New Threshold Approximation Method Dealing with Complex Intuitionistic Fuzzy Information

Abstract

Complex intuitionistic fuzzy sets (CIFSs) can represent problems in multicriteria decision-making that involve uncertainty and periodicity of semantics at the same time. CIFSs handle two-dimensional information that requires a long time and a huge number of calculations and processes. In this research, the complex intuitionistic shadowed set (CISHS) is defined as a novel threshold approximation method. CISHS is introduced to simplify processing and enhance the interpretation of outcomes obtained from CIFSs. The main idea of constructing CISHS is adding a complex non-membership function to the complex shadowed set (CSHS) range. Additionally, the basic operations, specifically complement, union, and intersection on CISHSs, are introduced. The properties of CISHSs are defined as null, absolute, and subset. Finally, an application to demonstrate the effectiveness of the proposed approach is employed by using an adaptive algorithm with CISHSs.

Keywords

fuzzy sets intuitionistic fuzzy sets shadowed sets intuitionistic shadowed sets complex fuzzy sets complex intuitionistic fuzzy sets complex intuitionistic shadowed sets

1. Introduction

Zadeh (1965) introduced the theory of fuzzy sets (FSs). FSs aid in formulating information from human knowledge into mathematical models and vice versa. FSs are mathematical tools used to explain vague and uncertain information. FSs are characterized by a membership function that assigns each object a membership value between 0 and 1. Narasimhan (1979) presented an application of FSs theory to the problem of locating gas stations. FSs have numerous applications in artificial intelligence by Bas and Egrioglu (2023), and in expert systems by Wójcik et al. (2023). Recently, many researchers applied FS and their generalizations to various fields of mathematics (Abuhijleh et al., 2021; Al-Masarwah et al., 2021; Al-Sharoa, 2023; Alsharo et al., 2024; Ramot et al., 2002; Rani & Garg, 2017).

Then, in 1986, intuitionistic fuzzy sets (IFSs) were introduced by Atanassov (2016) as an extension of FSs, by adding non-membership function. In an IFS, the membership function and the non-membership function are used to model hesitation, and it is more powerful to deal with the uncertainty and fuzziness. Biswas (1989) introduced the concept of intuitionistic fuzzy subgroup. Akram et al. (2013) proposed IF logic controller for heater fans. This concept has been applied in pattern recognition by Wang and Xin (2005), and in medical diagnosis by Kozae et al. (2020).

Existing theories are often inadequate for fully capturing information, and hence there is a loss of information during the process. To address this limitation, Ramot et al. (2002) introduced a new notion and named it complex fuzzy sets (CFSs) to model periodic problems or recurring problems. They extended the range of membership function to the unit disk in the complex plane. Ma et al. (2019) suggested a model based on CFS to identify the reference signal among multiple signals collected by a digital receiver. Sobhi and Dick (2023) proposed a new neuro-fuzzy architecture using CFS that has been designed for large-scale learning problems. In 2023, Dai (2023) proposed a new notion called linguistic complex fuzzy set.

Alkouri and Salleh (2012) extended the definition of CFSs to complex intuitionistic fuzzy sets (CIFSs) by adding a non-membership term to the notion of CFSs. Rani and Garg (2017) proposed a series of distance measures for CIFSs and applied them to problems in pattern recognition and medical diagnosis problems. Garg and Rani (2019) developed new averaging aggregation operators, called complex intuitionistic fuzzy weighted averaging. In 2023, Al-Sharoa (2023) presented the idea of complex intuitionistic fuzzy subgroups, and the properties were explored.

As a three-region approximation, Pedrycz (1998) defined the Shadowed sets (SHSs) as a tool for handling uncertainty. The main idea was transforming the range of FSs into three components, that is, 0, 1, and the unit interval [0,1]. The elements of the range of SHSs can be labeled using symbols, say, elevated, shadow, and reduced. Zhang and Yao (2020) proposed a game-theoretic approach to shadowed sets. Yue, Zhou, Yao, and Miao (2020) proposed a novel shadowed set method to construct shadowed neighborhoods for uncertain data classification.

