Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 845090, 18 pages
doi:10.1155/2012/845090
Research Article
Multivariate Extension Principle and Algebraic
Operations of Intuitionistic Fuzzy Sets
Yonghong Shen1 and Wei Chen2
1
2
School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China
School of Information, Capital University of Economics and Business, Beijing 100070, China
Correspondence should be addressed to Yonghong Shen, shenyonghong2008@hotmail.com
Received 31 July 2012; Revised 16 September 2012; Accepted 19 September 2012
Academic Editor: Chong Lin
Copyright q 2012 Y. Shen and W. Chen. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
This paper mainly focuses on multivariate extension of the extension principle of IFSs. Based
on the Cartesian product over IFSs, the multivariate extension principle of IFSs is established.
Furthermore, three kinds of representation of this principle are provided. Finally, a general
framework of the algebraic operation between IFSs is given by using the multivariate extension
principle.
1. Introduction
The concept of intuitionistic fuzzy sets was first proposed by Atanassov and Stoeva in 1983
1. However, this concept had not been widely concerned by many scholars, because it was
only presented in the symposium proceedings with regard to interval and fuzzy mathematics
in Poland. Until 1986, the notion of intuitionistic fuzzy sets was formally introduced by
Atanassov 2. Immediately, some new operators on intuitionistic fuzzy sets are defined and
the corresponding properties are studied in 1989 3.
The intuitionistic fuzzy sets can be viewed as an extension for fuzzy sets, which
is more objective and comprehensive to describe the uncertainty of the problem. In 1986,
Atanassov 2 established several different ways to change an intuitionistic fuzzy set into
a fuzzy set and defined an operator called Atanassov’s operator. Furthermore, the study
of the properties on this operator is carried out in 2, 4. Later, Burillo and Bustince 5
presented the Atanassov’s point operator and studied the construction of intuitionistic fuzzy
sets by using this type of operator. Meantime, they pointed out that it is possible to recover
a fuzzy set from an intuitionistic fuzzy set constructed by means of different operators.
Similarly, the intuitionistic fuzzy relations can also be seen as an extension for fuzzy
2
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relations, which were proposed by Bustince and Burillo in 6. Afterwards, Bustince studied
the construction of intuitionistic fuzzy relations with predetermined properties, which can
allow us to build reflexive, symmetric, antisymmetric, perfect antisymmetric, and transitive
intuitionistic fuzzy relations from fuzzy relations with the same properties on the basis of
the Atanassov’ operator in 7. As mentioned above, one can see that it is an important and
interesting research direction to extend some conclusions of fuzzy sets to intuitionistic fuzzy
environment. For this issue, in recent years, further studies have been completed by different
authors 6, 8–16.
In the theory of fuzzy sets, representation theorem, decomposition theorem, and
extension principle are regarded as three important basic theorems, which provide an
important theoretical basic for dealing with the fuzzy problems by the methods of classical
mathematics. Especially, the multivariate extension principle, which can be viewed as a
generalization of the extension principle of fuzzy sets, will provide a theoretical basic for
the operations between fuzzy sets. In addition, it should be pointed out that the Cartesian
product over fuzzy sets is an important tool to establish this principle. In 2007, Atanassova
17 constructed the extension principle of intuitionistic fuzzy sets. Meantime, several
types of Cartesian products over intuitionistic fuzzy sets were also introduced in 18. In
2008, Andonov 19 also introduced a Cartesian product over intuitionistic fuzzy sets and
explored some of properties. Based on these existing results, in this paper, the multivariate
extension principle of intuitionistic fuzzy sets will be considered. Meanwhile, some important
operations between intuitionistic fuzzy sets can be obtained by using this principle.
The rest of the paper is organized into five parts. In Section 2, some related concepts
and important conclusions on intuitionistic fuzzy sets are recalled. In Section 3, the Cartesian
product over intuitionistic fuzzy sets is reviewed and some related properties are discussed.
These results will establish a basis for the analysis and proof of the multivariate extension
principle of intuitionistic fuzzy sets. Section 4 establishes a multivariate extension principle
of intuitionistic fuzzy sets and provides three types of forms of this principle. In Section 5,
a general framework of the algebraic operation between intuitionistic fuzzy sets is proposed
by using the multivariate extension principle. Finally, a summarized conclusion is given in
Section 6.
