Hesitant interval-valued fuzzy sets: some new results

Abstract

Hesitant interval-valued fuzzy set is an extension of hesitant fuzzy set. It permits the membership degree of an element to a set to be represented as several possible interval numbers. The study of hesitant interval-valued fuzzy sets has opened a new area of research and applications. A number of mathematical results have been introduced and proved to enhance the applicability range of hesitant interval-valued fuzzy sets to wider areas. In this paper, we propose four new operations on hesitant interval valued fuzzy sets and study their properties and relations in details.

1 Introduction

Torra and Narukawa [21] and Torra [22] proposed the concept of hesitant fuzzy sets, as a new generalization of the notion of classical fuzzy sets [43], permitting an element to have a set of several possible membership values. They also discussed the relationships among hesitant fuzzy sets and other generalizations of fuzzy sets such as intuitionistic fuzzy sets [1, 2], type-2 fuzzy sets [8], and fuzzy multisets [41]. Hesitant fuzzy set can reflect the human’s hesitancy more objectively than the other classical extensions of fuzzy sets. Recently, a number of studies have focused on hesitant fuzzy sets and their applications [4, 9–20, 23–25, 30–40, 42, 44–53].

However, in many real decision making problems, the preference information provided by decision-maker is often imprecise or uncertain due to the increasing complexity of social-economic environment and lack of data about the problem domains. It may as well be due to an expert not having enough expertise or possess sufficient level of knowledge to precisely express their preferences over the objects considered. He/she has some uncertainty in providing his/her preferences. To handle such type of situations, Chen et al. [5, 7] introduced the notion of hesitant interval-valued fuzzy sets, permitting an element to have a set of several possible interval membership values. Some operations on hesitant interval valued fuzzy sets have been proposed by Chen et al. [5]. Wei et al. [29] developed some hesitant interval valued fuzzy information aggregation operators based on arithmetic and geometric means and then studied some desirable properties of these operators. Further, apart from study of aggregation operators, Chen at el. [7] studied preference relations under hesitant interval valued environment. Wei et al. [27] came out with a detailed study on distance and similarity measures for hesitant interval-valued fuzzy sets. Chen et al. [5] proposed correlation coefficients for hesitant fuzzy sets and hesitant interval valued fuzzy sets and studied their applications in clustering analysis. Bai [3] proposed some distance similarity measures for hesitant interval valued fuzzy sets and found their application in multicriteria decision making. Wei and Zhao [26] developed some hesitant interval valued fuzzy information aggregation operators based on Einstein operation on hesitant fuzzy set. Wei et al. [28] developed some models for hesitant interval valued fuzzy multiple attribute decision making based on correlation coefficient with incomplete weight information. Recently, Chen and Xu [6] gave a detailed study on properties of hesitant interval valued fuzzy sets. Broumi and Smarandache [4] proposed some new operations on interval valued intuitionistic hesitant fuzzy sets. Liao et al. [17] developed a qualitative decision making method based on hesitant fuzzy linguistic term sets. Merigó et al. [18] presented an overview of fuzzy research with bibliometric indicators.

These approaches employ different ways of defining measures in terms of aggregation operators. The important next step is the study of sets arises by coming set operations and then operations on elements of a set. In this paper, we first propose four new operations on hesitant interval valued fuzzy sets and study their properties.

The paper is organized as follows: In Sect. 2, some basic definitions related to hesitant fuzzy sets and hesitant interval valued fuzzy sets are briefly discussed. In Sect. 3, four new operations on hesitant interval valued fuzzy sets are proposed with their properties. Section 4 concludes the paper with some remarks.

2 Preliminaries

We start with the concept of hesitant fuzzy sets (HFSs) introduced by Torra and Narukawa [21] and Torra [22]. An HFS permits the membership degree of an element to be a set of several possible membership values between 0 and 1. This better describes the situations where a set of people have differing hesitancy in providing their preferences over objects in the process of decision making.

Definition 1

(Hesitant fuzzy set [22]): Let $X = \left\{ {x_{1} ,\,x_{2} , \ldots ,x_{n} } \right\}$ be a reference set, a set $E$ defined on $X$ given by

$$E = \left\{ \left\langle x,\, h_{E} \left( x \right) \right\rangle |x \in X \right\},$$

(1)

where $h_{E} \left( x \right)$ is a set of some different values in $\left[ {0,\,1} \right]$, denoting the possible membership degrees of the element $x \in X$ to the set $E$, is called a hesitant fuzzy set.

Example 1

Let $X = \left\{ {x_{1} ,x_{2} ,x_{3} } \right\}$ be a reference set. Also let $h_{{^{E} }} \left( {x_{1} } \right) = \left\{ {0.2,0.4} \right\}$, $h_{{^{E} }} \left( {x_{2} } \right) = \left\{ {0.5,0.7,0.8} \right\}$ and $h_{{^{{\tilde{E}}} }} \left( {x_{3} } \right) = \left\{ {0.3,0.6} \right\}$ denote the membership degree sets of $x_{i}$, $\left( {i = 1,2,3} \right)$ to the set $E$ respectively. Then $E$, an HFS is expressed as

$$E = \left\{ {\left\langle {x_{1} ,\left\{ {0.2,0.4} \right\}} \right\rangle ,\left\langle {x_{2} ,\left\{ {0.5,0.7,0.8} \right\}} \right\rangle ,\left\langle {x_{3} ,\left\{ {0.3,0.6} \right\}} \right\rangle } \right\}.$$

Further, Torra [22] defined the ‘empty hesitant fuzzy set’ and the ‘full hesitant fuzzy set’ as follows:

$$E^{ \circ } = \left\{ {\left\langle {x,\,h_{{E^{ \circ } }} \left( x \right)} \right\rangle |x \in X} \right\},\quad {\text{where }}h_{{E^{ \circ } }} \left( x \right) = \left\{ 0 \right\}\,\,\forall \,\,\,x \in X$$

(2)

$$E^{ * } = \left\{ {\left\langle {x,\,h_{{E^{ * } }} \left( x \right)} \right\rangle |x \in X} \right\},\quad {\text{where }}h_{{E^{ * } }} \left( x \right) = \left\{ 1 \right\}\,\,\forall \,\,\,x \in X.$$

(3)

For convenience, Xia and Xu [33] named the set $h = h_{E} \left( x \right)$ as the hesitant fuzzy element (HFE) and let $HFE\left( X \right)$ represents the family of all hesitant fuzzy elements defined on $X$.