In 2021, Yang and Yao (2021) exposed that shadowed sets can be constructed from IFSs. This approach is related to the notion of a three-way decision and is considered an approximation method for intuitionistic fuzzy information. In 2024, Hlael (2024) presented a new structure of intuitionistic fuzzy information, and the basic operations were introduced. Recently, Alsharo et al. (2024) introduced the notion of complex shadowed set (CSHS) by extending the range of SHSs from ${0, [0, 1], 1}$ to three values lies in the unit disk in the complex plane. These complex values are ${⟨ 0 e^{j θ^{φ}} ⟩, ⟨ a e^{j θ^{φ}} ⟩, ⟨ 1 e^{j θ^{φ}} ⟩}$ , which are characterized by a small disk and two rings inside the unit disk in the complex plane. Also, the basic operations, such as complement, union, and intersection were investigated. Furthermore, an application of CSHS in DM processes was illustrated and compared with complex fuzzy results.

The construction of complex intuitionistic shadow sets is mainly motivated by the desire to extend intuitionistic fuzzy logic and set theory into domains involving dynamic, oscillatory, or phase-based behavior, using the tools of complex numbers. Complex numbers allow for encoding both magnitude “indicating the degree to which an element belongs to a set” and Phase “representing when or in what direction this membership occurs”. In the case of complex intuitionistic fuzzy information, we have complex membership and non-membership values help us to identify the objects to be lying in rejection, accept, or shadow region.

In short, intuitionistic shadow sets simplify intuitionistic fuzzy systems, while complex membership and non-membership values generalize them for richer, more nuanced modeling. Therefore, combining the two offers a flexible, computationally manageable, and conceptually powerful framework for handling systems characterized by both vagueness and periodicity. In addition, complex intuitionistic shadow sets (CISHS) aim to simplify processing and enhance the interpretability of results derived from complex intuitionistic fuzzy sets (CIFSs). This extension enables more refined analysis of uncertainty across different phases and strengthens decision-making capabilities.

The idea of adding the complex non-membership function to the stricter of CFS introduced the notion of CIFS. The challenge of translating human knowledge involving periodicity semantics and uncertainty simultaneously into mathematical formulas and vice versa is addressed by CIFSs. So, the phase terms of CIFSs highlighted a new complication, which is adding enormous operations and calculations accomplished from the CIFSs and the explanation of acquired results.

To improve the understanding of CIFS findings and streamline processing, By appending a complex non-membership function to the range of complex shadowed sets, we provide the idea of the complex intuitionistic shadowed sets, which yields the subsequent new range from ${⟨ 0 e^{j θ^{φ}} ⟩, ⟨ a e^{j θ^{φ}} ⟩, ⟨ 1 e^{j θ^{φ}} ⟩}$ into ${⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩, ⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩, ⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩}$ . To illustrate our methodology, we have included a complex non-membership function in the CSH range for every item in the mathematical structure of CSHSs.

In this research, a literature review and relevant preliminaries are presented in Sections 2 and 3, respectively. Section 3 also introduces a new concept called complex intuitionistic shadow sets (CISHS), developed by adjoining a complex non-membership function to the range of complex shadowed sets (CSHS). Furthermore, several fundamental mathematical operations and their properties are defined and discussed. Section 4 then illustrates an application example in decision-making problems to validate the effectiveness of CISHSs in handling and approximating complex intuitionistic fuzzy information. Finally, Section 5 presents a discussion of the main contributions and directions for future work.

2. Preliminaries

Definition 2.1 (Zadeh 1965)

A fuzzy set $A$ in a universe of discourse $U$ , is defined by a membership function $μ_{A} (x)$ that assigns values within the interval [0,1].

Definition 2.2 (Atanassov 2016)

An intuitionistic fuzzy set $A$ in a universe of discourse $U$ , is an object with the form:

A = {⟨ x, μ_{A} (x), v_{A} (x) ⟩ | x \in U},

where the functions

μ_{A} (x) : U \to [0, 1]

and

v_{A} (x) : U \to [0, 1]

specify the degree of membership and degree of non-membership for every element

x \in U

to the set

A

respectively, with the condition

0 \leq μ_{A} (x) + v_{A} (x) \leq 1

for all

x \in U

Definition 2.3 (Pedrycz 1998)

let $U$ be a universe of discourse, and $μ_{A} (x) : U \to [0, 1]$ constructed the FS $(A) = f (x)$ over $U$ . A thresholds value $α$ and $β$ , where $0 \leq β < α \leq 1$ . The ${S H S}_{A} (U) = {⟨ x, B_{(α, β)} f (x) ⟩ | x \in U}$ where $B_{(α, β)} : U \to {0, a, 1}$ , given by

B_{(α, β)} f (x) = {\begin{cases} 0, i f μ_{A} (x) \leq β, \\ a, β < μ_{A} (x) < α, \\ 1, i f μ_{A} (x) \geq α, \end{cases}

where

a \in [0, 1] .