2. Preliminaries
For completeness and clarity, some basic notions and necessary conclusions on intuitionistic
fuzzy sets are reviewed in this section.
2.1. Intuitionistic Fuzzy Sets
Let X be the universe of discourse, an intuitionistic fuzzy set on X is an expression E given
by
2.1
E x, μE x, νE x : x ∈ X
with
μE : X −→ 0, 1,
such that 0 ≤ μE x νE x ≤ 1 for all x ∈ X.
νE : X −→ 0, 1
2.2
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Generally, μE x and νE x are called the degree of membership and the degree of
nonmembership of the element x in the set E, respectively. The complementary of E is denoted
by Ec {x, νE x, μE x : x ∈ X}. The symbol IFSsX denotes the set of all intuitionistic
fuzzy sets on X. Especially, if νE x 1 − μE x, the set E reduces to a fuzzy set. Meantime,
FSsX denotes the set of all fuzzy sets on X. In addition, intuitionistic fuzzy sets are
abbreviated as IFSs.
2.2. Cut Sets of IFSs and Its Properties
In this part, some concepts and conclusions associated with the cut sets of IFSs are summarized below.
Definition 2.1 see 18. Let E ∈ IFSsX, the parameters α, β satisfy the condition α β ≤ 1
for all α, β ∈ 0, 1, the following four sets:
Eα,β x ∈ X : μE x ≥ α, νE x ≤ β ,
Eα,β x ∈ X : μE x > α, νE x < β ,
· ·
2.3
Eα,β x ∈ X : μE x > α, νE x ≤ β ,
·
Eα,β x ∈ X : μE x ≥ α, νE x < β ,
·
are called the α, β-cut set, strong α, β-cut set, α, β-cut set, and α, β-cut set, respectively.
·
·
For convenience, the symbol I 2 is given to denote the set I 2 {α, β : α β ≤ 1, α, β ∈
0, 1}. For all α1 , β1 , α2 , β2 ∈ I 2 , the relations between them are defined as
a α1 , β1 α2 , β2 ⇔ α1 α2 and β1 β2 ;
b α1 , β1 ≤ α2 , β2 ⇔ α1 ≤ α2 and β1 ≥ β2 ;
c α1 , β1 < α2 , β2 ⇔ α1 , β1 ≤ α2 , β2 and α1 , β1 /
α2 , β2 .
Additionally, for the need of the following narrative, a particular operation between
the set I 2 and the IFSsX is quoted.
Definition 2.2 see 18. Let E ∈ IFSsX, α, β ∈ I 2 , the set α, βE ∈ IFSsX is defined as
follows
def α, β E x, α ∧ μE x, β ∨ νE x : x ∈ X .
Next, some properties of the cut set of IFSs are summarized.
Lemma 2.3 see 18. Let E ∈ IFSsX, α, β ∈ I 2 , then
i Eα,β ⊆ Eα,β ⊆ Eα,β ;
· ·
·
ii Eα,β ⊆ Eα,β ⊆ Eα,β ;
· ·
·
iii E0,1 X, E1,0 E1,0 E1,0 ∅.
·
·
· ·
2.4
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Lemma 2.4 see 18. Let E ∈ IFSsX, α1 , β1 , α2 , β2 ∈ I 2 , and α1 , β1 ≤ α2 , β2 , then
i Eα2 ,β2 ⊆ Eα1 ,β1 ;
ii Eα2 ,β2 ⊂ Eα1 ,β1 ;
·
·
·
·
iii Eα2 ,β2 ⊆ Eα1 ,β1 ;
·
·
iv Eα2 ,β2 ⊆ Eα1 ,β1 .
·
·
Based on the cut sets of IFSs and their properties, the basic theorems of IFSs are
developed by Atanassov 18 and Liu 14. In the following, some related theorems are
recalled for the needs of the proofs in Sections 3 and 4.
Lemma 2.5 decomposition theorem 18. Let E ∈ IFSsX, then
E
α,β∈I 2
α, β Eα,β .
2.5
Notice that the set E can also be decomposed by the rest of cut sets of E. This case is
similar with Lemma 2.5.
The concept of the binary nested set is introduced to give the representation theorem
of IFSs. The symbol P X denotes the power set of X.