Definition 2

(Operations on HFEs): Let $h,h_{1} \,{\text{and}}\,h_{2} \in HFE\left( X \right)$, Torra [22] and Xia and Xu [33] defined the following operations:

• (i) $h^{C} = \left\{ {1 - \gamma |\gamma \in h} \right\};$

• (ii) $h_{1} \cup h_{2} = \,\left\{ {\left( {\gamma_{1} \vee \gamma_{2} } \right)\,|\gamma_{1} \in h_{1} ,\gamma_{2} \in h_{2} } \right\};$

• (iii) $h_{1} \cap h_{2} = \,\left\{ {\left( {\gamma_{1} \wedge \gamma_{2} } \right)\,|\gamma_{1} \in h_{1} ,\gamma_{2} \in h_{2} } \right\};$

• (iv) $h^{\lambda } = \,\,\,\left\{ {\left( \gamma \right)^{\lambda } |\gamma \in h} \right\},\lambda > 0;$

• (v) $\lambda h = \,\,\,\left\{ {1 - \left( {1 - \gamma } \right)^{\lambda } |\gamma \in h} \right\},\lambda > 0;$

• (vi) $h_{1} \oplus h_{2} = \,\left\{ {\gamma_{1} + \gamma_{2} - \gamma_{1} \gamma_{2} \,|\gamma_{1} \in h_{1} ,\gamma_{2} \in h_{2} } \right\};$

• (vii) $h_{1} \otimes h_{2} = \,\left\{ {\gamma_{1} \gamma_{2} \,|\gamma_{1} \in h_{1} ,\gamma_{2} \in h_{2} } \right\}.$

Recently, Verma and Sharma [24] proposed the four new operations on HFEs.

Definition 3

Let $h,h_{1} \,{\text{and}}\,h_{2} \in HFE\left( X \right)$, Verma and Sharma [24] defined the following operations:

• (i) $h_{1} @h_{2} = \cup_{{\gamma_{1} \in h_{1},\gamma_{2} \in h_{2} }} \left\{ {\frac{{\left( {\gamma_{1} + \gamma_{2} } \right)}}{2}} \right\};$

• (ii) $h_{1} \$ h_{2} = \cup_{{\gamma_{1} \in h_{1},\gamma_{2} \in h_{2} }} \left\{ {\sqrt {\gamma_{1} \gamma_{2} } } \right\};$

• (iii) $h_{1} \# h_{2} = \,\,\, \cup_{{\gamma_{1} \in h_{1},\gamma_{2} \in h_{2} }} \left\{ {\frac{{2\gamma_{1} \gamma_{2} }}{{\left( {\gamma_{1} + \gamma_{2} } \right)}}} \right\};$

for which we shall accept that if $\gamma_{1} = \gamma_{2} = 0$, then $\frac{{2\gamma_{1} \gamma_{2} }}{{\gamma_{1} + \gamma_{2} }} = 0.$

• (iv) $h_{1} * h_{2} = \,\,\, \cup_{{\gamma \in h_{1} }} ,_{{\gamma_{2} \in h_{2} }} \left\{ {\frac{{\gamma_{1} + \gamma_{2} }}{{2\left( {\gamma_{1} \gamma_{2} + 1} \right)}}} \right\}\,\,.$

As explained in introduction, the precise membership degrees of an element to a set are sometimes hard to be specified. To overcome this issue, Chen et al. [5] introduced the concept of hesitant interval-valued fuzzy sets (HIVFSs) that represent the membership degrees of an element to a set with several possible interval values on $\left[ {0,\,1} \right]$.

Definition 4

(Hesitant interval-valued fuzzy set [5]): Let $X = \left\{ {x_{1} ,\,x_{2} , \ldots ,x_{n} } \right\}$ be a reference set and $D\left[ {0,\,1} \right]$ be the set of all closed subintervals of $\left[ {0,\,1} \right]$, a set $\tilde{E}$ defined on $X$ given by

$$\tilde{E} = \left\{ {\left\langle {x,\,\tilde{h}_{{\tilde{E}}} \left( x \right)} \right\rangle |x \in X} \right\}$$

(4)

where $\tilde{h}_{{\tilde{E}}} \left( x \right)$ is a set of some different interval values on $\left[ {0,\,1} \right]$, representing the possible membership degrees of the element $x \in X$ to the set $\tilde{E}$, is called an hesitant interval-valued fuzzy set.

Example 2

Let $X = \left\{ {x_{1} ,x_{2} ,x_{3} } \right\}$ be a reference set. Also let $\tilde{h}_{{^{{\tilde{E}}} }} \left( {x_{1} } \right) = \left\{ {\left[ {0.4,0.6} \right]} \right\}$, $\tilde{h}_{{^{{\tilde{E}}} }} \left( {x_{2} } \right) = \left\{ {\left[ {0.2,0.3} \right],\left[ {0.5,0.7} \right],\left[ {0.6,0.8} \right]} \right\}$ and $\tilde{h}_{{^{{\tilde{E}}} }} \left( {x_{3} } \right) = \left\{ {\left[ {0.3,0.4} \right],\left[ {0.5,0.7} \right]} \right\}$ denote the membership degree sets of $x_{i} \left( {i = 1,2,3} \right)$ to the set $\tilde{E}$ respectively. Then $\tilde{E}$ is an HIVFS and express it by

$$\tilde{E} = \left\{ {\left\langle {x_{1} ,\left\{ {\left[ {0.4,0.6} \right]} \right\}} \right\rangle ,\left\langle {x_{2} ,\left\{ {\left[ {0.2,0.3} \right],\left[ {0.5,0.7} \right],\left[ {0.6,0.8} \right]} \right\}} \right\rangle ,\left\langle {x_{3} ,\left\{ {\left[ {0.3,0.4} \right],\left[ {0.5,0.7} \right]} \right\}} \right\rangle } \right\}.$$

