Multivariate Extension Principle and Algebraic Operations of Intuitionistic Fuzzy Sets

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.

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 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 𝐸=𝑥,𝜇_𝐸(𝑥),𝜈_𝐸(𝑥)∶𝑥∈𝑋 (2.1) with 𝜇_𝐸∶𝑋→[0,1],𝜈_𝐸∶𝑋→[0,1] (2.2) such that 0≤𝜇_𝐸(𝑥)+𝜈_𝐸(𝑥)≤1 for all 𝑥∈𝑋.

Generally, 𝜇_𝐸(𝑥) and 𝜈_𝐸(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 𝐸_𝑐={⟨𝑥,𝜈_𝐸(𝑥),𝜇_𝐸(𝑥)⟩∶𝑥∈𝑋}. The symbol 𝐼𝐹𝑆𝑠(𝑋) denotes the set of all intuitionistic fuzzy sets on 𝑋. Especially, if 𝜈_𝐸(𝑥)=1−𝜇_𝐸(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 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∈𝐼𝐹𝑆𝑠(𝑋), the parameters 𝛼,𝛽 satisfy the condition 𝛼+𝛽≤1 for all 𝛼,𝛽∈[0,1], the following four sets: 𝐸_{⟨𝛼,𝛽⟩}=𝑥∈𝑋∶𝜇_𝐸(𝑥)≥𝛼,𝜈_𝐸(𝑥),𝐸≤𝛽_{⟨𝛼⋅,𝛽⋅⟩}=𝑥∈𝑋∶𝜇_𝐸(𝑥)>𝛼,𝜈_𝐸(𝑥),𝐸<𝛽_{⟨𝛼⋅,𝛽⟩}=𝑥∈𝑋∶𝜇_𝐸(𝑥)>𝛼,𝜈_𝐸(𝑥),𝐸≤𝛽_{⟨𝛼,𝛽⋅⟩}=𝑥∈𝑋∶𝜇_𝐸(𝑥)≥𝛼,𝜈_𝐸(𝑥)<𝛽, (2.3) 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 𝐼² 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 set 𝐼² and the IFSs(𝑋) is quoted.

Definition 2.2 (see [18]). Let 𝐸∈𝐼𝐹𝑆𝑠(𝑋), ⟨𝛼,𝛽⟩∈𝐼², the set ⟨𝛼,𝛽⟩𝐸∈𝐼𝐹𝑆𝑠(𝑋) is defined as follows 𝛼,𝛽𝐸=^def𝑥,𝛼∧𝜇_𝐸(𝑥),𝛽∨𝜈_𝐸(𝑥)∶𝑥∈𝑋. (2.4)
Next, some properties of the cut set of IFSs are summarized.

Lemma 2.3 (see [18]). Let 𝐸∈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]). Let 𝐸∈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 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∈𝐼𝐹𝑆𝑠(𝑋), then 𝐸=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝐸_{⟨𝛼,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∶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]). Let 𝑓 be a mapping from 𝐵𝑁(𝑋) to 𝐼𝐹𝑆𝑠(𝑋), that is, 𝑓∶𝐵𝑁(𝑋)→𝐼𝐹𝑆𝑠(𝑋), and 𝐻↦𝑓(𝐻)≜_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝐻𝛼,𝛽, (2.6) 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 notation 𝐵𝑁(𝑋) denotes the set of all the binary nested sets on 𝑋.

Lemma 2.7 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]). Suppose 𝑓∶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)𝑓:𝐼𝐹𝑆𝑠(𝑋)→𝐼𝐹𝑆𝑠(𝑌),𝐴↦𝑓(𝐴)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴_{⟨𝛼,𝛽⟩},𝑓⁻¹:𝐼𝐹𝑆𝑠(𝑌)→𝐼𝐹𝑆𝑠(𝑋),𝐵↦𝑓⁻¹(𝐵)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓⁻¹𝐵_{⟨𝛼,𝛽⟩}. (2.7) The membership and nonmembership functions of 𝑓,𝑓⁻¹ are defined, respectively, as follows 𝜇_𝑓(𝐴)𝑦=_{𝑓(𝑥)=𝑦}𝜇_𝐴(𝑥),𝜈_𝑓(𝐴)𝑦=_{𝑓(𝑥)=𝑦}𝜈_𝐴(𝑥),𝜇_𝑓⁻¹_(𝐵)(𝑥)=𝜇_𝐵𝑓(𝑥),𝜈_𝑓⁻¹_(𝐵)(𝑥)=𝜈_𝐵𝑓(𝑥), (2.8) where 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∶𝑥∈𝑋}∈𝐼𝐹𝑆𝑠(𝑋), 𝑓(𝐴)={⟨𝑦=𝑓(𝑥),𝜇_𝑓(𝐴)(𝑦),𝜈_𝑓(𝐴)(𝑦)⟩∶𝑥∈𝑋,𝑓(𝑥)⊆𝑌}∈𝐼𝐹𝑆𝑠(𝑌). 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 𝐴₁×𝐴₂×⋯×𝐴=_𝑛^def𝑥₁,𝑥₂,⋯,𝑥_𝑛∶𝑥_𝑖∈𝐴_𝑖,𝑖=1,2,…,𝑛, (3.1) the characteristic function is 𝜒_{𝐴1×𝐴2×⋯×𝐴𝑛}𝑥₁,𝑥₂,…,𝑥_𝑛=^𝑛_𝑘=1𝐴_𝑘𝑥_𝑘. (3.2)