Definition 2.6. Let f : I 2 → P X be a mapping, that is, α, β → Hα, β, for all
α1 , β1 , α2 , β2 ∈ I 2 , if α1 , β1 < α2 , β2 , there always has Hα2 , β2 ⊂ Hα1 , β1 , then the
set H is called the binary nested set on X.
Lemma 2.7 representation theorem 14. Let f be a mapping from BNX to IFSsX, that is,
f : BNX → IFSsX, and
H −→ fH α, β H α, β ,
α,β∈I 2
2.6
then
i f is a surjection from BNX, ∪, ∩, c to IFSsX, ∪, ∩, c ;
ii fHα,β ⊆ Hα, β ⊆ fHα,β ,
· ·
where the notation BNX denotes the set of all the binary nested sets on X.
Lemma 2.7 shows that there exists a unique fH such that fH
for all H ∈ BNX.
α,β∈I 2 α, βHα, β
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5
Lemma 2.8 extension principle 14. Suppose f : X → Y is a mapping from the ordinary
set X to the ordinary set Y , that is, x → fx, then the mapping f can induce into two mappings
f : IFSsX → IFSsY and f −1 : IFSsY → IFSsX
f:IFSsX −→ IFSsY ,
A −→ fA α, β f Aα,β ,
α,β∈I 2
B −→ f −1 B α, β f −1 Bα,β .
α,β∈I 2
f −1 :IFSsY −→ IFSsX,
2.7
The membership and nonmembership functions of f, f −1 are defined, respectively, as follows
μA x,
μfA y νA x,
νfA y fxy
μ
f −1 B
x μB
fxy
fx ,
ν
f −1 B
2.8
x νB fx ,
where A {x, μA x, νA x : x ∈ X} ∈ IFSsX, fA {y fx, μfA y, νfA y : x ∈
X, fx ⊆ Y } ∈ IFSsY . B and f −1 B can be given in a similar manner.
3. Cartesian Product over IFSs
The concept of Cartesian product over IFSs is introduced by Atanassov 18. Here we will
review this concept in detail and extend it to n arguments. It should be noted that the main
purpose of this section is to make a preparation for developing the multivariate extension
principle of IFSs.
As we all know, the ordinary Cartesian product is defined as
def
A1 × A2 × · · · × An {x1 , x2 , · · · , xn : xi ∈ Ai , i 1, 2, . . . , n},
3.1
the characteristic function is
χA1 ×A2 ×···×An x1 , x2 , . . . , xn n
Ak xk .
k1
3.2
The Cartesian product over FSs is obtained by extending the ordinary sets to fuzzy
sets.
Let Ak ∈ FSsXk k 1, 2, . . . , n. Based on the decomposition and representation
theorems of FSs, the Cartesian product can be obtained as follows:
A1 × A2 × · · · × An λ∈0,1
1
2
n
λ Aλ × Aλ × · · · × Aλ .
3.3
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In fact, it has been proved that the membership function of Cartesian product over FSs
is
μA1 ×A2 ×···×An x1 , x2 , . . . , xn n
k1
μAk xk .
3.4
Similarly, the Cartesian product over IFSs can also be obtained.
Definition 3.1. Let Ek ∈ IFSsXk k 1, 2, . . . , n, α, β ∈ I 2 , the following expression
def
E1 × E2 × · · · × En 1
2
n
α, β E α,β × E α,β × · · · × E α,β
α,β∈I 2
3.5
be called the Cartesian product over IFSs.
According to Lemma 2.4, we have
k
k
α1 , β1 < α2 , β2 ⇒ E α ,β ⊇ E α ,β
1 1
2 2
1
k 1, 2, . . . , n
2
n
1
2
n
⇒ E α ,β × E α ,β × · · · × E α ,β ⊇ E α ,β × E α ,β × · · · × E α ,β .
1 1
1 1
1 1
2 2
2 2
2 2
3.6
Hence, it is easy to see that the set
1
2
n
E α,β × E α,β × · · · × E α,β : α, β ∈ I 2
3.7
is a binary nested set on X X1 × X2 × · · · × Xn .
According to Lemma 2.7, we know that the above binary nested set can uniquely
identify an intuitionistic fuzzy set contained in IFSsX1 × X2 × · · · × Xn . Therefore, we have
1
2
n
∈ IFSsX1 × X2 × · · · × Xn ,
α, β E α,β × E α,β × · · · × E α,β
2
α,β
∈I
3.8
and it satisfies
E1 × E2 × · · · × En
1
α,β
· ·
2
n
⊆ E α,β × E α,β × . . . × E α,β
⊆ E1 × E2 × · · · × En
α,β
3.9
.