Chen and Xu [6] defined the ‘empty hesitant interval-valued fuzzy set’ and the ‘full hesitant interval-valued fuzzy set’ as follows:

$$\begin{aligned} \tilde{E}^{ \circ } &= \left\{ {\left\langle {x,\,\tilde{h}_{{\tilde{E}^{ \circ } }} \left( x \right)} \right\rangle |x \in X} \right\}, \nonumber\\ &\quad\quad\quad {\text{where }}\tilde{h}_{{\tilde{E}^{ \circ } }} \left( x \right) = \left\{ {\,\left[ {0,0} \right]\,} \right\} \quad \forall x \in X, \end{aligned}$$

(5)

$$\begin{aligned} \tilde{E}^{ * } &= \left\{ {\left\langle {x,\,\tilde{h}_{{\tilde{E}^{ * } }} \left( x \right)} \right\rangle |x \in X} \right\},\nonumber \\ &\quad\quad\quad {\text{where }}\tilde{h}_{{\tilde{E}^{ * } }} \left( x \right) = \left\{ {\,\left[ {1,1} \right]\,} \right\}\quad \forall x \in X. \end{aligned}$$

(6)

For convenience, Chen et al. [5] named the set $\tilde{h}_{{\tilde{E}}} \left( x \right)$ as the hesitant interval-valued fuzzy element (HIVFE) and denoted by

$$\tilde{h}_{{\tilde{E}}} \left( x \right) = \left\{ {\tilde{\gamma }|\tilde{\gamma } \in \tilde{h}_{{\tilde{E}}} \left( x \right)} \right\}$$

(7)

where $\tilde{\gamma } = \left[ {\tilde{\gamma }^{L} ,\tilde{\gamma }^{U} } \right]$ is an interval number. $\tilde{\gamma }^{L} = \inf \tilde{\gamma }$ and $\tilde{\gamma }^{L} = \sup \tilde{\gamma }$ represent the lower and upper limits of $\tilde{\gamma }$, respectively. An HIVFE is the basic unit of an HIVFS and it can be considered as a special case of the HIVFS. The relationship between HIVFE and HIVFS is similar to that between interval-valued fuzzy number and interval-valued fuzzy sets. Let $HIVFE\left( X \right)$ represents the family of all hesitant interval-valued fuzzy elements defined on $X$.

In addition, Chen et al. [5] defined the following seven operations on HIVFE and studied their properties.

Definition 5

(Operations on HIVFEs): Let $\tilde{h}$,$\tilde{h}_{1}$ and $\tilde{h}_{2}$ be three HIVFEs, then

• (i) $\tilde{h}^{C} = \left\{ {\left[ {1 - \tilde{\gamma }^{U} ,1 - \tilde{\gamma }^{L} } \right]|\tilde{\gamma } \in \tilde{h}} \right\};$

• (ii) $\begin{aligned} \tilde{h}_{1} \cup \tilde{h}_{2} &= \bigg\{ \left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{1}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{1}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} , \\ &\quad \tilde{\gamma }_{2} \in \tilde{h}_{2} \bigg\}; \end{aligned}$

• (iii) $\begin{aligned} \tilde{h}_{1} \cap \tilde{h}_{2} &= \bigg\{ \left[ {\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{1}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{1}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} , \\ &\quad \tilde{\gamma }_{2} \in \tilde{h}_{2} \bigg\}; \end{aligned}$

• (iv) $\tilde{h}^{\lambda } = \left\{ {\left[ {\left( {\tilde{\gamma }^{L} } \right)^{\lambda } ,\,\left( {\tilde{\gamma }^{U} } \right)^{\lambda } } \right]|\tilde{\gamma } \in \tilde{h}} \right\},\lambda > 0;$

• (v) $\begin{aligned} \lambda \tilde{h} &= \bigg\{ \left[ {1 - \left( {1 - \tilde{\gamma }^{L} } \right)^{\lambda } ,\,1 - \left( {1 - \tilde{\gamma }^{U} } \right)^{\lambda } } \right] \\ &\quad\quad\quad\quad\quad |\tilde{\gamma } \in \tilde{h} \bigg\},\lambda > 0; \end{aligned}$

• (vi) $\begin{aligned} \tilde{h}_{1} \oplus \tilde{h}_{2} &= \bigg\{ \left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{1}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{1}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{1}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{1}^{U} } \right] \\ &\quad |\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} \bigg\} \end{aligned}$

• (vii) $\tilde{h}_{1} \otimes \tilde{h}_{2} = \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{1}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{1}^{U} } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}.$

Remark:

It is noted that if $\tilde{\gamma }^{L} = \tilde{\gamma }^{U}$, then the operations in Definition 4 reduce to those of HFEs.

In the next section, we define four new operations on hesitant interval-valued fuzzy sets analogous to operations defined in Definition 3 and study their properties.

3 Four new operations on hesitant inter-valued fuzzy sets and their properties

Definition 6:

For $\tilde{h}_{1} ,\tilde{h}_{2} \in HIVFE\left( X \right)$, we propose the following operations on HIVFEs as follows

• (i) $\tilde{h}_{1} @\tilde{h}_{2} = \,\left\{ {\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$

• (ii) $\tilde{h}_{1} \$ \tilde{h}_{2} = \,\left\{ {\,\left[ {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } ,\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\};$

• (iii) $\tilde{h}_{1} \# \tilde{h}_{2} = \,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$

for which we shall accept that if $\tilde{\gamma }_{1} = \tilde{\gamma }_{2} = \left[ {0,0} \right]$, then $\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }} = 0$ and $\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }} = 0.$

• (iv) $\begin{aligned} \tilde{h}_{1} * \tilde{h}_{2} &= \left\{ \left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{{2\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} + 1} \right)}},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{{2\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} + 1} \right)}}} \right] \right. \\ &\quad \left. \vphantom{\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{{2\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} + 1} \right)}},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{{2\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} + 1} \right)}}} \right]} |\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} \right\}. \end{aligned}$

Obviously, for every two HIVFEs $\tilde{h}_{1} \,\,{\text{and}}\,\,\tilde{h}_{2}$, $\tilde{h}_{1} @\tilde{h}_{2} ,\,\tilde{h}_{1} \$ \tilde{h}_{2}$, $\tilde{h}_{1} \# \tilde{h}_{2} \,\,{\text{and}}\,\,\tilde{h}_{1} *\tilde{h}_{2}$ are also an HIVFE.

Remark:

It is noted that if $\tilde{\gamma }^{L} = \tilde{\gamma }^{U}$, then the operations in Definition 6 reduce to those of HFEs.