The Cartesian product over FSs is obtained by extending the ordinary sets to fuzzy sets.

Let 𝐴^(𝑘)∈FSs(𝑋_𝑘)(𝑘=1,2,…,𝑛). Based on the decomposition and representation theorems of FSs, the Cartesian product can be obtained as follows: 𝐴⁽¹⁾×𝐴⁽²⁾×⋯×𝐴^(𝑛)=_[]𝜆∈0,1𝜆𝐴_𝜆⁽¹⁾×𝐴_𝜆⁽²⁾×⋯×𝐴_𝜆^(𝑛). (3.3)

In fact, it has been proved that the membership function of Cartesian product over FSs is 𝜇_{𝐴(1)×𝐴(2)×⋯×𝐴(𝑛)}𝑥₁,𝑥₂,…,𝑥_𝑛=^𝑛_𝑘=1𝜇_𝐴(𝑘)𝑥_𝑘. (3.4)

Similarly, the Cartesian product over IFSs can also be obtained.

Definition 3.1. Let 𝐸^(𝑘)∈𝐼𝐹𝑆𝑠(𝑋_𝑘)(𝑘=1,2,…,𝑛), ⟨𝛼,𝛽⟩∈𝐼², the following expression 𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸=^(𝑛)def_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝐸⁽¹⁾_𝛼,𝛽×𝐸⁽²⁾_𝛼,𝛽×⋯×𝐸^(𝑛)_𝛼,𝛽 (3.5) be called the Cartesian product over IFSs.

According to Lemma 2.4, we have 𝛼₁,𝛽₁<𝛼₂,𝛽₂⇒𝐸_𝛼^(𝑘)_1,𝛽1⊇𝐸_𝛼^(𝑘)_2,𝛽2(𝑘=1,2,…,𝑛)⇒𝐸_𝛼⁽¹⁾_1,𝛽1×𝐸_𝛼⁽²⁾_1,𝛽1×⋯×𝐸_𝛼^(𝑛)_1,𝛽1⊇𝐸_𝛼⁽¹⁾_2,𝛽2×𝐸_𝛼⁽²⁾_2,𝛽2×⋯×𝐸_𝛼^(𝑛)_2,𝛽2. (3.6)

Hence, it is easy to see that the set 𝐸⁽¹⁾_𝛼,𝛽×𝐸⁽²⁾_𝛼,𝛽×⋯×𝐸^(𝑛)_𝛼,𝛽∶𝛼,𝛽∈𝐼² (3.7) is a binary nested set on 𝑋=𝑋₁×𝑋₂×⋯×𝑋_𝑛.

According to Lemma 2.7, we know that the above binary nested set can uniquely identify an intuitionistic fuzzy set contained in 𝐼𝐹𝑆𝑠(𝑋₁×𝑋₂×⋯×𝑋_𝑛). Therefore, we have 𝛼,𝛽𝐸_{𝛼,𝛽∈𝐼2}⁽¹⁾_𝛼,𝛽×𝐸⁽²⁾_𝛼,𝛽×⋯×𝐸^(𝑛)_𝛼,𝛽∈𝐼𝐹𝑆𝑠𝑋₁×𝑋₂×⋯×𝑋_𝑛, (3.8) and it satisfies 𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛)_{⟨𝛼⋅,𝛽⋅⟩}⊆𝐸⁽¹⁾_𝛼,𝛽×𝐸⁽²⁾_𝛼,𝛽×…×𝐸^(𝑛)_𝛼,𝛽⊆𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛)_{⟨𝛼,𝛽⟩}. (3.9)

For the membership function and nonmembership function of the Cartesian product over IFSs, we have the following theorem.

