# Multivariate Extension Principle and Algebraic Operations of Intuitionistic Fuzzy Sets

Authors: Yonghong Shen; Wei Chen
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

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.


---

## Body

## 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 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 Section2, 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

LetX 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
(2.2)μE:X→[0,1],νE:X→[0,1]
such that 0≤μE(x)+νE(x)≤1 for all x∈X.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 IFSs(X) 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, FSs(X) denotes the set of all fuzzy sets on X. In addition, intuitionistic fuzzy sets are abbreviated as IFSs.


### 2.2. Cut Sets ofIFSs and Its Properties

In this part, some concepts and conclusions associated with the cut sets ofIFSs are summarized below.Definition 2.1 (see [18]).
LetE∈IFSs(X), the parameters α,β satisfy the condition α+β≤1 for all α,β∈[0,1], the following four sets:
(2.3)E〈α,β〉={x∈X:μE(x)≥α,νE(x)≤β},E〈α·,β·〉={x∈X:μE(x)>α,νE(x)<β},E〈α·,β〉={x∈X:μE(x)>α,νE(x)≤β},E〈α,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 symbolI2 is given to denote the set I2={〈α,β〉:α+β≤1,α,β∈[0,1]}. For all 〈α1,β1〉,〈α2,β2〉∈I2, 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 setI2 and the IFSs(X) is quoted.Definition 2.2 (see [18]).
LetE∈IFSs(X), 〈α,β〉∈I2, the set 〈α,β〉E∈IFSs(X) is defined as follows
(2.4)〈α,β〉E≝{〈x,α∧μE(x),β∨νE(x)〉:x∈X}.
Next, some properties of the cut set ofIFSs are summarized.Lemma 2.3 (see [18]).
LetE∈IFSs(X), 〈α,β〉∈I2, then(i)
E〈α·,β·〉⊆E〈α·,β〉⊆E〈α,β〉;(ii)
E〈α·,β·〉⊆E〈α,β·〉⊆E〈α,β〉;(iii)
E〈0,1〉=X, E〈1·,0〉=E〈1,0·〉=E〈1·,0·〉=∅.Lemma 2.4 (see [18]).
LetE∈IFSs(X), 〈α1,β1〉,〈α2,β2〉∈I2, and 〈α1,β1〉≤〈α2,β2〉, then(i)
E〈α2,β2〉⊆E〈α1,β1〉;(ii)
E〈α2·,β2·〉⊂E〈α1·,β1·〉;(iii)
E〈α2·,β2〉⊆E〈α1·,β1〉;(iv)
E〈α2,β2·〉⊆E〈α1,β1·〉.Based on the cut sets ofIFSs 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]).
LetE∈IFSs(X), then
(2.5)E=⋃〈α,β〉∈I2〈α,β〉E〈α,β〉.Notice that the setE 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 ofIFSs. The symbol P(X) denotes the power set of X.Definition 2.6.
Letf:I2→P(X) be a mapping, that is, 〈α,β〉↦H(α,β), for all 〈α1,β1〉,〈α2,β2〉∈I2, if 〈α1,β1〉<〈α2,β2〉, there always has H(α2,β2)⊂H(α1,β1), then the set H is called the binary nested set on X.Lemma 2.7 (representation theorem [14]).
Letf be a mapping from BN(X) to IFSs(X), that is, f:BN(X)→IFSs(X), and
(2.6)H↦f(H)≜⋃〈α,β〉∈I2〈α,β〉H(α,β),
then(i)
f is a surjection from (BN(X),∪,∩,c) to (IFSs(X),∪,∩,c);(ii)
(f(H))〈α·,β·〉⊆H(α,β)⊆(f(H))〈α,β〉,
where the notationBN(X) denotes the set of all the binary nested sets on X.Lemma2.7 shows that there exists a unique f(H) such that f(H)=⋃〈α,β〉∈I2〈α,β〉H(α,β) for all H∈BN(X).Lemma 2.8 (extension principle [14]).
Supposef:X→Y is a mapping from the ordinary set X to the ordinary set Y, that is, x↦f(x), then the mapping f can induce into two mappings f:IFSs(X)→IFSs(Y) and f-1:IFSs(Y)→IFSs(X)(2.7)f:IFSs(X)→IFSs(Y),A↦f(A)=⋃〈α,β〉∈I2〈α,β〉f(A〈α,β〉),f-1:IFSs(Y)→IFSs(X),B↦f-1(B)=⋃〈α,β〉∈I2〈α,β〉f-1(B〈α,β〉).
The membership and nonmembership functions of f,f-1 are defined, respectively, as follows
(2.8)μf(A)(y)=⋁f(x)=yμA(x),νf(A)(y)=⋀f(x)=yνA(x),μf-1(B)(x)=μB(f(x)),νf-1(B)(x)=νB(f(x)),
where A={〈x,μA(x),νA(x)〉:x∈X}∈IFSs(X), f(A)={〈y=f(x),μf(A)(y),νf(A)(y)〉:x∈X,f(x)⊆Y}∈IFSs(Y). B and f-1(B) can be given in a similar manner.