Definition 2.4 (Ramot et al. 2002)

A complex fuzzy set $S$ , which is defined in a discourse‘s universe $U$ , is characterized by a membership function $μ_{S} (x)$ , that assigns a complex-valued degree to any element $x \in U$ in $S$ . The values of $μ_{S} (x)$ may all, by definition, lie within the unit circle in the complex plane, and are thus have the form $φ_{s} (x) . e^{j θ_{s} (x)}$ , where $j = \sqrt{- 1}$ , both of $φ_{s} (x) \in [0, 1]$ and $θ_{s} \in [0, 2 π] .$ The complex fuzzy set $S$ can be represented as a set of ordered pairs

S = {⟨ x, μ_{s} (x) ⟩ | x \in U} .

Definition 2.5 (Alkouri and Salleh 2012)

A CIFS $A$ defined on $U$ is given as

A = {⟨ x, μ_{A} (x), v_{A} (x) ⟩ | x \in U},

where

μ_{A}, v_{A} : U \to {a : a \in C, | a | \leq 1}

are complex—valued membership and non-membership functions, respectively, and are represented as

μ_{A} (x) = φ_{A} (x) e^{j θ_{A}^{φ} (x)}

and

v_{A} (x) = ϑ_{A} (x) e^{j θ_{A}^{ϑ} (x)}

, where

0 \leq φ_{A} (x), ϑ_{A} (x) \leq 1

;

0 \leq φ_{A} (x) + ϑ_{A} (x) \leq 1

and

0 \leq θ_{A}^{φ} (x), θ_{A}^{ϑ} (x) \leq 2 π

;

0 \leq θ_{A}^{φ} (x) + θ_{A}^{ϑ} (x) \leq 2 π

for all

x \in U

Definition 2.6 (Alsharo et al. 2024)

Let $U$ be a universe of discourse. The set ${C S H S}_{A}$ is said to be a CSHS of the universe U if it is a function mapping as follows:

{C S H S}_{A} : U \to {0 e^{j θ}, [0, 1] e^{j θ}, 1 e^{j θ}},

where the codomain of

{C S H S}_{A}

is denoted as polar form in the complex plane by

{C S H S}_{A} (x) = φ_{{C S H S}_{A}} (x) e^{j θ_{{C S H S}_{A}} (x)} .

We may denote the phase term

θ_{{C S H S}_{A}} (x)

by using the property of periodic in complex number as

θ_{{C S H S}_{A}} (x) = θ_{{C S H S}_{A}} (x) + 2 k π, k = 0, \pm 1, \pm 2, \dots .

Based on proper values of threshold

α_{φ}

and

β_{φ}

, where

0 \leq α_{φ} < β_{φ} \leq 1,

the codomain function of CSHS divides the codomain of CFSs into three disjoint regions; the disk and two rings in the unit disk in the complex plane are as follows:

{C S H S}_{A} (x) = {\begin{matrix} 0 e^{j θ (x)}, i f | φ (x) e^{j θ (x)} | = φ (x) \leq α_{φ} \\ 1 e^{j θ (x)}, i f | φ (x) e^{j θ (x)} | = φ (x) \geq β_{φ} \\ [0, 1] e^{j θ (x)}, i f α_{φ} < | φ (x) e^{j θ (x)} | = φ (x) < β_{φ} \end{matrix},

where

x \in U

φ (x) \in [0, 1]

, and

θ (x)

has any value in the interval [0, 2

π

]

Some basic properties of CSHSs are presented: If

{C S H S}_{A}

and

{C S H S}_{B}

are two CSHSs on

U

with memberships

μ_{{C S H S}_{A}} (x) = φ_{{C S H S}_{A}} (x) e^{j θ_{{C S H S}_{A}} (x)}

μ_{{C S H S}_{B}} (x) = φ_{{C S H S}_{B}} (x) e^{j θ_{{C S H S}_{B}} (x)}

1)
The union of ${C S H S}_{A}$ and ${C S H S}_{B}$ , ${C S H S}_{A} \cup {C S H S}_{B}$ is defined as
${C S H S}_{A} \cup {C S H S}_{B} (x) = m a x {φ_{{C S H S}_{A}} (x), φ_{{C S H S}_{B}} (x)} e^{j m a x {θ_{{C S H S}_{A}} (x), θ_{{C S H S}_{B}} (x)}}$