Theorem 3.2. Let 𝐸^(𝑘)∈𝐼𝐹𝑆𝑠(𝑋_𝑘)(𝑘=1,2,…,𝑛), then 𝜇_{𝐸(1)×𝐸(2)×⋯×𝐸(𝑛)}𝑥₁,𝑥₂,…,𝑥_𝑛=^𝑛_𝑘=1𝜇_𝐸(𝑘)𝑥_𝑘,𝜈_𝐸⁽¹⁾_×𝐸⁽²⁾_×⋯×𝐸^(𝑛)𝑥₁,𝑥₂,…,𝑥_𝑛=^𝑛_𝑘=1𝜈_𝐸^(𝑘)𝑥_𝑘. (3.10)(3.11)

Proof. First of all, we prove the membership function of Cartesian product over IFSs. For all (𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋=𝑋₁×𝑋₂×⋯×𝑋_𝑛, according to Definitions 2.2 and 3.1, we can obtain 𝜇_{𝐸(1)×𝐸(2)×⋯×𝐸(𝑛)}𝑥₁,𝑥₂,…,𝑥_𝑛=_{𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸⁽¹⁾_{⟨𝛼,𝛽⟩×𝐸}⁽²⁾_{⟨𝛼,𝛽⟩×⋯×𝐸}^(𝑛)_{⟨𝛼,𝛽⟩}𝑥₁,𝑥₂,…,𝑥_𝑛=_{𝛼,𝛽∈𝐼2}𝛼∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.12) Now, we will prove that _{𝛼,𝛽∈𝐼2}𝛼∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘=^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.13) Since _{𝛼,𝛽∈𝐼2}𝛼∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≤_{𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘,⇒_{𝛼,𝛽∈𝐼2}𝛼∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≤^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.14) Assume that ⋁_{⟨𝛼,𝛽⟩∈𝐼2}(𝛼∧(⋀^𝑛_𝑘=1(𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}(𝑥_𝑘))))<⋀^𝑛_𝑘=1(⋁_{⟨𝛼,𝛽⟩∈𝐼2}(𝛼∧𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}(𝑥_𝑘))), then there exists a constant 𝜆 such that _{𝛼,𝛽∈𝐼2}𝛼∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘<𝜆<^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.15) Since ⋀^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝜆,𝛽⟩}(𝑥_𝑘)={0,1}, if ⋀^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝜆,𝛽⟩}(𝑥_𝑘)=1, then 𝜆>_{𝛼,𝛽∈𝐼2}𝛼∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≥𝜆∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝜆,𝛽⟩}𝑥_𝑘=𝜆. (3.16) In addition, if ⋀^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝜆,𝛽⟩}(𝑥_𝑘)=0, then there exists a constant 𝑘₀(1≤𝑘₀≤𝑛) such that 𝜇_𝐸0^)(𝑘_{⟨𝜆,𝛽⟩}(𝑥_𝑘0)=0. By Definition 2.1, we know that 𝜇_𝐸0)(𝑘(𝑥_𝑘0)<𝜆 or 𝜈_𝐸0^)(𝑘(𝑥_𝑘0)>𝛽, and then for all 𝛼>𝜆, we have 𝜇_𝐸0^)(𝑘_{⟨𝛼,𝛽⟩}(𝑥_𝑘0)=0.
Hence, 𝜆<^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≤_{𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸0^)(𝑘_{⟨𝛼,𝛽⟩}𝑥_𝑘0≤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 𝜇_{𝐸(1)×𝐸(2)×⋯×𝐸(𝑛)}𝑥₁,𝑥₂,…,𝑥_𝑛=_{𝛼,𝛽∈𝐼2}𝛼∧^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘=^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛼∧𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘=^𝑛_𝑘=1𝜇_𝐸(𝑘)𝑥_𝑘. (3.18) Now we start to prove the expression (3.11). Analogously, we have 𝜈_𝐸⁽¹⁾_×𝐸⁽²⁾_×⋯×𝐸^(𝑛)𝑥₁,𝑥₂,…,𝑥_𝑛=_{𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸⁽¹⁾_{⟨𝛼,𝛽⟩×𝐸}⁽²⁾_{⟨𝛼,𝛽⟩×⋯×𝐸}^(𝑛)_{⟨𝛼,𝛽⟩}𝑥₁,𝑥₂,…,𝑥_𝑛=_{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.19) Next, we will prove that _{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘=^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.20) Since _{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≥_{𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘⇒_{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≥^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.21) Similarly, we assume that _{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘>^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘, (3.22) then there exists a constant 𝛾 such that _{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘>𝛾>^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘. (3.23) Since ⋁^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛾⟩}(𝑥_𝑘)={0,1}, if ⋁^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛾⟩}(𝑥_𝑘)=0, then 𝛾<_{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≤𝛾∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘=𝛾. (3.24) On the other hand, if ⋁^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛾⟩}(𝑥_𝑘)=1, then there exists a constant 𝑘₁(1≤𝑘₁≤𝑛) such that 𝜈_𝐸1^)(𝑘_{⟨𝛼,𝛾⟩}(𝑥_𝑘1)=1, that is, 𝜇_𝐸1)(𝑘(𝑥_𝑘1)>𝛼 and 𝜈_𝐸1^)(𝑘(𝑥_𝑘1)<𝛾. Therefore, for all 𝛽>𝛾, we have 𝜈_𝐸1^)(𝑘_{⟨𝛼,𝛽⟩}(𝑥_𝑘1)=1.
So we can obtain 𝛾>^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘≥_{𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸1^)(𝑘_{⟨𝛼,𝛽⟩}𝑥_𝑘1≥𝛾. (3.25) 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 𝜈_𝐸⁽¹⁾_×𝐸⁽²⁾_×⋯×𝐸^(𝑛)𝑥₁,𝑥₂,…,𝑥_𝑛=_{𝛼,𝛽∈𝐼2}𝛽∨^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘=^𝑛_{𝑘=1𝛼,𝛽∈𝐼2}𝛽∨𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}𝑥_𝑘=^𝑛_𝑘=1𝜈_𝐸^(𝑘)𝑥_𝑘. (3.26)