Example 3

Let $\tilde{h}_{1} \left( x \right) = \left\{ {\left[ {0.3,0.5} \right],\left[ {0.2,0.6} \right]} \right\}$, and $\tilde{h}_{2} \left( x \right) = \left\{ {\left[ {0.6,0.8} \right]} \right\}$ be two hesitant interval valued fuzzy elements. Then we have

• $\begin{aligned} \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)\left( x \right) &= \left\{\left[ {\frac{0.3 + 0.6}{2},\frac{0.5 + 0.8}{2}} \right], \right. \\ &\quad \left. \left[ {\frac{0.2 + 0.6}{2},\frac{0.6 + 0.8}{2}} \right] \right\} \\ & = \left\{ {\left[ {0.4500,0.6500} \right],\left[ {0.4000,0.7000} \right]} \right\}. \hfill \\ \end{aligned}$

• $\begin{aligned} \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)\left( x \right) &= \left\{ \left[ {\sqrt {0.3 \times 0.6} ,\sqrt {0.5 \times 0.8} } \right], \right. \\ &\quad \left. \left[ {\sqrt {0.2 \times 0.6} ,\sqrt {0.6 \times 0.8} } \right] \right\} \hfill \\ & = \left\{ {\left[ {0.4243,0.6325} \right],\left[ {0.3464,0.6928} \right]} \right\}. \hfill \\ \end{aligned}$

• $\begin{aligned} \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)\left( x \right) &= \left\{ \left[ {\frac{2 \times 0.3 \times 0.6}{0.3 + 0.6},\frac{2 \times 0.5 \times 0.8}{0.5 + 0.8}} \right], \right. \\ &\quad \left. \left[ {\frac{2 \times 0.2 \times 0.6}{0.2 + 0.6},\frac{2 \times 0.6 \times 0.8}{0.6 + 0.8}} \right] \right\} \hfill \\ &= \left\{ {\left[ {0.4000,0.6154} \right],\left[ {0.3000,0.6857} \right]} \right\}. \hfill \\ \end{aligned}$

• $\begin{aligned} \left( {\tilde{h}_{1} * \tilde{h}_{2} } \right)\left( x \right) &= \left\{ \left[ {\frac{0.3 + 0.6}{{2 \times \left( {0.3 \times 0.6 + 1} \right)}},\frac{0.5 + 0.8}{{2 \times \left( {0.5 \times 0.8 + 1} \right)}}} \right], \right. \\ &\quad \left. \left[ {\frac{0.2 + 0.6}{{2 \times \left( {0.2 \times 0.6 + 1} \right)}},\frac{0.6 + 0.8}{{2 \times \left( {0.6 \times 0.8 + 1} \right)}}} \right] \right\} \hfill \\ &= \left\{ {\left[ {0.3814,0.4643} \right],\left[ {0.3571,0.4730} \right]} \right\}. \hfill \\ \end{aligned}$

Using the operations defined in Definitions 5 and 6, we now present some properties and relations between the operations on HIVFEs:

Theorem 1

For every HIVFE $\tilde{h}$ and for every $\lambda ,\,\lambda_{1} ,\lambda_{2} > 0$, the following properties hold:

• (a) $\tilde{h}\,@\tilde{h} = \tilde{h};$

• (b) $\tilde{h}\,\$ \tilde{h} = \tilde{h};$

• (c) $\tilde{h}\,\# \tilde{h} = \tilde{h};$

• (d) $\tilde{h}\,\$ \tilde{h}^{\circ} = \tilde{h}^{\circ};$

• (e) $\tilde{h}\,\$ \tilde{h}^{*} = \tilde{h}$ ^½;

• (f) $\tilde{h}\,\# \tilde{h}^{\circ} = \tilde{h}^{\circ};$

• (g) $\lambda_{1} \tilde{h}\, \cup \lambda_{2} \tilde{h} = \lambda_{1} \tilde{h}$ when $\lambda_{1} > \lambda_{2}$ and $\lambda_{1} \tilde{h}\, \cup \lambda_{2} \tilde{h} = \lambda_{2} \tilde{h}$ when $\lambda_{1} < \lambda_{2} ;$

• (h) $\lambda_{1} \tilde{h}\, \cap \lambda_{2} \tilde{h} = \lambda_{2} \tilde{h}$ when $\lambda_{1} > \lambda_{2}$ and $\lambda_{1} \tilde{h}\, \cap \lambda_{2} \tilde{h} = \lambda_{1} \tilde{h}$ when $\lambda_{1} < \lambda_{2} ;$

• (i) $\tilde{h}^{{\lambda_{1} }} \, \cup \tilde{h}^{{\lambda_{2} }} = \tilde{h}^{{\lambda_{2} }}$ when $\lambda_{1} > \lambda_{2}$ and $\tilde{h}^{{\lambda_{1} }} \, \cup \tilde{h}^{{\lambda_{2} }} = \tilde{h}^{{\lambda_{1} }}$ when $\lambda_{1} < \lambda_{2} ;$

• (j) $\tilde{h}^{{\lambda_{1} }} \, \cap \tilde{h}^{{\lambda_{2} }} = \tilde{h}^{{\lambda_{1} }}$ when $\lambda_{1} > \lambda_{2}$ and $\tilde{h}^{{\lambda_{1} }} \, \cup \tilde{h}^{{\lambda_{2} }} = \tilde{h}^{{\lambda_{2} }}$ when $\lambda_{1} < \lambda_{2} .$

where $h^{ \circ }$ and $\tilde{h}^{ * }$ are respectively the empty hesitant interval-valued fuzzy element and full hesitant interval-valued fuzzy element.