2)
The intersection of ${C S H S}_{A}$ and ${C S H S}_{B}$ , ${C S H S}_{A} \cap {C S H S}_{B}$ is defined as
${C S H S}_{A} \cap {C S H S}_{B} (x) = m i n {φ_{{C S H S}_{A}} (x), φ_{{C S H S}_{B}} (x)} e^{j m i n {θ_{{C S H S}_{A}} (x), θ_{{C S H S}_{B}} (x)}},$

3)
The complement of ${C S H S}_{A}$ , $\bar{{C S H S}_{A}}$ is defined as
$μ_{\bar{{C S H S}_{A}}} (x) = φ_{\bar{{C S H S}_{A}}} (x) e^{j θ_{\bar{{C S H S}_{A}}} (x)} = (1 - φ_{{C S H S}_{A}} (x)) e^{j 2 π - θ_{{C S H S}_{A}} (x)} = (1 - φ_{{C S H S}_{A}} (x)) e^{j θ_{{C S H S}_{A}} (x)} .$

Proposition 2.1 (Ramot et al. 2002)

If ${C S H S}_{A}$ , ${C S H S}_{B}$ and ${C S H S}_{C}$ are CSHSs on $U$ , then the following properties are satisfied:

a.
${C S H S}_{A} \cup {C S H S}_{B} = {C S H S}_{B} \cup {C S H S}_{A},$
b.
${C S H S}_{A} \cap {C S H S}_{B} = {C S H S}_{B} \cap {C S H S}_{A},$
c.
${C S H S}_{A} \cup ({C S H S}_{B} \cap {C S H S}_{C}) = ({C S H S}_{A} \cup {C S H S}_{B}) \cap ({C S H S}_{A} \cup {C S H S}_{C}),$
d.
${C S H S}_{A} \cap ({C S H S}_{B} \cup {C S H S}_{C}) = ({C S H S}_{A} \cap {C S H S}_{B}) \cup ({C S H S}_{A} \cap {C S H S}_{C}),$
e.
$\bar{({C S H S}_{A} \cup {C S H S}_{B})} = \bar{{C S H S}_{A}} \cap \bar{{C S H S}_{B}},$
f.
$\bar{({C S H S}_{A} \cap {C S H S}_{B})} = \bar{A {C S H S}_{A}} \cup \bar{B {C S H S}_{B}},$
g.
$\bar{\bar{{C S H S}_{A}}} = {C S H S}_{A} .$

Table 1 summarizes the symbols used in this research along with their associated constraints, for improved clarity and ease of reference.

Table 1.
Main Symbols with their Constraints and Notations.

Symbol Meaning / notations Conditions / Notes

$μ_{A} (x)$ Membership function of the FS and IFS A $\in [0, 1]$

$ν_{A} (x)$ Non-membership function of the IFS A ${0 \leq μ}_{A}$ ( $x$ )+ $ν_{A}$ ( $x$ ) $\leq$ 1

$θ$ Phase/level term in Polar exponential form for both CFS and CIFS. $θ$ $\in$ [0, 2 $π$ ]

$α, β$ Thresholds for SHS and CSHS $0 \leq β < α \leq 1$

$φ_{A} (x)$ Amplitude of complex membership for both CFS/CIFS $\in [0, 1]$

$ϑ_{A} (x)$ Amplitude term of CIFS, complex non-membership function. $\in$ [0,1], and ${0 \leq φ}_{A} (x) + ϑ_{A} (x) \leq 1$

${\hat{φ}}_{A} (x) {\hat{ϑ}}_{A}$ ( $x$ ), CISHS amplitude terms of complex membership and complex non-membership functions respectively. ${\hat{φ}}_{A} (x)$ ${\hat{ϑ}}_{A}$ ( $x$ ) $\in$ [0,1], and $0 \leq {\hat{φ}}_{A} (x)$ + ${\hat{ϑ}}_{A} (x) \leq 1$

$θ^{φ}, θ^{ϑ}$ CISHS phase terms of membership and non-membership functions respectively. $θ^{φ}$ , $θ^{ϑ} \in [0, 2 π]$ and $0 \leq θ^{φ} + θ^{ϑ} \leq 2 π$ .