Remark 3.3. According to the poof, one can see that the notations ⋁_{⟨𝛼,𝛽⟩} and ⋀^𝑛_𝑘=1, ⋀_{⟨𝛼,𝛽⟩} and ⋁^𝑛_𝑘=1 are commutative, respectively. The reason is that ⋀^𝑛_𝑘=1𝜇_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}(𝑥_𝑘) and ⋁^𝑛_𝑘=1𝜈_𝐸^(𝑘)_{⟨𝛼,𝛽⟩}(𝑥_𝑘) 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 𝐸^(𝑘)∈𝐼𝐹𝑆𝑠(𝑋_𝑘)(𝑘=1,2,…,𝑛), ⟨𝛼,𝛽⟩∈𝐼², then(i)(𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛))_{⟨𝛼,𝛽⟩}=𝐸⁽¹⁾_{⟨𝛼,𝛽⟩}×𝐸⁽²⁾_{⟨𝛼,𝛽⟩}×⋯×𝐸^(𝑛)_{⟨𝛼,𝛽⟩}; (ii)(𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛))_{⟨𝛼⋅,𝛽⋅⟩}=𝐸⁽¹⁾_{⟨𝛼⋅,𝛽⋅⟩}×𝐸⁽²⁾_{⟨𝛼⋅,𝛽⋅⟩}×⋯×𝐸^(𝑛)_{⟨𝛼⋅,𝛽⋅⟩}; (iii)(𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛))_{⟨𝛼⋅,𝛽⟩}=𝐸⁽¹⁾_{⟨𝛼⋅,𝛽⟩}×𝐸⁽²⁾_{⟨𝛼⋅,𝛽⟩}×⋯×𝐸^(𝑛)_{⟨𝛼⋅,𝛽⟩}; (iv)(𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛))_{⟨𝛼,𝛽⋅⟩}=𝐸⁽¹⁾_{⟨𝛼,𝛽⋅⟩}×𝐸⁽²⁾_{⟨𝛼,𝛽⋅⟩}×⋯×𝐸^(𝑛)_{⟨𝛼,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,…,𝑥_𝑛∈𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛)_{⟨𝛼,𝛽⟩}⟺^𝑛_𝑘=1𝜇_𝐸(𝑘)𝑥_𝑘≥𝛼,^𝑛_𝑘=1𝜈_𝐸^(𝑘)𝑥_𝑘≤𝛽⟺𝜇_𝐸(𝑘)𝑥_𝑘≥𝛼,𝜈_𝐸^(𝑘)𝑥_𝑘≤𝛽(𝑘=1,2,…,𝑛)⟺𝑥_𝑘∈𝐸^(𝑘)_𝛼,𝛽()𝑥𝑘=1,2,…,𝑛₁,𝑥₂,…,𝑥_𝑛∈𝐸⁽¹⁾_𝛼,𝛽×𝐸⁽²⁾_𝛼,𝛽×⋯×𝐸^(𝑛)_𝛼,𝛽. (3.27) Hence, the equality holds, namely, 𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛)_{⟨𝛼,𝛽⟩}=𝐸⁽¹⁾_𝛼,𝛽×𝐸⁽²⁾_𝛼,𝛽×⋯×𝐸^(𝑛)_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 𝐸^(𝑘)∈𝐼𝐹𝑆𝑠(𝑋_𝑘)(𝑘=1,2,…,𝑛), then the operation of Cartesian product may be viewed as a mapping, which is defined as follows ×∶𝐼𝐹𝑆𝑠𝑋₁×𝐼𝐹𝑆𝑠𝑋₂×⋯×𝐼𝐹𝑆𝑠𝑋_𝑛→𝐼𝐹𝑆𝑠𝑋₁×𝑋₂×⋯×𝑋_𝑛,𝐸⁽¹⁾,𝐸⁽²⁾,…,𝐸^(𝑛)↦𝐸⁽¹⁾×𝐸⁽²⁾×⋯×𝐸^(𝑛). (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×⋯×𝑋_𝑛→𝑌=𝑌₁×𝑌₂×⋯×𝑌_𝑚,𝑥₁,𝑥₂,…,𝑥_𝑛↦𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑦=𝑦₁,𝑦₂,…,𝑦_𝑚. (4.2)