For the membership function and nonmembership function of the Cartesian product
over IFSs, we have the following theorem.
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Theorem 3.2. Let Ek ∈ IFSsXk k 1, 2, . . . , n, then
μE1 ×E2 ×···×En x1 , x2 , . . . , xn νE1 ×E2 ×···×En x1 , x2 , . . . , xn n
k1
n
k1
μEk xk ,
3.10
νEk xk .
3.11
Proof. First of all, we prove the membership function of Cartesian product over IFSs. For all
x1 , x2 , . . . , xn ∈ X X1 × X2 × · · · × Xn , according to Definitions 2.2 and 3.1, we can obtain
μE1 ×E2 ×···×En x1 , x2 , . . . , xn α ∧ μE1 ×E2 ×···×En x1 , x2 , . . . , xn α,β
α,β
α,β
α,β∈I 2
n
.
α∧
μEk xk α,β
k1
α,β∈I 2
3.12
Now, we will prove that
α∧
α,β∈I 2
n k1
⎞
⎛
n
⎝
μEk xk α ∧ μEk xk ⎠.
α,β
α,β
2
k1
α,β∈I
3.13
Since
α∧
α,β∈I 2
α∧
α,β∈I 2
α,β∈I 2
α ∧ μEk xk ,
α,β
⎞
⎛
n n
≤ ⎝
μEk xk α ∧ μEk xk ⎠.
α,β
α,β
2
k1
k1
α,β∈I
α,β∈I 2 α ∧ a constant λ such that
α,β∈I 2
Assume that
α,β
k1
⇒
n ≤
μEk xk n
k xk k1 μEα,β
<
n
k1 α,β∈I 2 α ∧ μEk
α,β
3.14
xk , then there exists
⎞
⎛
n n
<λ< ⎝
μEk xk α ∧ μEk xk ⎠.
α,β
α,β
2
k1
k1
α,β∈I
α∧
3.15
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Since
n
k xk k1 μEλ,β
λ>
k xk k1 μEλ,β
α∧
α,β∈I 2
In addition, if
n
{0, 1}, if
n k1
n
k xk k1 μEλ,β
1, then
≥λ∧
μEk xk α,β
n λ.
μEk xk k1
λ,β
3.16
0, then there exists a constant k0 1 ≤ k0 ≤ n such that
μEk0 xk0 0. By Definition 2.1, we know that μEk0 xk0 < λ or νEk0 xk0 > β, and then for
λ,β
all α > λ, we have μEk0 xk0 0.
α,β
Hence,
λ<
n
k1
⎛
⎝
α,β
∈I 2
⎞
α ∧ μEk xk ⎠ ≤
α,β
α,β
∈I 2
α ∧ μEk0 xk0 α,β
≤ λ.
3.17
Obviously, the expressions 3.16 and 3.17 are contradictory. Consequently, the previous
hypothesis does not hold and the expression 3.13 holds.
By the expression 3.13, we can obtain
μE1 ×E2 ×···×En x1 , x2 , . . . , xn n
α∧
μEk xk α,β
k1
α,β∈I 2
⎞
⎛
n
⎝
α ∧ μEk xk ⎠
α,β
k1
α,β∈I 2
n
3.18
μEk xk .
k1
Now we start to prove the expression 3.11. Analogously, we have
νE1 ×E2 ×···×En x1 , x2 , . . . , xn β ∨ νE1 ×E2 ×···×En x1 , x2 , . . . , xn α,β
α,β
α,β
α,β∈I 2
n
.
β∨
νEk xk α,β
k1
α,β∈I 2
3.19
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9
Next, we will prove that
⎞
⎛
n n
νEk xk β ∨ νEk xk ⎠.
⎝
α,β
α,β
2
k1
k1
α,β∈I
β∨
α,β
∈I 2
3.20
Since
β∨
α,β∈I 2
β∨
α,β∈I 2
n ≥
νEk xk α,β
k1
⇒
n ≥
νEk xk α,β
k1
α,β
α,β∈I 2
β ∨ νEk xk ⎛
n
⎝
α,β∈I 2
k1
⎞
3.21
β ∨ νEk xk ⎠.