## 2.1. Intuitionistic Fuzzy Sets

LetX 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
(2.2)μE:X→[0,1],νE:X→[0,1]
such that 0≤μE(x)+νE(x)≤1 for all x∈X.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 IFSs(X) 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, FSs(X) denotes the set of all fuzzy sets on X. In addition, intuitionistic fuzzy sets are abbreviated as IFSs.


## 2.2. Cut Sets ofIFSs and Its Properties

In this part, some concepts and conclusions associated with the cut sets ofIFSs are summarized below.Definition 2.1 (see [18]).
LetE∈IFSs(X), the parameters α,β satisfy the condition α+β≤1 for all α,β∈[0,1], the following four sets:
(2.3)E〈α,β〉={x∈X:μE(x)≥α,νE(x)≤β},E〈α·,β·〉={x∈X:μE(x)>α,νE(x)<β},E〈α·,β〉={x∈X:μE(x)>α,νE(x)≤β},E〈α,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 symbolI2 is given to denote the set I2={〈α,β〉:α+β≤1,α,β∈[0,1]}. For all 〈α1,β1〉,〈α2,β2〉∈I2, 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 setI2 and the IFSs(X) is quoted.Definition 2.2 (see [18]).
LetE∈IFSs(X), 〈α,β〉∈I2, the set 〈α,β〉E∈IFSs(X) is defined as follows
(2.4)〈α,β〉E≝{〈x,α∧μE(x),β∨νE(x)〉:x∈X}.
Next, some properties of the cut set ofIFSs are summarized.Lemma 2.3 (see [18]).
LetE∈IFSs(X), 〈α,β〉∈I2, then(i)
E〈α·,β·〉⊆E〈α·,β〉⊆E〈α,β〉;(ii)
E〈α·,β·〉⊆E〈α,β·〉⊆E〈α,β〉;(iii)
E〈0,1〉=X, E〈1·,0〉=E〈1,0·〉=E〈1·,0·〉=∅.Lemma 2.4 (see [18]).
LetE∈IFSs(X), 〈α1,β1〉,〈α2,β2〉∈I2, and 〈α1,β1〉≤〈α2,β2〉, then(i)
E〈α2,β2〉⊆E〈α1,β1〉;(ii)
E〈α2·,β2·〉⊂E〈α1·,β1·〉;(iii)
E〈α2·,β2〉⊆E〈α1·,β1〉;(iv)
E〈α2,β2·〉⊆E〈α1,β1·〉.Based on the cut sets ofIFSs 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]).
LetE∈IFSs(X), then
(2.5)E=⋃〈α,β〉∈I2〈α,β〉E〈α,β〉.Notice that the setE 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 ofIFSs. The symbol P(X) denotes the power set of X.Definition 2.6.
Letf:I2→P(X) be a mapping, that is, 〈α,β〉↦H(α,β), for all 〈α1,β1〉,〈α2,β2〉∈I2, if 〈α1,β1〉<〈α2,β2〉, there always has H(α2,β2)⊂H(α1,β1), then the set H is called the binary nested set on X.Lemma 2.7 (representation theorem [14]).
Letf be a mapping from BN(X) to IFSs(X), that is, f:BN(X)→IFSs(X), and
(2.6)H↦f(H)≜⋃〈α,β〉∈I2〈α,β〉H(α,β),
then(i)
f is a surjection from (BN(X),∪,∩,c) to (IFSs(X),∪,∩,c);(ii)
(f(H))〈α·,β·〉⊆H(α,β)⊆(f(H))〈α,β〉,
where the notationBN(X) denotes the set of all the binary nested sets on X.Lemma2.7 shows that there exists a unique f(H) such that f(H)=⋃〈α,β〉∈I2〈α,β〉H(α,β) for all H∈BN(X).Lemma 2.8 (extension principle [14]).
Supposef:X→Y is a mapping from the ordinary set X to the ordinary set Y, that is, x↦f(x), then the mapping f can induce into two mappings f:IFSs(X)→IFSs(Y) and f-1:IFSs(Y)→IFSs(X)(2.7)f:IFSs(X)→IFSs(Y),A↦f(A)=⋃〈α,β〉∈I2〈α,β〉f(A〈α,β〉),f-1:IFSs(Y)→IFSs(X),B↦f-1(B)=⋃〈α,β〉∈I2〈α,β〉f-1(B〈α,β〉).
The membership and nonmembership functions of f,f-1 are defined, respectively, as follows
(2.8)μf(A)(y)=⋁f(x)=yμA(x),νf(A)(y)=⋀f(x)=yνA(x),μf-1(B)(x)=μB(f(x)),νf-1(B)(x)=νB(f(x)),
where A={〈x,μA(x),νA(x)〉:x∈X}∈IFSs(X), f(A)={〈y=f(x),μf(A)(y),νf(A)(y)〉:x∈X,f(x)⊆Y}∈IFSs(Y). B and f-1(B) can be given in a similar manner.


