7. The hybrid fuzzy addition
and a real number y is a special
in U
fuzzy number x
3.2 The Arithmetic Operators on Fuzzy Numbers
case of Definition 6 as follows:
On the basis of the generic abstract model of arithmetical
operators on fuzzy numbers, the instances of fuzzy addition,
y {((x - e ) + y, m (x - e )),
x +
x
x
X
(x + y, 1.0),
, , , } ,
subtraction, multiplication, and division, i.e., {
will be formally derived based on Theorem 1 in this
subsection with specific elaborations.
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,
Definition 6. The arithmetic operator of fuzzy addition
can be
, on two fuzzy numbers x and y in U
((x + ex ) + y, mX (x + ex ))
}
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Advances in Applied and Pure Mathematics
3.2.3 Fuzzy Multiplication of Fuzzy Numbers
3.2.2 Fuzzy Subtraction of Fuzzy Numbers
Definition 8. The arithmetic operator of fuzzy subtraction
, on two fuzzy numbers x and y in U can
,
Definition
y {([x - e ] - [y - e ], min(m (x - e ), m (y - e ))),
x
x
y
x
y
X
Y
(x - y, 1.0),
([x + ex ] - [y + ey ], min(mX (x + ex ), mY (y + ey )))
operator
of
fuzzy
y {([x - e ]´ [y - e ], min(m (x - e ), m (y - e ))),
x ´
x
y
x
y
X
Y
(x ´ y, 1.0),
([x + ex ]´ [y + ey ], min(mX (x + ex ), mY (y + ey )))
}
(21)
}
(19)
and b as
Example 7. Applying the fuzzy numbers a
a can
given in Examples 2 and 3, the fuzzy subtraction b
be carried out according to Definition 8 as follows:
and b as
Example 8. Applying the fuzzy numbers a
b can be
given in Examples 2 and 3, the fuzzy addition a
carried out according to Definition 10 as follows:
a = {([b - e ] - [a - e ], min(m (a - e ), m (b - e ))),
b
b
a
a
b
A
B
(b - a, 1.0),
b = {([a - e ]´[b - e ], min(m (a - e ), m (b - e ))),
a ´
a
b
a
b
A
B
([b + eb ] - [a + ea ], min(mA (a + ea ), mB (b + eb )))
}
(a ´b, 1.0),
([a + ea ]´[b + eb ], min(mA (a + ea ), mB (b + eb )))
= {(4.7 - 1.8, min(0.5, 0.8)),
}
= {(1.8 ´ 4.7, min(0.5, 0.8)),
(5.0 - 2.0, 1.0),
(5.3 - 2.2, min(0.7, 0.4))
(2.0 ´ 5.0, 1.0),
}
(2.2 ´ 5.3, min(0.7, 0.4))
}
= {(2.9, 0.5),(3.0, 1.0),(3.1, 0.4)}
b yields different result as follows:
However, a
= {(8.5, 0.5),(10.0, 1.0),(11.7, 0.4)}
b = {([a - e ] - [b - e ], min(m (a - e ), m (b - e ))),
a
a
b
a
b
A
B
,
,
Definition 11. The hybrid fuzzy multiplication
and a real number y is a
in U
on a fuzzy number x
(a - b, 1.0),
([a + ea ] - [b + eb ], min(mA (a + ea ), mB (b + eb )))
}
= {(1.8 - 4.7, min(0.5, 0.8)),
(2.0 - 5.0, 1.0),
(2.2 - 5.3, min(0.7, 0.4))
}
= {(-2.9, 0.5),(-3.0, 1.0),(-3.1, 0.4)}
The
comparative
results
indicate
that
b a b ( a ) (a b ) .
special case of Definition 10 as follows:
y {((x - e ) ´ y, m (x - e )),
x ´
x
x
X
(x + y, 1.0),
((x + ex ) ´ y, mX (x + ex ))
(22)
}
3.2.4 Fuzzy Division of Fuzzy Numbers
Definition 12. The arithmetic operator of fuzzy division
,
, on
Definition 9. The hybrid fuzzy subtraction
and a real number y is a
in U
a fuzzy number x
, on two fuzzy numbers x and y in U can
,
be derived from Theorem 1 as follows:
special case of Definition 8 as follows:
ISBN: 978-960-474-380-3
arithmetic
can be derived from Theorem 1 as follows:
U
be derived from Theorem 1 as follows:
y {((x - e ) - y, m (x - e )),
x
x
x
X
(x - y, 1.0),
((x + ex ) - y, mX (x + ex ))
}
The
10.