α,β
Similarly, we assume that
⎞
⎛
n n
> ⎝
νEk xk β ∨ νEk xk ⎠,
α,β
α,β
2
k1
k1
α,β
∈I
β∨
α,β∈I 2
3.22
then there exists a constant γ such that
β∨
α,β
n
k xk k1 νEα,γ
γ<
β∨
α,β∈I 2
On the other hand, if
k xk k1 νEα,γ
n k1
n
n
⎛
⎝
k1
n
{0, 1}, if
>γ >
νEk xk α,β
k1
∈I 2
Since
n α,β
∈I 2
k xk k1 νEα,γ
α,β
α,β
3.23
0, then
νEk xk β ∨ νEk
⎞
xk ⎠.
≤γ∨
n k1
νEk xk α,β
γ.
3.24
1, then there exists a constant k1 1 ≤ k1 ≤ n such
that νEk1 xk1 1, that is, μEk1 xk1 > α and νEk1 xk1 < γ. Therefore, for all β > γ, we have
α,γ
νEk1 xk1 1.
α,β
So we can obtain
⎞
γ> ⎝
β ∨ νEk xk ⎠ ≥
β ∨ νEk1 xk1 ≥ γ.
α,β
α,β
k1
α,β∈I 2
α,β∈I 2
n
⎛
3.25
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Notice that the expressions 3.24 and 3.25 are also contradictory. So we can see that the
expression 3.20 holds, and then we can obtain
νE1 ×E2 ×···×En x1 , x2 , . . . , xn n
β∨
νEk xk α,β
k1
α,β∈I 2
⎞
⎛
n
⎝
β ∨ νEk xk ⎠
α,β
k1
α,β∈I 2
n
3.26
νEk xk .
k1
Remark 3.3. According to the poof, one can see that the notations α,β and nk1 , α,β
n
n
n
and k1 are commutative, respectively. The reason is that k1 μEk xk and k1 νEk xk α,β
α,β
are equal to {0, 1}. Notice that these relations are not necessarily satisfied under general
conditions.
Based on Theorem 3.2, we can obtain some properties of cut sets of the Cartesian
product over IFSs.
Theorem 3.4. Let Ek ∈ IFSsXk k 1, 2, . . . , n, α, β ∈ I 2 , then
1
2
n
1
2
n
i E1 × E2 × · · · × En α,β Eα,β × Eα,β × · · · × Eα,β ;
ii E1 × E2 × · · · × En α,β Eα,β × Eα,β × · · · × Eα,β ;
· ·
iii E1 × E2 × · · · × En α,β ·
· ·
1
Eα,β
·
1
· ·
×
2
Eα,β
·
· ·
× ··· ×
2
n
Eα,β ;
·
n
iv E1 × E2 × · · · × En α,β Eα,β × Eα,β × · · · × Eα,β .
·
·
·
·
Proof. We only prove the first equality, the remaining equalities can be proved in a similar
way.
According to Definition 2.1 and Theorem 3.2, for all
x1 , x2 , . . . , xn ∈ E1 × E2 × · · · × En
⇐⇒
n
μEk xk ≥ α,
k1
⇐⇒ μEk xk ≥ α,
k
⇐⇒ xk ∈ E α,β
n
α,β
νEk xk ≤ β
k1
νEk xk ≤ β
k 1, 2, . . . , n
k 1, 2, . . . , n
1
2
n
x1 , x2 , . . . , xn ∈ E α,β × E α,β × · · · × E α,β .
3.27
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11
Hence, the equality holds, namely,
E1 × E2 × · · · × En
1
α,β
2
n
E α,β × E α,β × · · · × E α,β .
3.28
4. Multivariate Extension Principle of IFSs
Based on the Cartesian product over IFSs, we will discuss the multivariate extension principle
of IFSs.
Let Ek ∈ IFSsXk k 1, 2, . . . , n, then the operation of Cartesian product may be
viewed as a mapping, which is defined as follows
× : IFSsX1 × IFSsX2 × · · · × IFSsXn −→ IFSsX1 × X2 × · · · × Xn ,
E1 , E2 , . . . , En −→ E1 × E2 × · · · × En .