$a$ The amplitude term of complex membership function in the shadowed region. $a \in$ [0, 1]

$a^{'}$ The amplitude term of the non-membership function in the shadowed region. $a^{'}$ $\in$ [0, 1]

$⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩$ Complex membership and complex non-membership values represent the shadowed region in CISHS. Used in CIFS, CISHS, $0 \leq a + a^{'} \leq 1$

$j$ Imaginary unit $j = \sqrt{- 1}$

$w, p$ Threshold values for absolute value of complex membership and complex non-membership in CISHS $0 \leq w < p \leq 1$

3. Complex Intuitionistic Shadowed Sets

Symbol	Meaning / notations	Conditions / Notes
$μ_{A} (x)$	Membership function of the FS and IFS A	$\in [0, 1]$
$ν_{A} (x)$	Non-membership function of the IFS A	${0 \leq μ}_{A}$ ( $x$ )+ $ν_{A}$ ( $x$ ) $\leq$ 1
$θ$	Phase/level term in Polar exponential form for both CFS and CIFS.	$θ$ $\in$ [0, 2 $π$ ]
$α, β$	Thresholds for SHS and CSHS	$0 \leq β < α \leq 1$
$φ_{A} (x)$	Amplitude of complex membership for both CFS/CIFS	$\in [0, 1]$
$ϑ_{A} (x)$	Amplitude term of CIFS, complex non-membership function.	$\in$ [0,1], and ${0 \leq φ}_{A} (x) + ϑ_{A} (x) \leq 1$
${\hat{φ}}_{A} (x) {\hat{ϑ}}_{A}$ ( $x$ ),	CISHS amplitude terms of complex membership and complex non-membership functions respectively.	${\hat{φ}}_{A} (x)$ ${\hat{ϑ}}_{A}$ ( $x$ ) $\in$ [0,1], and $0 \leq {\hat{φ}}_{A} (x)$ + ${\hat{ϑ}}_{A} (x) \leq 1$
$θ^{φ}, θ^{ϑ}$	CISHS phase terms of membership and non-membership functions respectively.	$θ^{φ}$ , $θ^{ϑ} \in [0, 2 π]$ and $0 \leq θ^{φ} + θ^{ϑ} \leq 2 π$ .
$a$	The amplitude term of complex membership function in the shadowed region.	$a \in$ [0, 1]
$a^{'}$	The amplitude term of the non-membership function in the shadowed region.	$a^{'}$ $\in$ [0, 1]
$⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩$	Complex membership and complex non-membership values represent the shadowed region in CISHS.	Used in CIFS, CISHS, $0 \leq a + a^{'} \leq 1$
$j$	Imaginary unit	$j = \sqrt{- 1}$
$w, p$	Threshold values for absolute value of complex membership and complex non-membership in CISHS	$0 \leq w < p \leq 1$

In this section, we define the notion CISHS, which is a generalization of CSHS. The transition from CSHS to CISHS provides a more expressive and informative framework for handling uncertainty, especially in domains involving contradictory or incomplete information. While CSHS approximates the degree of membership using complex numbers, it lacks the ability to express opposition (non-membership) and hesitation. CISHS overcomes this by approximating a complex-valued non-membership function, enabling the representation of both support and rejection levels, along with the inherent hesitation. Figure 1 presents the three regions of membership and non-membership functions for CISHS instead of only one function in CSHS and Tables 11 summarizes their core differences. Some definitions and properties are given.

Figure 1.

The Three Regions of Membership and Non-membership Function for CISHS.

Table 2.

$\bar{{C I S H S}_{A} (U)}$

$C {I S H S}_{A} (U)$	$\bar{{C I S H S}_{A} (U)}$
$⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩$	$⟨ 0 e^{j 2 π - θ^{φ}}, 1 e^{j 2 π - θ^{ϑ}} ⟩$
$⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩$	$⟨ 1 e^{j 2 π - θ^{φ}}, 0 e^{j 2 π - θ^{ϑ}} ⟩$
$⟨ a e^{j θ^{φ}}, a^{^{'}} e^{j θ^{ϑ}} ⟩$	$⟨ a e^{j 2 π - θ^{φ}}, a^{'} e^{j 2 π - θ^{ϑ}} ⟩$

Table 3.

$C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U)$ .

$C {I S H S}_{A, B} (U)$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{φ}} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j m i n ((θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x ((θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x ((θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m a x ((θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x ((θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j m a x ((θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$

Table 4.

$C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U) \hat{=} {C I S H S}_{B} (U) \hat{\cup} C {I S H S}_{A} (U)$ .