## 3. Cartesian Product overIFSs

The concept of Cartesian product overIFSs 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(3.1)A1×A2×⋯×An≝{(x1,x2,⋯,xn):xi∈Ai,i=1,2,…,n},
the characteristic function is
(3.2)χA1×A2×⋯×An(x1,x2,…,xn)=⋀k=1nAk(xk).The Cartesian product over FSs is obtained by extending the ordinary sets to fuzzy sets.LetA(k)∈FSs(Xk)(k=1,2,…,n). Based on the decomposition and representation theorems of FSs, the Cartesian product can be obtained as follows:
(3.3)A(1)×A(2)×⋯×A(n)=⋃λ∈[0,1]λ(Aλ(1)×Aλ(2)×⋯×Aλ(n)).In fact, it has been proved that the membership function of Cartesian product over FSs is(3.4)μA(1)×A(2)×⋯×A(n)(x1,x2,…,xn)=⋀k=1nμA(k)(xk).Similarly, the Cartesian product overIFSs can also be obtained.Definition 3.1.
LetE(k)∈IFSs(Xk)(k=1,2,…,n), 〈α,β〉∈I2, the following expression
(3.5)E(1)×E(2)×⋯×E(n)≝⋃〈α,β〉∈I2〈α,β〉(E〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n))
be called the Cartesian product over IFSs.According to Lemma2.4, we have
(3.6)〈α1,β1〉<〈α2,β2〉⇒E〈α1,β1〉(k)⊇E〈α2,β2〉(k)(k=1,2,…,n)⇒E〈α1,β1〉(1)×E〈α1,β1〉(2)×⋯×E〈α1,β1〉(n)⊇E〈α2,β2〉(1)×E〈α2,β2〉(2)×⋯×E〈α2,β2〉(n).Hence, it is easy to see that the set(3.7){E〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n):〈α,β〉∈I2}
is a binary nested set on X=X1×X2×⋯×Xn.According to Lemma2.7, we know that the above binary nested set can uniquely identify an intuitionistic fuzzy set contained in IFSs(X1×X2×⋯×Xn). Therefore, we have
(3.8)⋃〈α,β〉∈I2〈α,β〉(E〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n))∈IFSs(X1×X2×⋯×Xn),
and it satisfies
(3.9)(E(1)×E(2)×⋯×E(n))〈α·,β·〉⊆E〈α,β〉(1)×E〈α,β〉(2)×…×E〈α,β〉(n)⊆(E(1)×E(2)×⋯×E(n))〈α,β〉.For the membership function and nonmembership function of the Cartesian product overIFSs, we have the following theorem.Theorem 3.2.
LetE(k)∈IFSs(Xk)(k=1,2,…,n), then
(3.10)μE(1)×E(2)×⋯×E(n)(x1,x2,…,xn)=⋀k=1nμE(k)(xk),(3.11)νE(1)×E(2)×⋯×E(n)(x1,x2,…,xn)=⋁k=1nνE(k)(xk).Proof.
First of all, we prove the membership function of Cartesian product overIFSs. For all (x1,x2,…,xn)∈X=X1×X2×⋯×Xn, according to Definitions 2.2 and 3.1, we can obtain