4.1
Now we will define the multivariate extension principle. It is assumed that f is a
mapping from X to Y , namely,
f : X X1 × X2 × · · · × Xn −→ Y Y1 × Y2 × · · · × Ym ,
x1 , x2 , . . . , xn −→ fx1 , x2 , . . . , xn y y1 , y2 , . . . , ym .
4.2
By the extension principle of IFSs Lemma 2.8, the mapping f can be induced to the
following two mappings.
f:IFSsX −→ IFSsY ,
f −1 :IFSsY −→ IFSsX,
A −→ fA α, β f Aα,β ,
α,β∈I 2
B −→ f −1 B α, β f −1 Bα,β .
α,β∈I 2
4.3
Next, we use the two mappings f, f −1 to make compound operations with the
following two Cartesian products of IFSs, respectively.
×1 :IFSsX1 × IFSsX2 × · · · × IFSsXn −→ IFSsX,
A1 , A2 , . . . , An −→ A1 × A2 × · · · × An ,
×2 :IFSsY1 × IFSsY2 × · · · × IFSsYm −→ IFSsY ,
B1 , B2 , . . . , Bm −→ B1 × B2 × · · · × Bm .
4.4
Based on the above compound operations, we will get the following definition about
multivariate extension principle of IFSs.
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Journal of Applied Mathematics
Definition 4.1. Let Ak ∈ IFSsXk k 1, 2, . . . , n, Bl ∈ IFSsYl l 1, 2, . . . , m, and f be
a mapping from X to Y , that is,
f:X X1 × X2 × · · · × Xn −→ Y Y1 × Y2 × · · · × Ym ,
x1 , x2 , . . . , xn −→ fx1 , x2 , . . . , xn y y1 , y2 , . . . , ym .
4.5
The two induced mappings of f can be defined as follows:
f:IFSsX1 × IFSsX2 × · · · × IFSsXn −→ IFSsY ,
def A1 , A2 , . . . , An −→ f A1 , A2 , . . . , An f A1 × A2 × · · · × An ,
f −1 :IFSsY1 × IFSsY2 × · · · × IFSsYm −→ IFSsX,
4.6
B1 , B2 , . . . , Bm −
→ f −1 B1 , B2 , . . . , Bm
f −1 B1 × B2 × · · · × Bm .
def
It is obvious that the above result is an extension of Lemma 2.8 of the Cartesian product
over IFSs. Hence, it is said to be the multivariate extension principle of IFSs.
According to Lemma 2.8 and Theorem 3.2, we can obtain the membership function
and nonmembership function of f and f −1 . The corresponding conclusions are given by the
following theorem.
Theorem 4.2. Let f and f −1 be two induced mappings, which are given by Definition 4.1, then the
membership function and nonmembership function of f and f −1 are given by
μfA1 ,A2 ,...,An y νfA1 ,A2 ,...,An y fx1 ,x2 ,...,xn y
fx1 ,x2 ,...,xn y
μAk xk ,
k1
n
n
νAk xk ,
k1
m
μf −1 B1 ,B2 ,...,Bm x μBl yl ,
l1
νf −1 B1 ,B2 ,...,Bm x where y1 , y2 , . . . , ym fx.
m
νBl yl ,
l1
4.7
Journal of Applied Mathematics
13
Below we will discuss some other forms of the multivariate extension principle of IFSs.
Theorem 4.3 multivariate extension principle I.
f A1 , A2 , . . . , An f
−1
1
2
B ,B ,...,B
m
1
2
n
α, β f A α,β , A α,β , . . . , A α,β ,
α,β∈I 2
1
2
m
α, β f −1 B α,β , B α,β , . . . , B α,β .
α,β∈I 2
4.8
Proof. By Lemma 2.8 and Definition 4.1, we have
f A1 , A2 , . . . , An f A1 × A2 × · · · × An
α, β f A1 × A2 × · · · × An
α,β∈I 2
α,β∈I 2
α,β
4.9
1
2
n
α, β f A α,β × A α,β × · · · × A α,β .
Since
1
2
n
f A α,β , A α,β , . . . , A α,β
k
y : ∃xk ∈ A α,β k 1, 2, . . . , n, fx1 , x2 , . . . , xn y
1
2
n
y : ∃x1 , x2 , . . . , xn ∈ A α,β × A α,β × · · · × A α,β , fx1 , x2 , . . . , xn y
4.10
1
2
n
f A α,β × A α,β × · · · × A α,β ,
we know that
1
2
n
f A ,A ,...,A
1
2
n
α, β f A α,β , A α,β , . . . , A α,β .