Proof

These follow directly from the definitions. □

Theorem 2

For $\tilde{h}_{1} ,\tilde{h}_{2} \in HIVFE\left( X \right),$

• (a) $\tilde{h}_{1} @\tilde{h}_{2} = \tilde{h}_{2} @\tilde{h}_{1} ;$

• (b) $\tilde{h}_{1} \$ \tilde{h}_{2} = \tilde{h}_{2} \$ \,\tilde{h}_{1} ;$

• (c) $\tilde{h}_{1} \# \tilde{h}_{2} = \tilde{h}_{2} \# \tilde{h}_{1} ;$

• (d) $\tilde{h}_{1} * \tilde{h}_{2} = \tilde{h}_{2} * \,\tilde{h}_{1} .$

Proof

These also simply follow from definitions. □

Note

One can easily verify that,

• (a) $\tilde{h}_{1} @\left( {\tilde{h}_{2} @\tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)@\tilde{h}_{3} ;$

• (b) $\tilde{h}_{1} \# \left( {\tilde{h}_{2} \# \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)\,\# \tilde{h}_{3} ;$

• (c) $\tilde{h}_{1} \$ \left( {\tilde{h}_{2} \$ \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)\,\$ \,\tilde{h}_{3} ;$

• (d) $\tilde{h}_{1} * \left( {\tilde{h}_{2} * \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) * \tilde{h}_{3} .$

Theorem 3

For $\tilde{h}_{1} ,\tilde{h}_{2} \,{\text{and}}\,\tilde{h}_{3} \in HIVFE\left( X \right),$

• (a) $\tilde{h}_{1} @\left( {\tilde{h}_{2} \cup \tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right);$

• (b) $\tilde{h}_{1} @\left( {\tilde{h}_{2} \cap \tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right);$

• (c) $\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} \cup \tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right);$

• (d) $\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} \cap \tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right);$

• (e) $\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} \cup \tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right);$

• (f) $\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} \cap \tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right);$

• (g) $\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} @\tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right);$

• (h) $\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} @\tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right);$

• (i) $\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} \$ \tilde{h}_{3} } \right) = \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \otimes \,\tilde{h}_{3} } \right);$

• (j) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)@\tilde{h}_{3} = \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right) \cup \left( {\tilde{h}_{2} @\tilde{h}_{3} } \right);$

• (k) $\left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)@\tilde{h}_{3} = \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right) \cap \left( {\tilde{h}_{2} @\tilde{h}_{3} } \right);$

• (l) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \tilde{h}_{3} = \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right) \cup \left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right);$

• (m) $\left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) \oplus \tilde{h}_{3} = \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right) \cap \left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right);$

• (n) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \tilde{h}_{3} = \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right) \cup \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right);$

• (o) $\left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) \otimes \tilde{h}_{3} = \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right) \cap \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right);$

• (p) $\left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \oplus \tilde{h}_{3} = \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right)@\left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right);$

• (q) $\left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \otimes \tilde{h}_{3} = \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right)@\left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right);$

• (r) $\left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) \otimes \tilde{h}_{3} = \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right)\$ \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right).$

Proof

We prove (a), (c), (e), (g), (i), (k), (m), (o) and (q) results (b), (d), (f), (h), (j), (l), (n) and (r) can be proved analogously.

(a) Using Definitions in 5 and 6, we have

$$\begin{aligned} & \tilde{h}_{1} @\left( {\tilde{h}_{2} \cup \tilde{h}_{3} } \right) \\ &\quad = \tilde{h}_{1} @\left( {\left\{ {\left[ {\left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \\ &\quad = \left\{ \left[ {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right)}}{2}} \right),\left( {\frac{{\tilde{\gamma }_{1}^{L} + \left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)}}{2}} \right)} \right] \right. \\ &\qquad \left. \vphantom{\left[ {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right)}}{2}} \right),\left( {\frac{{\tilde{\gamma }_{1}^{L} + \left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)}}{2}} \right)} \right]} |\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} \right\}, \\ &\quad= \left\{ \left[ \begin{aligned} \left( {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right) \vee \left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right)} \right) \hfill \\ ,\left( {\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right) \vee \left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right)} \right) \hfill \\ \end{aligned} \right] \right. \\ &\qquad \left. \vphantom{\left[ \begin{aligned} \left( {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right) \vee \left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right)} \right) \hfill \\ ,\left( {\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right) \vee \left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right)} \right) \hfill \\ \end{aligned} \right]} |\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} \right\}, \\ &\quad = \left( {\left\{ {\,\left[ {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right),\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \hfill \\ &\qquad \cup \left( {\left\{ {\,\left[ {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right),\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \\&\quad = \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right). \end{aligned}$$

This proves (a).

(c) From Definitions in 5 and 6, we have

$$\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} \cup \tilde{h}_{3} } \right)$$

$$= \tilde{h}_{1} \oplus \left( {\left\{ {\left[ {\left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right),$$

$$= \left\{ {\,\left[ \begin{aligned} \tilde{\gamma }_{1}^{L} + \left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right) - \tilde{\gamma }_{1}^{L} \left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\tilde{\gamma }_{1}^{U} + \left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right) - \tilde{\gamma }_{1}^{U} \left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right) \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$= \left\{ {\,\left[ \begin{aligned} \left( {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \vee \left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right)} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \vee \left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right) \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$\begin{aligned} = \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \cup \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \hfill \\ \end{aligned}$$

$$= \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right).$$

This proves (c).

(e) Using definitions in (5) and (6), we have

$$\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} \cup \tilde{h}_{3} } \right)$$

$$= \tilde{h}_{1} \oplus \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right),$$

$$= \left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \left( {\tilde{\gamma }_{2}^{L} \vee \tilde{\gamma }_{3}^{L} } \right),\tilde{\gamma }_{1}^{U} \left( {\tilde{\gamma }_{2}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$= \left\{ {\,\left[ {\left( {\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \vee \left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right)} \right),\,\left( {\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \vee \left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$\begin{aligned} = \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \cup \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \hfill \\ \end{aligned}$$

$$= \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right).$$

This proves (e).

(g) From definitions in (5) and (6), we have

$$\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} @\tilde{h}_{3} } \right)$$

$$= \tilde{h}_{1} \oplus \left( {\left\{ {\,\left[ {\left( {\frac{{\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right),\left( {\frac{{\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right)} \right]\,|\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right),$$

$$\begin{aligned} &\quad = \left\{ \left[ \begin{aligned} \tilde{\gamma }_{1}^{L} + \left( {\frac{{\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right) - \tilde{\gamma }_{1}^{L} \left( {\frac{{\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right), \hfill \\ \tilde{\gamma }_{1}^{U} + \left( {\frac{{\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right) - \tilde{\gamma }_{1}^{U} \left( {\frac{{\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right) \hfill \\ \end{aligned} \right] \right. \\ &\qquad \left. \vphantom{\left[ \begin{aligned} \tilde{\gamma }_{1}^{L} + \left( {\frac{{\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right) - \tilde{\gamma }_{1}^{L} \left( {\frac{{\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right), \hfill \\ \tilde{\gamma }_{1}^{U} + \left( {\frac{{\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right) - \tilde{\gamma }_{1}^{U} \left( {\frac{{\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right) \hfill \\ \end{aligned} \right]} |\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} \right\}, \\ &\quad = \left\{ \left[ \begin{aligned} \frac{{\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right)}}{2}, \hfill \\ \,\,\,\,\,\,\,\frac{{\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)}}{2} \hfill \\ \end{aligned} \right] \right. \\ & \qquad \left. \vphantom{\left[ \begin{aligned} \frac{{\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right)}}{2}, \hfill \\ \,\,\,\,\,\,\,\frac{{\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)}}{2} \hfill \\ \end{aligned} \right]} |\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} \right\}, \\ &\quad = \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \hfill \\ &\qquad @\left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \\ &\quad = \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right). \end{aligned}$$

This proves (g).