(3.12)μE(1)×E(2)×⋯×E(n)(x1,x2,…,xn)=⋁〈α,β〉∈I2(α∧(μE〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n))(x1,x2,…,xn))=⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk)))).
Now, we will prove that
(3.13)⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk))))=⋀k=1n(⋁〈α,β〉∈I2(α∧μE〈α,β〉(k)(xk))).
Since
(3.14)⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk))))≤⋁〈α,β〉∈I2(α∧μE〈α,β〉(k)(xk)),⇒⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk))))≤⋀k=1n(⋁〈α,β〉∈I2(α∧μE〈α,β〉(k)(xk))).
Assume that ⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk))))<⋀k=1n(⋁〈α,β〉∈I2(α∧μE〈α,β〉(k)(xk))), then there exists a constant λ such that
(3.15)⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk))))<λ<⋀k=1n(⋁〈α,β〉∈I2(α∧μE〈α,β〉(k)(xk))).
Since ⋀k=1nμE〈λ,β〉(k)(xk)={0,1}, if ⋀k=1nμE〈λ,β〉(k)(xk)=1, then
(3.16)λ>⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk))))≥λ∧(⋀k=1n(μE〈λ,β〉(k)(xk)))=λ.
In addition, if ⋀k=1nμE〈λ,β〉(k)(xk)=0, then there exists a constant k0(1≤k0≤n) such that μE〈λ,β〉(k0)(xk0)=0. By Definition 2.1, we know that μE(k0)(xk0)<λ or νE(k0)(xk0)>β, and then for all α>λ, we have μE〈α,β〉(k0)(xk0)=0.
Hence,(3.17)λ<⋀k=1n(⋁〈α,β〉∈I2(α∧μE〈α,β〉(k)(xk)))≤⋁〈α,β〉∈I2(α∧μE〈α,β〉(k0)(xk0))≤λ.
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
(3.18)μE(1)×E(2)×⋯×E(n)(x1,x2,…,xn)=⋁〈α,β〉∈I2(α∧(⋀k=1n(μE〈α,β〉(k)(xk))))=⋀k=1n(⋁〈α,β〉∈I2(α∧μE〈α,β〉(k)(xk)))=⋀k=1nμE(k)(xk).
Now we start to prove the expression (3.11). Analogously, we have
(3.19)νE(1)×E(2)×⋯×E(n)(x1,x2,…,xn)=⋀〈α,β〉∈I2(β∨(νE〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n))(x1,x2,…,xn))=⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk)))).
Next, we will prove that
(3.20)⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk))))=⋁k=1n(⋀〈α,β〉∈I2(β∨νE〈α,β〉(k)(xk))).
Since
(3.21)⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk))))≥⋀〈α,β〉∈I2(β∨νE〈α,β〉(k)(xk))⇒⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk))))≥⋁k=1n(⋀〈α,β〉∈I2(β∨νE〈α,β〉(k)(xk))).
Similarly, we assume that