α,β∈I 2
Similarly, we can prove the second equality.
4.11
14
Journal of Applied Mathematics
Theorem 4.4 multivariate extension principle II.
1
2
n
f A ,A ,...,A
⎞
⎛
1
2
n
α, β f ⎝A , A , . . . , A ⎠
α,β
α,β
α,β
2
· ·
· ·
· ·
α,β∈I
⎞
⎛
1
2
n
α, β f ⎝A , A , . . . , A ⎠
α
α
α,β
,β
,β
·
·
·
α,β∈I 2
⎞
⎛
1
2
n
α, β f ⎝A , A , . . . , A ⎠,
α,β
α,β
α,β
2
α,β
∈I
·
·
·
⎛
f −1 B1 , B2 , . . . , Bm ⎞
4.12
1
2
m
α, β f −1 ⎝B , B , . . . , B ⎠
α,β
α,β
α,β
2
· ·
· ·
· ·
α,β∈I
⎛
⎞
−1
1
2
m
α, β f ⎝B , B , . . . , B ⎠
α,β
α,β
α,β
2
α,β
∈I
·
·
·
⎛
⎞
−1
1
2
m
α, β f ⎝B , B , . . . , B ⎠.
α,β
α,β
α,β
2
α,β∈I
·
·
·
Proof. The proof method is similar to that of Theorem 4.3. Thus, we omit it here.
Theorem 4.5 multivariate extension principle III.
1 2 n α, β f HA α, β , HA α, β , . . . , HA α, β ,
α,β∈I 2
f A1 , A2 , . . . , An k
k
4.13
k
where Aα,β ⊆ HA α, β ⊆ Aα,β k 1, 2, . . . , n.
· ·
f −1 B1 , B2 , . . . , Bm l
l
1 2 m α, β f −1 HB α, β , HB α, β , . . . , HB α, β ,
α,β∈I 2
l
where Bα,β ⊆ HB α, β ⊆ Bα,β l 1, 2, . . . , m.
· ·
4.14
Journal of Applied Mathematics
15
Proof. Since
k
A
α,β
k ⊆ HA
· ·
1
⇒ A
α,β
k
α, β ⊆ A α,β k 1, 2, . . . , n
2
× A
· ·
1
α,β
n
× · · · × A
· ·
α,β
1 2 n ⊆ HA α, β × HA α, β × · · · × HA α, β
· ·
2
n
⊆ A α,β × A α,β × · · · × A α,β
2 1
2
n
1 n ⇒ f A α,β , A α,β , . . . , A α,β
⊆ f HA α, β , HA α, β , . . . , HA α, β
1
2
n
⊆ f A α,β , A α,β , . . . , A α,β
⇒ f A1 , A2 , . . . , An 4.15
2
n
, A , . . . , A ⎠
α,β
α,β
α,β
α,β∈I 2
⊆
α,β∈I 2
⊆
⎞
⎛
1
α, β f ⎝A
· ·
· ·
· ·
1 2 n α, β f HA α, β , HA α, β , . . . , HA α, β
1
2
n
α, β f A α,β , A α,β , . . . , A α,β
α,β∈I 2
f A1 , A2 , . . . 0, An .
Similarly, the second equality can be proved by the above method.
5. Algebraic Operations of the IFSs(R)
In this section, we will define several algebraic operations between IFSs on real number field
R using the multivariate extension principle, such as the arithmetic operations , −, ·, ÷,
conjunction and disjunction operations ∧, ∨, and so forth. In general, we denote the set
of all intuitionistic fuzzy sets on real number field R by the symbol IFSsR.
Let ∗ be an algebraic operation on R, that is,
∗ : R × R −→ R,
x, y −
→ z x ∗ y.
5.1
According to the multivariate extension principle of IFSs, we can define several
algebraic operations on IFSsR.