By the extension principle of IFSs (Lemma 2.8), the mapping 𝑓 can be induced to the following two mappings. 𝑓:𝐼𝐹𝑆𝑠(𝑋)→𝐼𝐹𝑆𝑠(𝑌),𝐴↦𝑓(𝐴)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴_{⟨𝛼,𝛽⟩},𝑓⁻¹:𝐼𝐹𝑆𝑠(𝑌)→𝐼𝐹𝑆𝑠(𝑋),𝐵↦𝑓⁻¹(𝐵)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓⁻¹𝐵_{⟨𝛼,𝛽⟩}. (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×⋯×𝐼𝐹𝑆𝑠𝑋_𝑛(𝑋),𝐴→𝐼𝐹𝑆𝑠₍₁₎,𝐴⁽²⁾,…,𝐴^(𝑛)↦𝐴⁽¹⁾×𝐴⁽²⁾×⋯×𝐴^(𝑛),×₂:𝐼𝐹𝑆𝑠𝑌₁×𝐼𝐹𝑆𝑠𝑌₂×⋯×𝐼𝐹𝑆𝑠𝑌_𝑚(𝑌),𝐵→𝐼𝐹𝑆𝑠₍₁₎,𝐵⁽²⁾,…,𝐵^(𝑚)↦𝐵⁽¹⁾×𝐵⁽²⁾×⋯×𝐵^(𝑚). (4.4)

Based on the above compound operations, we will get the following definition about multivariate extension principle of IFSs.

Definition 4.1. Let 𝐴^(𝑘)∈𝐼𝐹𝑆𝑠(𝑋_𝑘)(𝑘=1,2,…,𝑛), 𝐵^(𝑙)∈𝐼𝐹𝑆𝑠(𝑌_𝑙)(𝑙=1,2,…,𝑚), and 𝑓 be a mapping from 𝑋 to 𝑌, that is, 𝑓:𝑋=𝑋₁×𝑋₂×⋯×𝑋_𝑛→𝑌=𝑌₁×𝑌₂×⋯×𝑌_𝑚,𝑥₁,𝑥₂,…,𝑥_𝑛↦𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑦=𝑦₁,𝑦₂,…,𝑦_𝑚. (4.5) The two induced mappings of 𝑓 can be defined as follows: 𝑓:𝐼𝐹𝑆𝑠𝑋₁×𝐼𝐹𝑆𝑠𝑋₂×⋯×𝐼𝐹𝑆𝑠𝑋_𝑛(𝑌),𝐴→𝐼𝐹𝑆𝑠₍₁₎,𝐴⁽²⁾,…,𝐴^(𝑛)↦𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴=^(𝑛)def𝑓𝐴⁽¹⁾×𝐴⁽²⁾×⋯×𝐴^(𝑛),𝑓⁻¹:𝐼𝐹𝑆𝑠𝑌₁×𝐼𝐹𝑆𝑠𝑌₂×⋯×𝐼𝐹𝑆𝑠𝑌_𝑚(𝑋),𝐵→𝐼𝐹𝑆𝑠₍₁₎,𝐵⁽²⁾,…,𝐵^(𝑚)↦𝑓⁻¹𝐵⁽¹⁾,𝐵⁽²⁾,…,𝐵^(𝑚)_def=𝑓⁻¹𝐵⁽¹⁾×𝐵⁽²⁾×⋯×𝐵^(𝑚). (4.6)
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 𝜇_{𝑓(𝐴(1),𝐴(2),…,𝐴(𝑛))}𝑦=_{𝑓(𝑥1,𝑥2,…,𝑥𝑛)=𝑦}^𝑛_𝑘=1𝜇_𝐴(𝑘)𝑥_𝑘,𝜈_𝑓(𝐴⁽¹⁾_,𝐴⁽²⁾_,…,𝐴^(𝑛)₎𝑦=_{𝑓𝑥1,𝑥2,…,𝑥𝑛=𝑦}^𝑛_𝑘=1𝜈_𝐴^(𝑘)𝑥_𝑘,𝜇_𝑓⁻¹_{(𝐵(1),𝐵(2),…,𝐵(𝑚))}(𝑥)=^𝑚_𝑙=1𝜇_𝐵(𝑙)𝑦_𝑙,𝜈_𝑓⁻¹_{(𝐵(1),𝐵(2),…,𝐵(𝑚))}(𝑥)=^𝑚_𝑙=1𝜈_𝐵^(𝑙)𝑦_𝑙, (4.7) where (𝑦₁,𝑦₂,…,𝑦_𝑚)=𝑓(𝑥).