(3.22)⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk))))>⋁k=1n(⋀〈α,β〉∈I2(β∨νE〈α,β〉(k)(xk))),
then there exists a constant γ such that
(3.23)⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk))))>γ>⋁k=1n(⋀〈α,β〉∈I2(β∨νE〈α,β〉(k)(xk))).
Since ⋁k=1nνE〈α,γ〉(k)(xk)={0,1}, if ⋁k=1nνE〈α,γ〉(k)(xk)=0, then
(3.24)γ<⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk))))≤γ∨(⋁k=1n(νE〈α,β〉(k)(xk)))=γ.
On the other hand, if ⋁k=1nνE〈α,γ〉(k)(xk)=1, then there exists a constant k1(1≤k1≤n) such that νE〈α,γ〉(k1)(xk1)=1, that is, μE(k1)(xk1)>α and νE(k1)(xk1)<γ. Therefore, for all β>γ, we have νE〈α,β〉(k1)(xk1)=1.
So we can obtain(3.25)γ>⋁k=1n(⋀〈α,β〉∈I2(β∨νE〈α,β〉(k)(xk)))≥⋀〈α,β〉∈I2(β∨νE〈α,β〉(k1)(xk1))≥γ.
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
(3.26)νE(1)×E(2)×⋯×E(n)(x1,x2,…,xn)=⋀〈α,β〉∈I2(β∨(⋁k=1n(νE〈α,β〉(k)(xk))))=⋁k=1n(⋀〈α,β〉∈I2(β∨νE〈α,β〉(k)(xk)))=⋁k=1nνE(k)(xk).Remark 3.3.
According to the poof, one can see that the notations⋁〈α,β〉 and ⋀k=1n, ⋀〈α,β〉 and ⋁k=1n are commutative, respectively. The reason is that ⋀k=1nμE〈α,β〉(k)(xk) and ⋁k=1nνE〈α,β〉(k)(xk) are equal to {0,1}. Notice that these relations are not necessarily satisfied under general conditions.
Based on Theorem3.2, we can obtain some properties of cut sets of the Cartesian product over IFSs.Theorem 3.4.
LetE(k)∈IFSs(Xk)(k=1,2,…,n), 〈α,β〉∈I2, then(i)
(E(1)×E(2)×⋯×E(n))〈α,β〉=E〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n);(ii)
(E(1)×E(2)×⋯×E(n))〈α·,β·〉=E〈α·,β·〉(1)×E〈α·,β·〉(2)×⋯×E〈α·,β·〉(n);(iii)
(E(1)×E(2)×⋯×E(n))〈α·,β〉=E〈α·,β〉(1)×E〈α·,β〉(2)×⋯×E〈α·,β〉(n);(iv)
(E(1)×E(2)×⋯×E(n))〈α,β·〉=E〈α,β·〉(1)×E〈α,β·〉(2)×⋯×E〈α,β·〉(n).Proof.
We only prove the first equality, the remaining equalities can be proved in a similar way.
According to Definition2.1 and Theorem 3.2, for all
(3.27)(x1,x2,…,xn)∈(E(1)×E(2)×⋯×E(n))〈α,β〉⟺⋀k=1nμE(k)(xk)≥α,⋁k=1nνE(k)(xk)≤β⟺μE(k)(xk)≥α,νE(k)(xk)≤β(k=1,2,…,n)⟺xk∈E〈α,β〉(k)(k=1,2,…,n)(x1,x2,…,xn)∈E〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n).
Hence, the equality holds, namely,
(3.28)(E(1)×E(2)×⋯×E(n))〈α,β〉=E〈α,β〉(1)×E〈α,β〉(2)×⋯×E〈α,β〉(n).