(i) Using definitions in (5) and (6), we have

$$\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} \$ \tilde{h}_{3} } \right)$$

$$= \tilde{h}_{1} \otimes \left( {\left\{ {\,\left[ {\sqrt {\tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } ,\sqrt {\tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } } \right]\,|\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right),$$

$$= \left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \sqrt {\tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } ,\tilde{\gamma }_{1}^{U} \sqrt {\tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

On the other hand, we have

$$\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right)$$

$$\begin{aligned} &= \left( {\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \hfill \\ &\quad \$ \left( {\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \hfill \\ \end{aligned}$$

$$= \left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \sqrt {\tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } ,\tilde{\gamma }_{1}^{U} \sqrt {\tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}.$$

This proves (i)

(k) From definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)@\tilde{h}_{3}$$

$$= \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right)@\tilde{h}_{3} ,$$

$$= \left\{ {\,\left[ {\frac{{\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \tilde{\gamma }_{3}^{L} }}{2},\frac{{\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \tilde{\gamma }_{3}^{U} }}{2}} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}$$

$$= \left\{ {\,\left[ \begin{aligned} \left( {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right) \vee \left( {\frac{{\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right)} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right) \vee \left( {\frac{{\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right)} \right) \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}$$

$$\begin{aligned} &= \left( {\left\{ {\,\left[ {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right),\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \hfill \\ &\quad \cup \left( {\left\{ {\,\left[ {\left( {\frac{{\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} }}{2}} \right),\left( {\frac{{\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} }}{2}} \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \hfill \\ \end{aligned}$$

$$= \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right) \cup \left( {\tilde{h}_{2} @\tilde{h}_{3} } \right).$$

This proves (k).

(m) From definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \tilde{h}_{3}$$

$$= \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \oplus \tilde{h}_{3} ,$$

$$= \left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \tilde{\gamma }_{3}^{L} - \left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\tilde{\gamma }_{3}^{L} , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \tilde{\gamma }_{3}^{U} - \left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\tilde{\gamma }_{3}^{U} \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$= \left\{ {\,\left[ \begin{aligned} \left( {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right) \vee \left( {\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } \right)} \right), \hfill \\ \,\,\left( {\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right) \vee \left( {\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right) \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$\begin{aligned} = \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \cup \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \hfill \\ \end{aligned}$$

$$= \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right) \cup \left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right).$$

This proves (m).

(o) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \tilde{h}_{3}$$

$$= \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \otimes \tilde{h}_{3} ,$$

$$= \left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\tilde{\gamma }_{3}^{L} ,\,\,\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\tilde{\gamma }_{3}^{U} } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$= \left\{ {\,\left[ {\left( {\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right) \vee \left( {\tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } \right)} \right),\left( {\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right) \vee \left( {\tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\},$$

$$= \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right)\,\, \cup \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right),$$

$$= \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right) \cup \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right).$$

This proves (o).

(q) From definitions in (5) and (6), we have

$$\begin{aligned} & \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \oplus \tilde{h}_{3} \\ &\quad = \left( {\left\{ {\,\left[ {\left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right),\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \oplus \tilde{h}_{3} , \\ &\quad = \left\{ {\,\left[ \begin{aligned} \left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right) + \tilde{\gamma }_{3}^{L} - \left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right)\tilde{\gamma }_{3}^{L} , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right) + \tilde{\gamma }_{3}^{U} - \left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right)\tilde{\gamma }_{3}^{U} \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}, \\ &\quad = \left\{ {\,\left[ \begin{aligned} \frac{{\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right) + \left( {\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } \right)}}{2}, \hfill \\ \,\,\,\,\,\,\,\frac{{\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right) + \left( {\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } \right)}}{2} \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}, \\ &\quad = \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \hfill \\ &\qquad @\left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{3}^{L} - \tilde{\gamma }_{2}^{L} \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{3}^{U} - \tilde{\gamma }_{2}^{U} \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\}} \right), \\ &\quad= \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right)@\left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right).\end{aligned}$$

This proves (q).

This proves the theorem. □

Note

It can be easily verify that,

• (a) $\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \oplus \tilde{h}_{3} } \right);$

• (b) $\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \otimes \tilde{h}_{3} } \right);$

• (c) $\tilde{h}_{1} @\left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right);$

• (d) $\tilde{h}_{1} @\left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} @\tilde{h}_{3} } \right);$

• (e) $\tilde{h}_{1} \$ \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \$ \,\tilde{h}_{3} } \right);$

• (f) $\tilde{h}_{1} \$ \left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \$ \tilde{h}_{3} } \right);$

• (g) $\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} \$ \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \oplus \,\tilde{h}_{3} } \right);$

• (h) $\tilde{h}_{1} \# \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \# \,\tilde{h}_{3} } \right);$

• (i) $\tilde{h}_{1} \# \left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \# \tilde{h}_{3} } \right);$

• (j) $\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} \# \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)\# \left( {\tilde{h}_{1} \otimes \,\tilde{h}_{3} } \right);$

• (k) $\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} \# \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)\# \left( {\tilde{h}_{1} \oplus \,\tilde{h}_{3} } \right);$

• (l) $\tilde{h}_{1} * \left( {\tilde{h}_{2} \otimes \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} * \,\tilde{h}_{3} } \right);$

• (m) $\tilde{h}_{1} * \left( {\tilde{h}_{2} \oplus \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} * \tilde{h}_{3} } \right);$