16
Journal of Applied Mathematics
Definition 5.1. Let ∗ be an algebraic operation on R, then the corresponding algebraic
operation on IFSsR is defined as
∗ : IFSsR × IFSsR −→ IFSsR,
α, β Aα,β ∗ Bα,β ,
α,β∈I 2
z : ∃ x, y ∈ Aα,β × Bα,β , x ∗ y z ,
def
A, B −→ A ∗ B Aα,β ∗ Bα,β
5.2
the membership function and nonmembership function of the operational result are given,
respectively, by
⎧
μA x ∧ μB y ,
⎪
⎨ μA∗B z x∗yz
∗:
⎪
ν
νA x ∨ νB y .
z
⎩ A∗B
5.3
x∗yz
By Definition 5.1, we can obtain the following some algebraic operations between IFSs
on R.
a Arithmetic operation
⎧
μ
∧
μ
μ
y μA x ∧ μB z − x ,
z
x
AB
A
B
⎪
⎨
xyz
x∈R
:
⎪
ν
∨
ν
ν
z
x
νA x ∨ νB z − x,
A
B y
⎩ AB
xyz
x∈R
⎧
μA x ∧ μB y μA x ∧ μB x − z ,
⎪
⎨ μA−B z x−yz
x∈R
−:
⎪
νA x ∨ νB y νA x ∨ νB x − z,
⎩ νA−B z x−yz
x∈R
⎧
⎪
μA x ∧ μB y
μA·B z ⎪
⎪
⎪
x·yz
⎪
⎧
⎪
z ⎪
⎪
⎪
⎪
μA x ∧ μB
⎪
⎪
⎪
⎪
⎪
⎪
x
⎨⎛
⎪
x∈R
⎪
⎛
⎞⎞ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎝μA 0 ∧ ⎝
μB y ⎠⎠ ∨ μB 0 ∧
μA x
⎪
⎪
⎪
⎪
⎩
⎨
y∈R
x∈R
·:
⎪
νA·B z νA x ∨ νB y
⎪
⎪
⎪
x·yz
⎪
⎧
⎪
z ⎪
⎪
⎪
⎪
νA x ∨ νB
⎪
⎪
⎪
⎪
⎪
⎪
x
⎨⎛
⎪
x∈R ⎞⎞
⎪
⎛
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎝νA 0 ∨ ⎝ νB y ⎠⎠ ∧ μB 0 ∨
νA x
⎪
⎪
⎪
⎪
⎩
⎩
y∈R
x∈R
z / 0,
z 0,
z / 0,
z 0.
Journal of Applied Mathematics
⎧
⎪
μA x ∧ μB y μA y · z ∧ μB y ,
μ
⎪
⎨ A÷B z
x÷yz
/0
y
÷:
⎪
∨
ν
ν
ν
y
νA y · z ∨ νB y .
z
A x
B
⎪
⎩ A÷B
x÷yz
17
y/
0
5.4
b Conjunction and disjunction operations
⎧
μ
∧
μ
μ
y ,
z
x
A∧B
A
B
⎪
⎨
x∧yz
∧:
⎪
νA x ∨ νB y ,
⎩ νA∧B z x∧yz
⎧
μ
μA x ∧ μB y ,
z
A∨B
⎪
⎨
x∨yz
∨:
⎪
ν
νA x ∨ νB y .
z
⎩ A∨B
5.5
x∨yz
6. Conclusion
In this paper, we presented the multivariate extension principle of IFSs. First of all, we
reviewed the notion of Cartesian product over IFSs proposed by Atanassov 18 and defined
the membership function and nonmembership function of Cartesian product in intuitionistic
fuzzy environment. In addition, some relevant properties were discussed. Afterwards, the
multivariate extension principle of IFSs was presented. Finally, we provided a general
framework of the algebraic operation between IFSs and gave several common operations. In
short, this paper not only provides a theoretical foundation for algebraic operations between
IFSs, but also enriches the theory of IFSs.
Acknowledgments
The authors would like to express their sincere thanks to the editor and reviewers for their
valuable suggestions. This work was supported by “Qing Lan” Talent Engineering Funds
by Tianshui Normal University. The second author acknowledge the support of the Beijing
Municipal Education Commission Foundation of China no. KM201210038001, the Funding
Project for Academic Human Resources Development in Institutions of Higher Learning
Under the Jurisdiction of Beijing Municipality no. PHR201007117, PHR201108333, and the
Beijing Natural Science Foundation no. 9123025.
References
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Sciences, vol. 41, no. 5, pp. 35–38, 1988.
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