## 4. Multivariate Extension Principle ofIFSs

Based on the Cartesian product overIFSs, we will discuss the multivariate extension principle of IFSs.LetE(k)∈IFSs(Xk)(k=1,2,…,n), then the operation of Cartesian product may be viewed as a mapping, which is defined as follows
(4.1)×:IFSs(X1)×IFSs(X2)×⋯×IFSs(Xn)→IFSs(X1×X2×⋯×Xn),(E(1),E(2),…,E(n))↦E(1)×E(2)×⋯×E(n).Now we will define the multivariate extension principle. It is assumed thatf is a mapping from X to Y, namely,
(4.2)f:X=X1×X2×⋯×Xn→Y=Y1×Y2×⋯×Ym,(x1,x2,…,xn)↦(f(x1,x2,…,xn))=y=(y1,y2,…,ym).By the extension principle ofIFSs (Lemma 2.8), the mapping f can be induced to the following two mappings.
(4.3)f:IFSs(X)→IFSs(Y),A↦f(A)=⋃‍〈α,β〉∈I2〈α,β〉f(A〈α,β〉),f-1:IFSs(Y)→IFSs(X),B↦f-1(B)=⋃‍〈α,β〉∈I2〈α,β〉f-1(B〈α,β〉).Next, we use the two mappingsf,f-1 to make compound operations with the following two Cartesian products of IFSs, respectively. 						(4.4)×1:IFSs(X1)×IFSs(X2)×⋯×IFSs(Xn)→IFSs(X),(A(1),A(2),…,A(n))↦A(1)×A(2)×⋯×A(n),×2:IFSs(Y1)×IFSs(Y2)×⋯×IFSs(Ym)→IFSs(Y),(B(1),B(2),…,B(m))↦B(1)×B(2)×⋯×B(m).Based on the above compound operations, we will get the following definition about multivariate extension principle ofIFSs.Definition 4.1.
LetA(k)∈IFSs(Xk)(k=1,2,…,n), B(l)∈IFSs(Yl)(l=1,2,…,m), and f be a mapping from X to Y, that is,
(4.5)f:X=X1×X2×⋯×Xn→Y=Y1×Y2×⋯×Ym,(x1,x2,…,xn)↦f(x1,x2,…,xn)=y=(y1,y2,…,ym).
The two induced mappings of f can be defined as follows:
(4.6)f:IFSs(X1)×IFSs(X2)×⋯×IFSs(Xn)→IFSs(Y),(A(1),A(2),…,A(n))↦f(A(1),A(2),…,A(n))≝f(A(1)×A(2)×⋯×A(n)),f-1:IFSs(Y1)×IFSs(Y2)×⋯×IFSs(Ym)→IFSs(X),(B(1),B(2),…,B(m))↦f-1(B(1),B(2),…,B(m))≝f-1(B(1)×B(2)×⋯×B(m)).
It is obvious that the above result is an extension of Lemma2.8 of the Cartesian product over IFSs. Hence, it is said to be the multivariate extension principle of IFSs.According to Lemma2.8 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.
Letf 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
(4.7)μf(A(1),A(2),…,A(n))(y)=⋁f(x1,x2,…,xn)=y(⋀k=1nμA(k)(xk)),νf(A(1),A(2),…,A(n))(y)=⋀f(x1,x2,…,xn)=y(⋁k=1nνA(k)(xk)),μf-1(B(1),B(2),…,B(m))(x)=⋀l=1mμB(l)(yl),νf-1(B(1),B(2),…,B(m))(x)=⋁l=1mνB(l)(yl),
where (y1,y2,…,ym)=f(x).Below we will discuss some other forms of the multivariate extension principle ofIFSs.Theorem 4.3 (multivariate extension principle I).
(4.8)
f
(
A
(
1
)
,
A
(
2
)
,
…
,
A
(
n
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
(
A
〈
α
,
β
〉
(
1
)
,
A
〈
α
,
β
〉
(
2
)
,
…
,
A
〈
α
,
β
〉
(
n
)
)
,
f
-
1
(
B
(
1
)
,
B
(
2
)
,
…
,
B
(
m