Below we will discuss some other forms of the multivariate extension principle of IFSs.

Theorem 4.3 (multivariate extension principle I). 𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴^(𝑛)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴⁽¹⁾_𝛼,𝛽,𝐴⁽²⁾_𝛼,𝛽,…,𝐴^(𝑛)_𝛼,𝛽,𝑓⁻¹𝐵⁽¹⁾,𝐵⁽²⁾,…,𝐵^(𝑚)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓⁻¹𝐵⁽¹⁾_𝛼,𝛽,𝐵⁽²⁾_𝛼,𝛽,…,𝐵^(𝑚)_𝛼,𝛽. (4.8)

Proof. By Lemma 2.8 and Definition 4.1, we have 𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴^(𝑛)=𝑓𝐴⁽¹⁾×𝐴⁽²⁾×⋯×𝐴^(𝑛)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴⁽¹⁾×𝐴⁽²⁾×⋯×𝐴^(𝑛)_{⟨𝛼,𝛽⟩}=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴⁽¹⁾_𝛼,𝛽×𝐴⁽²⁾_𝛼,𝛽×⋯×𝐴^(𝑛)_𝛼,𝛽. (4.9) Since 𝑓𝐴⁽¹⁾_𝛼,𝛽,𝐴⁽²⁾_𝛼,𝛽,…,𝐴^(𝑛)_𝛼,𝛽=𝑦∶∃𝑥_𝑘∈𝐴^(𝑘)_𝛼,𝛽(𝑘=1,2,…,𝑛),𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑥=𝑦𝑦∶∃₁,𝑥₂,…,𝑥_𝑛∈𝐴⁽¹⁾_𝛼,𝛽×𝐴⁽²⁾_𝛼,𝛽×⋯×𝐴^(𝑛)_𝛼,𝛽,𝑓𝑥₁,𝑥₂,…,𝑥_𝑛𝐴=𝑦=𝑓⁽¹⁾_𝛼,𝛽×𝐴⁽²⁾_𝛼,𝛽×⋯×𝐴^(𝑛)_𝛼,𝛽, (4.10) we know that 𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴^(𝑛)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴⁽¹⁾_𝛼,𝛽,𝐴⁽²⁾_𝛼,𝛽,…,𝐴^(𝑛)_𝛼,𝛽. (4.11) Similarly, we can prove the second equality.

Theorem 4.4 (multivariate extension principle II). 𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴^(𝑛)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴_𝛼⁽¹⁾_⋅,𝛽⋅,𝐴_𝛼⁽²⁾_⋅,𝛽⋅,…,𝐴_𝛼^(𝑛)_⋅,𝛽⋅=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴_𝛼⁽¹⁾_⋅,𝛽,𝐴_𝛼⁽²⁾_⋅,𝛽,…,𝐴_𝛼^(𝑛)_⋅,𝛽=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴⁽¹⁾_𝛼,𝛽⋅,𝐴⁽²⁾_𝛼,𝛽⋅,…,𝐴^(𝑛)_𝛼,𝛽⋅,𝑓⁻¹𝐵⁽¹⁾,𝐵⁽²⁾,…,𝐵^(𝑚)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓⁻¹𝐵_𝛼⁽¹⁾_⋅,𝛽⋅,𝐵_𝛼⁽²⁾_⋅,𝛽⋅,…,𝐵_𝛼^(𝑚)_⋅,𝛽⋅=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓⁻¹𝐵_𝛼⁽¹⁾_⋅,𝛽,𝐵_𝛼⁽²⁾_⋅,𝛽,…,𝐵_𝛼^(𝑚)_⋅,𝛽=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓⁻¹𝐵⁽¹⁾_𝛼,𝛽⋅,𝐵⁽²⁾_𝛼,𝛽⋅,…,𝐵^(𝑚)_𝛼,𝛽⋅. (4.12)

Proof. The proof method is similar to that of Theorem 4.3. Thus, we omit it here.