• (n) $\tilde{h}_{1} \otimes \left( {\tilde{h}_{2} * \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) * \left( {\tilde{h}_{1} \otimes \,\tilde{h}_{3} } \right)$

• (o) $\tilde{h}_{1} \oplus \left( {\tilde{h}_{2} * \tilde{h}_{3} } \right) \ne \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) * \left( {\tilde{h}_{1} \oplus \,\tilde{h}_{3} } \right).$

Theorem 4

For $\tilde{h}_{1} ,\tilde{h}_{2} \in HIVFE\left( X \right),$

• (a) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) = \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right);$

• (b) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) = \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right);$

• (c) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (d) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (e) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)\# \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) = \tilde{h}_{1} \# \tilde{h}_{2} ;$

• (f) $\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) * \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) = \tilde{h}_{1} * \tilde{h}_{2} .$

Proof

We shall use the fact that for every two real numbers $a\,\,{\text{and}}\,\,b$ it follows

$$\left( {a \vee b} \right) + \left( {a \wedge b} \right) = a + b,$$

$$\left( {a \vee b} \right) \times \left( {a \wedge b} \right) = a \times b.$$

(a) Using definitions in (5), we have

$$\begin{aligned} &\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) \\ &\quad = \left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\qquad \oplus \left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\} \hfill \\ &\quad = \left( {\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right) - \left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right), \hfill \\ \,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right) - \left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right) \hfill \\ \end{aligned} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right)\, \\ &\quad = \left( {\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right), \\ &\quad = \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \end{aligned}$$

This proves (a).

(b) From definitions in (5), we have

$$\begin{aligned} & \left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) \\ &\quad = \left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\qquad \otimes \left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{3}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{3}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{3} \in \tilde{h}_{3} } \right\} \hfill \\ &\quad = \left( {\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)} \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right) \\ &\quad = \left( {\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]\,|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}} \right), \\ &\quad = \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right). \end{aligned}$$

This proves (b).

(c) Using definitions in (5) and (6), we have

$$\begin{aligned} & \left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) \\ &\quad = \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\qquad @\,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ &\quad = \,\left\{ {\,\left[ {\frac{{\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right)}}{2},\frac{{\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)}}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \\ &\quad = \,\left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \\ &\quad = \tilde{h}_{1} @\tilde{h}_{2} . \end{aligned}$$

This proves (c).

(d) From definitions in (5) and (6), we have

$$\begin{aligned} & \left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)\,\$ \,\left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right) \\ &\quad = \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\qquad \$ \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ &\quad = \,\left\{ {\,\left[ {\sqrt {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right)} ,\sqrt {\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \\ &\quad = \,\left\{ {\,\left[ {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } ,\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \\ &\quad = \tilde{h}_{1} \$ \tilde{h}_{2} . \end{aligned}$$

This proves (d).

(e) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)\# \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)$$

$$\begin{aligned} = \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\# \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ {\frac{{2\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right)}}{{\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right)}},\frac{{2\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)}}{{\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)}}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \,\left\{ {\,\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} \# \tilde{h}_{2} .$$

This proves (e).

(f) From definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) * \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)$$

$$\begin{aligned} = \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, * \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$\begin{aligned} &= \left\{ \left[ {\frac{{\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right)}}{{2\left( {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right) + 1} \right)}},\frac{{\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)}}{{2\left( {\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right) + } \right)1}}} \right] \right. \\ &\quad \left. \vphantom{\left[ {\frac{{\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right)}}{{2\left( {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right) + 1} \right)}},\frac{{\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)}}{{2\left( {\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right) + } \right)1}}} \right]} |\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} \right\}, \end{aligned}$$

$$= \left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{{2\left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} + 1} \right)}},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{{2\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} + 1} \right)}}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} * \tilde{h}_{2} .$$

This proves (f).

This proves the theorem. □

Theorem 5

For $\tilde{h}_{1} ,\tilde{h}_{2} \in HIVFE\left( X \right)$, then the following relations are valid

• (a) $\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (b) $\left( {\tilde{h}_{1} * \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) = \tilde{h}_{1} * \tilde{h}_{2} ;$

• (c) $\left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (d) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) = \tilde{h}_{1} \oplus \tilde{h}_{2} ;$

• (e) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) = \tilde{h}_{1} \otimes \tilde{h}_{2} ;$

• (f) $\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \# \tilde{h}_{2} ;$

• (g) $\left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (h) $\left( {\tilde{h}_{1} * \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) = \tilde{h}_{1} * \tilde{h}_{2} ;$

• (i) $\left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (j) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) = \tilde{h}_{1} \oplus \tilde{h}_{2} ;$

• (k) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) = \tilde{h}_{1} \otimes \tilde{h}_{2} .$

Proof

In the following, we prove (a) and (f), other results can be proved analogously.

(a) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)\$ \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad \$ \,\,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ {\sqrt {\left( {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}} \right)\left( {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}} \right)} ,\sqrt {\left( {\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right)\left( {\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right)} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right).$$

This proves (a).

(f) From definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad @\,\,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ {\frac{{\left( {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}} \right) + \left( {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}} \right)}}{2},\frac{{\left( {\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right) + \left( {\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right)}}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right).$$

This proves (f).

This proves the theorem. □

Theorem 6

For $\tilde{h}_{1} ,\tilde{h}_{2} \in HIVFE\left( X \right)$, then the following relations are valid:

• (a) $\left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (b) $\left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (c) $\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \# \tilde{h}_{2} ;$

• (d) $\left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \# \tilde{h}_{2} ;$

• (e) $\left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (f) $\left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (g) $\left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) = \tilde{h}_{1} * \tilde{h}_{2} ;$

• (h) $\left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} * \tilde{h}_{2} } \right) = \tilde{h}_{1} * \tilde{h}_{2} .$

Proof

These results follow directly from definitions.

Theorem 7

For $\tilde{h}_{1} ,\tilde{h}_{2} \in HIVFE\left( X \right)$, we have the following identities:

• (a) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) = \tilde{h}_{1} \otimes \tilde{h}_{2} ;$

• (b) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) = \tilde{h}_{1} \oplus \tilde{h}_{2} ;$

• (c) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (d) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} \oplus \tilde{h}_{2} ;$

• (e) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} \otimes \tilde{h}_{2} ;$

• (f) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (g) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (h) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) = \tilde{h}_{1} \oplus \tilde{h}_{2} ;$

• (i) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) = \tilde{h}_{1} \otimes \tilde{h}_{2} ;$

• (j) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (k) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \# \tilde{h}_{2} ;$

• (l) $\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \oplus \tilde{h}_{2} ;$

• (m) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \otimes \tilde{h}_{2} ;$

• (n) $\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \# \tilde{h}_{2} .$

Proof

We prove (a), (c), (e), (g), (i), (k) and (m), other results can be proved analogously.