,
, on two fuzzy numbers x and y in
multiplication
y {([x - e ] ¸ [y - e ], min(m (x - e ), m (y - e ))),
x ¸
x
y
x
y
X
Y
(x ¸ y, 1.0),
([x + ex ] ¸ [y + ey ], min(mX (x + ex ), mY (y + ey )))
(20)
}
(23)
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Advances in Applied and Pure Mathematics
and b as
Example 9. Applying the fuzzy numbers a
y {((x - e ) ¸ y, m (x - e )),
x ¸
x
x
X
b can be
given in Examples 2 and 3, the fuzzy division a
carried out according to Definition 12 as follows:
(x ¸ y, 1.0),
((x + ex ) ¸ y, mX (x + ex ))
}
b = {([a - e ] ¸ [b - e ], min(m (a - e ), m (b - e ))),
a ¸
a
b
a
b
A
B
Examples 5 through 10 provided in this subsection
demonstrates that the formal fuzzy arithmetical operations can
be much efficiently manipulated by manual calculations as for
those of real numbers.
(a ¸ b, 1.0),
([a + ea ] ¸ [b + eb ], min(mA (a + ea ), mB (b + eb )))
}
= {(1.8 ¸ 4.7, min(0.5, 0.8)),
(2.0 ¸ 5.0, 1.0),
4. Properties of Fuzzy Algebra on Fuzzy
Numbers and Variables
(2.2 ¸ 5.3, min(0.7, 0.4))
}
= {(0.38, 0.5),(0.4, 1.0),(0.42, 0.4)}
The basic properties of fuzzy algebra on fuzzy numbers in
can be expressed by the commutative, associative,
U
a can be
Example 10. Similarly, the fuzzy division b
carried out according to Definition 12 as follows:
distributive, inversive, idempotent, and identity laws as
summarized in Table 1. In Table 1, the first category of
properties is applied to fuzzy arithmetic on all fuzzy numbers;
while the second category is for hybrid fuzzy operations
between mixed fuzzy and real numbers.
a = {((5 - 0.3) ¸ (2 - 0.2)), min(0.5, 0.8)),
b ¸
(5.0 ¸ 2.0, 1.0),
(((5 + 0.3) ¸ (2 + 0.2)), min(0.7, 0.4))
}
It is noteworthy that and do not obey the
commutative and associative laws of fuzzy arithmetic. With
' share the
regards to the identity law of fuzzy arithmetic, a
= {(2.6, 0.5),(2.5,1.0),(2.4, 0.8)}
The
comparative
1
a b a b
(b a )1 .
results
indicate
. However, its membership value is
same base value of a
constrained by the minimum, i.e.:
that
a' {(a, min( a (a a ), 0 (0 0 )))}
,
, on a
Definition 13. The hybrid fuzzy division
and a real number y is a special
in U
fuzzy number x
{(a, min( a (a a ), 1 (0 1 )))}
case of Definition 12 as follows:
Table 1. Properties of Fuzzy Arithmetic
No.
Property
1
Commutative
a b b a
a b b a
a b b a
a b b a
2
Associative
a (b c) (a b ) c
a (b c) (a b ) c
a (b c ) (a b ) c
a (b c ) (a b ) c
3
Distributive
a (b c) a b a c
(b c) / a b / a c / a
a (b c) a b a c
(b c) / a b / a c / a
4
Inversive
b a b ( a ) (a b )
1
b a b a
(a b )1
b a b ( a) (a b )
b a b a1 (a b )1
5
Idempotent
a a 0 {(0, min(A (a a ), A (a a )))}
a a 0 {(0, min(A (a a ), A (a a )))}
a a 1 {(1, min(A (a a ), A (a a )))}
a a 1 {(1, min(A (a a ), A (a a )))}
a 0 a', a 1 a'
a 1 a'
a 0 0',
a 0 a , a 1 a
a 1 a
a 0 0,
6
Identity
ISBN: 978-960-474-380-3
(24)
Fuzzy algebra
Hybrid fuzzy algebra
88
(25)
Advances in Applied and Pure Mathematics
So do
References
0' {(0, min( a ( a a ), 0 (0 0 )))}
(26)
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A theory of fuzzy numbers, fuzzy arithmetic, and their
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The denotational mathematical structure of fuzzy
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Acknowledgement
A number of notions in this work have been inspired by
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This work is supported in part by a discovery fund granted
by the Natural Sciences and Engineering Research Council of
Canada (NSERC). The author would like to thank the
anonymous reviewers for their valuable comments on the
previous version of this paper.
ISBN: 978-960-474-380-3
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