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
-
1
(
B
〈
α
,
β
〉
(
1
)
,
B
〈
α
,
β
〉
(
2
)
,
…
,
B
〈
α
,
β
〉
(
m
)
)
.Proof.
By Lemma2.8 and Definition 4.1, we have
(4.9)f(A(1),A(2),…,A(n))=f(A(1)×A(2)×⋯×A(n))=⋃〈α,β〉∈I2〈α,β〉f((A(1)×A(2)×⋯×A(n))〈α,β〉)=⋃〈α,β〉∈I2〈α,β〉f(A〈α,β〉(1)×A〈α,β〉(2)×⋯×A〈α,β〉(n)).
Since
(4.10)f(A〈α,β〉(1),A〈α,β〉(2),…,A〈α,β〉(n))={y:∃xk∈A〈α,β〉(k)(k=1,2,…,n),f(x1,x2,…,xn)=y}={y:∃(x1,x2,…,xn)∈A〈α,β〉(1)×A〈α,β〉(2)×⋯×A〈α,β〉(n),f(x1,x2,…,xn)=y}=f(A〈α,β〉(1)×A〈α,β〉(2)×⋯×A〈α,β〉(n)),
we know that
(4.11)f(A(1),A(2),…,A(n))=⋃〈α,β〉∈I2〈α,β〉f(A〈α,β〉(1),A〈α,β〉(2),…,A〈α,β〉(n)).
Similarly, we can prove the second equality.Theorem 4.4 (multivariate extension principle II).
(4.12)
f
(
A
(
1
)
,
A
(
2
)
,
…
,
A
(
n
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
(
A
〈
α
·
,
β
·
〉
(
1
)
,
A
〈
α
·
,
β
·
〉
(
2
)
,
…
,
A
〈
α
·
,
β
·
〉
(
n
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
(
A
〈
α
·
,
β
〉
(
1
)
,
A
〈
α
·
,
β
〉
(
2
)
,
…
,
A
〈
α
·
,
β
〉
(
n
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
(
A
〈
α
,
β
·
〉
(
1
)
,
A
〈
α
,
β
·
〉
(
2
)
,
…
,
A
〈
α
,
β
·
〉
(
n
)
)
,
f
-
1
(
B
(
1
)
,
B
(
2
)
,
…
,
B
(
m
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
-
1
(
B
〈
α
·
,
β
·
〉
(
1
)
,
B
〈
α
·
,
β
·
〉
(
2
)
,
…
,
B
〈
α
·
,
β
·
〉
(
m
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
-
1
(
B
〈
α
·
,
β
〉
(
1
)
,
B
〈
α
·
,
β
〉
(
2
)
,
…
,
B
〈
α
·
,
β
〉
(
m
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
-
1
(
B
〈
α
,
β
·
〉
(
1
)
,
B
〈
α
,
β
·
〉
(
2
)
,
…
,
B
〈
α
,
β
·
〉
(
m
)
)
.Proof.
The proof method is similar to that of Theorem4.3. Thus, we omit it here.Theorem 4.5 (multivariate extension principle III).
(4.13)
f
(
A
(
1
)
,
A
(
2
)
,
…
,
A
(
n
)
)
=
⋃
〈
α
,
β
〉
∈
I
2
〈
α
,
β
〉
f
(
H
A
(
1
)
(
α
,
β
)
,
H
A
(
2
)
(
α
,
β
)
,
…
,
H
A
(
n
)
(
α
,
β
)
)
,
where A〈α·,β·〉(k)⊆HA(k)(α,β)⊆A〈α,β〉(k)(k=1,2,…,n). 							(4.14)f-1(B(1),B(2),…,B(m))=⋃〈α,β〉∈I2〈α,β〉f-1(HB(1)(α,β),HB(2)(α,β),…,HB(m)(α,β)),
where B〈α·,β·〉(l)⊆HB(l)(α,β)⊆B〈α,β〉(l)(l=1,2,…,m).Proof.
Since(4.15)A〈α·,β·〉(k)⊆HA(k)(α,β)⊆A〈α,β〉(k)(k=1,2,…,n)⇒A〈α·,β·〉(1)×A〈α·,β·〉(2)×⋯×A〈α·,β·〉(n)⊆HA(1)(α,β)×HA(2)(α,β)×⋯×HA(n)(α,β)⊆A〈α,β〉(1)×A〈α,β〉(2)×⋯×A〈α,β〉(n)⇒f(A〈α,β〉(1),A〈α,β〉(2),…,A〈α,β〉(n))⊆f(HA(1)(α,β),HA(2)(α,β),…,HA(n)(α,β))⊆f(A〈α,β〉(1),A〈α,β〉(2),…,A〈α,β〉(n))⇒f(A(1),A(2),…,A(n))=⋃〈α,β〉∈I2〈α,β〉f(A〈α·,β·〉(1),A〈α·,β·〉(2),…,A〈α·,β·〉(n))⊆⋃〈α,β〉∈I2〈α,β〉f(HA(1)(α,β),HA(2)(α,β),…,HA(n)(α,β))⊆⋃〈α,β〉∈I2〈α,β〉f(A〈α,β〉(1),A〈α,β〉(2),…,A〈α,β〉(n))=f(A(1),A(2),…,A(n)).
Similarly, the second equality can be proved by the above method.