(a) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \&\quad \, \cap \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right).$$

This proves (a).

(c) From definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad \cap \,\,\left\{ {\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} @\tilde{h}_{2} .$$

This proves (c).

(e) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\&\quad \cap \,\,\left\{ {\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2}} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right).$$

This proves (v).

(g) From definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \&\quad \cap \,\,\left\{ {\left[ {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } ,\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } } \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \left\{ {\,\left[ {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } ,\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} \$ \tilde{h}_{2} .$$

This proves (g).

(i) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\&\quad \cap \,\,\left\{ {\left[ {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } ,\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } } \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} \otimes \tilde{h}_{2} .$$

This proves (i).

(k) From definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad\cap \,\,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} \# \tilde{h}_{2} .$$

This proves (k).

(m) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad \cap \,\,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} \otimes \tilde{h}_{2} .$$

This proves (m).

This proves the theorem. □

Theorem 8

For $\tilde{h}_{1} ,\tilde{h}_{2} \in HIVFE\left( X \right)$, then the following relations are valid

• (a) $\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (b) $\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)\# \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right)\$ \left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right)@\left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (c) $\left( {\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right)@\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)} \right)\$ \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (d) $\left( {\left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} \$ \tilde{h}_{2} ;$

• (e) $\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (f) $\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (g) $\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (h) $\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \$ \tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} @\tilde{h}_{2} ;$

• (i) $\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} @\tilde{h}_{2} } \right)} \right)\$ \left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)} \right) = \tilde{h}_{1} \$ \tilde{h}_{2} .$

Proof

In the following, we prove (a), (c) and (e), other results can be proved analogously.

(a) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)$$

$$\begin{aligned} = \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \oplus \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right) + \left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right) - \left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right) + \left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right) - \left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} ,$$

(8)

and

$$\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)$$

$$\begin{aligned} = \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \otimes \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

(9)

Now taking @ with (8) and (9), we get

$$\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right)$$

$$\begin{aligned} &= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad @\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{2}, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{2} \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \,\left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} @\tilde{h}_{2} .$$

This proves (a).

(c) From definitions in (5) and (6), we have

$$\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \otimes \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right)@\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left( \begin{aligned} \left\{ {\left[ {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \otimes \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ \end{aligned} \right) \hfill \\ &\quad @\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$\begin{aligned} &= \,\left\{ {\,\left[ {\left( {\tilde{\gamma }_{1}^{L} \vee \tilde{\gamma }_{2}^{L} } \right)\left( {\tilde{\gamma }_{1}^{L} \wedge \tilde{\gamma }_{2}^{L} } \right),\left( {\tilde{\gamma }_{1}^{U} \vee \tilde{\gamma }_{2}^{U} } \right)\left( {\tilde{\gamma }_{1}^{U} \wedge \tilde{\gamma }_{2}^{U} } \right)} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad @\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$\begin{aligned} &= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad @\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \,\left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

(10)

and

$$\tilde{h}_{1} \# \tilde{h}_{2} = \,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

(11)

Now taking $ with (10) and (11), we get

$$\left( {\left( {\left( {\tilde{h}_{1} \cup \tilde{h}_{2} } \right) \oplus \left( {\tilde{h}_{1} \cap \tilde{h}_{2} } \right)} \right)@\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right)} \right)\$ \left( {\tilde{h}_{1} \# \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad \$ \,\left\{ {\left[ {\frac{{2\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }},\frac{{2\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ {\sqrt {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } ,\,\sqrt {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} \$ \tilde{h}_{2} .$$

This proves (c).

(e) Using definitions in (5) and (6), we have

$$\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad \cup \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \vee \left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right)} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \vee \left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right)} \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

(12)

and

$$\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)} \right)$$

$$\begin{aligned} &= \,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &\quad \cap \,\,\left\{ {\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \left( {\left( {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right) \wedge \left( {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} } \right)} \right), \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\left( {\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right) \wedge \left( {\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right)} \right) \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}$$

$$= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} ,$$

(13)

Now taking @ with (12) and (13), we get

$$\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cup \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)} \right)@\left( {\left( {\tilde{h}_{1} \oplus \tilde{h}_{2} } \right) \cap \left( {\tilde{h}_{1} \otimes \tilde{h}_{2} } \right)} \right)$$

$$\begin{aligned} &= \,\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\} \hfill \\ &quad @\left\{ {\,\left[ {\tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} ,\tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} } \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\}, \hfill \\ \end{aligned}$$

$$= \,\left\{ {\,\left[ \begin{aligned} \frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} - \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} + \tilde{\gamma }_{1}^{L} \tilde{\gamma }_{2}^{L} }}{2}, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} - \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} + \tilde{\gamma }_{1}^{U} \tilde{\gamma }_{2}^{U} }}{2} \hfill \\ \end{aligned} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \,\left\{ {\,\left[ {\frac{{\tilde{\gamma }_{1}^{L} + \tilde{\gamma }_{2}^{L} }}{2},\frac{{\tilde{\gamma }_{1}^{U} + \tilde{\gamma }_{2}^{U} }}{2}} \right]|\tilde{\gamma }_{1} \in \tilde{h}_{1} ,\tilde{\gamma }_{2} \in \tilde{h}_{2} } \right\},$$

$$= \tilde{h}_{1} @\tilde{h}_{2} .$$

This proves (e).

This proves the theorem. □

4 Conclusions

Hesitant interval-valued fuzzy sets present a powerful tool for study of human’s hesitancy more objectively and precisely. This paper introduced four new operations on hesitant interval-valued fuzzy sets. Then, a number of results have been proved, some of which follow simply and bring out the characteristics of the hesitant inter-valued fuzzy sets. The novelty of these operations is that they enriched and developed the theory of hesitant interval valued fuzzy sets and prepare a strong background for applications of these sets in different situations.