Theorem 4.5 (multivariate extension principle III). 𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴^(𝑛)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐻_𝐴⁽¹⁾𝛼,𝛽,𝐻_𝐴⁽²⁾𝛼,𝛽,…,𝐻_𝐴^(𝑛)𝛼,𝛽, (4.13) where 𝐴^(𝑘)_{⟨𝛼⋅,𝛽⋅⟩}⊆𝐻_𝐴^(𝑘)(𝛼,𝛽)⊆𝐴^(𝑘)_{⟨𝛼,𝛽⟩}(𝑘=1,2,…,𝑛). 𝑓⁻¹𝐵⁽¹⁾,𝐵⁽²⁾,…,𝐵^(𝑚)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓⁻¹𝐻_𝐵⁽¹⁾𝛼,𝛽,𝐻_𝐵⁽²⁾𝛼,𝛽,…,𝐻_𝐵^(𝑚)𝛼,𝛽, (4.14) where 𝐵^(𝑙)_{⟨𝛼⋅,𝛽⋅⟩}⊆𝐻_𝐵^(𝑙)(𝛼,𝛽)⊆𝐵^(𝑙)_{⟨𝛼,𝛽⟩}(𝑙=1,2,…,𝑚).

Proof. Since 𝐴_𝛼^(𝑘)_⋅,𝛽⋅⊆𝐻_𝐴^(𝑘)𝛼,𝛽⊆𝐴^(𝑘)_𝛼,𝛽(𝑘=1,2,…,𝑛)⇒𝐴_𝛼⁽¹⁾_⋅,𝛽⋅×𝐴_𝛼⁽²⁾_⋅,𝛽⋅×⋯×𝐴_𝛼^(𝑛)_⋅,𝛽⋅⊆𝐻_𝐴⁽¹⁾𝛼,𝛽×𝐻_𝐴⁽²⁾𝛼,𝛽×⋯×𝐻_𝐴^(𝑛)𝛼,𝛽⊆𝐴⁽¹⁾_𝛼,𝛽×𝐴⁽²⁾_𝛼,𝛽×⋯×𝐴^(𝑛)_𝛼,𝛽⇒𝑓𝐴⁽¹⁾_𝛼,𝛽,𝐴⁽²⁾_𝛼,𝛽,…,𝐴^(𝑛)_𝛼,𝛽⊆𝑓𝐻_𝐴⁽¹⁾𝛼,𝛽,𝐻_𝐴⁽²⁾𝛼,𝛽,…,𝐻_𝐴^(𝑛)𝐴𝛼,𝛽⊆𝑓⁽¹⁾_𝛼,𝛽,𝐴⁽²⁾_𝛼,𝛽,…,𝐴^(𝑛)_𝛼,𝛽⇒𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴^(𝑛)=_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴_𝛼⁽¹⁾_⋅,𝛽⋅,𝐴_𝛼⁽²⁾_⋅,𝛽⋅,…,𝐴_𝛼^(𝑛)_⋅,𝛽⋅⊆_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐻_𝐴⁽¹⁾𝛼,𝛽,𝐻_𝐴⁽²⁾𝛼,𝛽,…,𝐻_𝐴^(𝑛)⊆𝛼,𝛽_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝑓𝐴⁽¹⁾_𝛼,𝛽,𝐴⁽²⁾_𝛼,𝛽,…,𝐴^(𝑛)_𝛼,𝛽=𝑓𝐴⁽¹⁾,𝐴⁽²⁾,…,𝐴^(𝑛). (4.15) 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 𝐼𝐹𝑆𝑠(𝑅).

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 𝐼𝐹𝑆𝑠(𝑅).