## 5.  Algebraic Operations of theIFSs(R)

In this section, we will define several algebraic operations betweenIFSs 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 IFSs(R).Let* be an algebraic operation on R, that is,
(5.1)*:R×R→R,(x,y)↦z=x*y.According to the multivariate extension principle ofIFSs, we can define several algebraic operations on IFSs(R).Definition 5.1.
Let* be an algebraic operation on R, then the corresponding algebraic operation on IFSs(R) is defined as
(5.2)*:IFSs(R)×IFSs(R)→IFSs(R),(A,B)↦A*B≝⋃‍〈α,β〉∈I2〈α,β〉(A〈α,β〉*B〈α,β〉),A〈α,β〉*B〈α,β〉={z:∃(x,y)∈A〈α,β〉×B〈α,β〉,x*y=z},
the membership function and nonmembership function of the operational result are given, respectively, by
(5.3)*:{μA*B(z)=⋁x*y=z(μA(x)∧μB(y)),νA*B(z)=⋀x*y=z(νA(x)∨νB(y)).By Definition5.1, we can obtain the following some algebraic operations between IFSs on R.(a)Arithmetic operation(5.4)+:{μA+B(z)=⋁x+y=z(μA(x)∧μB(y))=⋁x∈R(μA(x)∧μB(z-x)),νA+B(z)=⋀x+y=z(νA(x)∨νB(y))=⋀x∈R(νA(x)∨νB(z-x)),-:{μA-B(z)=⋁x-y=z(μA(x)∧μB(y))=⋁x∈R(μA(x)∧μB(x-z)),νA-B(z)=⋀x-y=z(νA(x)∨νB(y))=⋀x∈R(νA(x)∨νB(x-z)),·:{μA·B(z)=⋁x·y=z(μA(x)∧μB(y))={⋁x∈R(μA(x)∧μB(zx))(z≠0),(μA(0)∧(⋁y∈RμB(y)))∨(μB(0)∧(⋁x∈RμA(x)))(z=0),νA·B(z)=⋀x·y=z(νA(x)∨νB(y))={⋀x∈R(νA(x)∨νB(zx))(z≠0),(νA(0)∨(⋀y∈RνB(y)))∧(μB(0)∨(⋀x∈RνA(x)))(z=0).÷:{μA÷B(z)=⋁x÷y=z(μA(x)∧μB(y))=⋁y≠0(μA(y·z)∧μB(y)),νA÷B(z)=⋀x÷y=z(νA(x)∨νB(y))=⋀y≠0(νA(y·z)∨νB(y)).(b)Conjunction and disjunction operations(5.5)∧:{μA∧B(z)=⋁x∧y=z(μA(x)∧μB(y)),νA∧B(z)=⋀x∧y=z(νA(x)∨νB(y)),∨:{μA∨B(z)=⋁x∨y=z(μA(x)∧μB(y)),νA∨B(z)=⋀x∨y=z(νA(x)∨νB(y)).


## 6. Conclusion

In this paper, we presented the multivariate extension principle ofIFSs. 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.



---

*Source: 845090-2012-10-18.xml*