Definition 5.1. Let ∗ be an algebraic operation on 𝑅, then the corresponding algebraic operation on 𝐼𝐹𝑆𝑠(𝑅) is defined as (𝑅)(𝑅)(𝑅),()∗∶𝐼𝐹𝑆𝑠×𝐼𝐹𝑆𝑠→𝐼𝐹𝑆𝑠𝐴,𝐵↦𝐴∗𝐵=^def_{𝛼,𝛽∈𝐼2}𝛼,𝛽𝐴_{⟨𝛼,𝛽⟩}∗𝐵_{⟨𝛼,𝛽⟩},𝐴_{⟨𝛼,𝛽⟩}∗𝐵_{⟨𝛼,𝛽⟩}=𝑧∶∃𝑥,𝑦∈𝐴_{⟨𝛼,𝛽⟩}×𝐵_{⟨𝛼,𝛽⟩},𝑥∗𝑦=𝑧, (5.2) the membership function and nonmembership function of the operational result are given, respectively, by 𝜇∗:^𝐴∗𝐵(𝑧)=⋁_{𝑥∗𝑦=𝑧}𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦,𝜈_𝐴∗𝐵(𝑧)=⋀_{𝑥∗𝑦=𝑧}𝜈_𝐴(𝑥)∨𝜈_𝐵𝑦. (5.3)

By Definition 5.1, we can obtain the following some algebraic operations between IFSs on 𝑅.

(a) Arithmetic operation𝜇+:^𝐴+𝐵(𝑧)=⋁_{𝑥+𝑦=𝑧}𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦=⋁_𝑥∈𝑅𝜇_𝐴(𝑥)∧𝜇_𝐵(),𝜈𝑧−𝑥_𝐴+𝐵(𝑧)=⋀_{𝑥+𝑦=𝑧}𝜈_𝐴(𝑥)∨𝜈_𝐵𝑦=⋀_𝑥∈𝑅𝜈_𝐴(𝑥)∨𝜈_𝐵(,𝜇𝑧−𝑥)−:^𝐴−𝐵(𝑧)=⋁_{𝑥−𝑦=𝑧}𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦=⋁_𝑥∈𝑅𝜇_𝐴(𝑥)∧𝜇_𝐵(),𝜈𝑥−𝑧_𝐴−𝐵(𝑧)=⋀_{𝑥−𝑦=𝑧}𝜈_𝐴(𝑥)∨𝜈_𝐵𝑦=⋀_𝑥∈𝑅𝜈_𝐴(𝑥)∨𝜈_𝐵(),𝜇𝑥−𝑧⋅:^𝐴⋅𝐵(𝑧)=⋁_{𝑥⋅𝑦=𝑧}𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦=_𝑥⋁_∈𝑅𝜇_𝐴(𝑥)∧𝜇_𝐵𝑧𝑥(),𝜇𝑧≠0_𝐴(0)∧⋁_𝑦∈𝑅𝜇_𝐵𝑦∨𝜇_𝐵(0)∧⋁_𝑥∈𝑅𝜇_𝐴(𝑥)(),𝜈𝑧=0_𝐴⋅𝐵(𝑧)=⋀_{𝑥⋅𝑦=𝑧}𝜈_𝐴(𝑥)∨𝜈_𝐵𝑦=_𝑥⋀_∈𝑅𝜈_𝐴(𝑥)∨𝜈_𝐵𝑧𝑥(),𝜈𝑧≠0_𝐴(0)∨⋀_𝑦∈𝑅𝜈_𝐵𝑦∧𝜇_𝐵(0)∨⋀_𝑥∈𝑅𝜈_𝐴(𝑥)().𝜇𝑧=0÷:^𝐴÷𝐵(𝑧)=⋁_{𝑥÷𝑦=𝑧}𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦=⋁_𝑦≠0𝜇_𝐴𝑦⋅𝑧∧𝜇_𝐵𝑦,𝜈_𝐴÷𝐵(𝑧)=⋀_{𝑥÷𝑦=𝑧}𝜈_𝐴(𝑥)∨𝜈_𝐵𝑦=⋀_𝑦≠0𝜈_𝐴𝑦⋅𝑧∨𝜈_𝐵𝑦. (5.4)

(b) Conjunction and disjunction operations𝜇∧:^𝐴∧𝐵(𝑧)=⋁_{𝑥∧𝑦=𝑧}𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦,𝜈_𝐴∧𝐵(𝑧)=⋀_{𝑥∧𝑦=𝑧}𝜈_𝐴(𝑥)∨𝜈_𝐵𝑦,𝜇∨:^𝐴∨𝐵(𝑧)=⋁_{𝑥∨𝑦=𝑧}𝜇_𝐴(𝑥)∧𝜇_𝐵𝑦,𝜈_𝐴∨𝐵(𝑧)=⋀_{𝑥∨𝑦=𝑧}𝜈_𝐴(𝑥)∨𝜈_𝐵𝑦. (5.5)

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).