C ${I S H S}_{A} (U)$	$C {I S H S}_{B} (U)$	C ${I S H S}_{A} (U) \hat{\cup} {C I S H S}_{B} (U)$	C ${I S H S}_{B} (U) \hat{\cup} {C I S H S}_{A} (U)$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 0 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 1 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 0 e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, 1 e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, 0 e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, a^{'} e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, 0 e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, 0 e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, 0 e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, a^{'} e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, 0 e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m a x (θ_{B}^{φ}, θ_{A}^{φ})}, a^{'} e^{j m i n (θ_{B}^{ϑ}, θ_{A}^{ϑ})} ⟩$

Table 5.

$C {I S H S}_{A} (U) \hat{\cap} C {I S H S}_{B} (U)$ .

${C I S H S}_{A, B} (U)$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j m a x ((θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$

Table 6.

$C {I S H S}_{A} (U) \hat{\cap} {C I S H S}_{\emptyset} (U) = {C I S H S}_{\emptyset} (U)$ .

$C {I S H S}_{A} (U)$	$({C I S H S}_{\emptyset} (U))$	$C {I S H S}_{A} (U) \hat{\cap} ({C I S H S}_{\emptyset} (U))$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j 0}, 1 e^{j 2 π} ⟩$	$⟨ 0 e^{j m i n (θ_{A}^{φ}, 0)}, 1 e^{j m a x (θ_{A}^{ϑ}, 2 π)} ⟩ = ⟨ 0 e^{j 0}, 1 e^{j 2 π} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j 0}, 1 e^{j 2 π} ⟩$	$⟨ 0 e^{j m i n (θ_{A}^{φ}, 0)}, 1 e^{j m a x (θ_{A}^{ϑ}, 2 π)} ⟩ = ⟨ 0 e^{j 0}, 1 e^{j 2 π} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j 0}, 1 e^{j 2 π} ⟩$	$⟨ 0 e^{j m i n (θ_{A}^{φ}, 0)}, 1 e^{j m a x (θ_{A}^{ϑ}, 2 π)} ⟩ = ⟨ 0 e^{j 0}, 1 e^{j 2 π} ⟩$

Table 7.

$\bar{\bar{C {I S H S}_{A} (U)}} = C {I S H S}_{A} (U)$ .

$C {I S H S}_{A} (U)$	$\bar{C {I S H S}_{A} (U)}$	$\bar{\bar{C {I S H S}_{A} (U)}}$
$⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩$	$⟨ 0 e^{j 2 π - θ^{φ}}, 1 e^{j 2 π - θ^{ϑ}} ⟩$	$⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩$
$⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩$	$⟨ 1 e^{j 2 π - θ^{φ}}, 0 e^{j 2 π - θ^{ϑ}} ⟩$	$⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩$
$⟨ a e^{j θ^{φ}}, a^{^{'}} e^{j θ^{ϑ}} ⟩$	$⟨ a e^{j 2 π - θ^{φ}}, a^{^{'}} e^{j 2 π - θ^{ϑ}} ⟩$	$⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩$

Table 8.

LHS= $\bar{C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U)}$ .

C ${I S H S}_{A} (U)$	$C {I S H S}_{B} (U)$	${C I S H S}_{A} (U) \hat{\cup} {C I S H S}_{B} (U)$	$(C {{I S H S}_{A} (U) \hat{\cup} {C I S H S}_{B} (U))}^{c}$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ 1 e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j 2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \\ a^{'} e^{j 2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} \end{matrix} ⟩$

Table 9.

RHS= $\bar{C {I S H S}_{A} (U)} \hat{\cap} \bar{C {I S H S}_{B} (U)}$ .

$\bar{C {I S H S}_{A} (U)}$	$\bar{C {I S H S}_{B} (U)}$	$\bar{C {I S H S}_{A} (U)} \hat{\cap} \bar{C {I S H S}_{B} (U)}$
$⟨ 1 e^{j (2 π - θ_{A}^{φ})}, 0 e^{j (2 π - θ_{A)}^{ϑ}} ⟩$	$⟨ 1 e^{j (2 π - θ_{B}^{φ})}, 0 e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, 0 e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ 1 e^{j (2 π - θ_{A}^{φ})}, 0 e^{j (2 π - θ_{A)}^{ϑ}} ⟩$	$⟨ 0 e^{j (2 π - θ_{B}^{φ})}, 1 e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ 0 e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, 1 e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ 1 e^{j (2 π - θ_{A}^{φ})}, 0 e^{j (2 π - θ_{A)}^{ϑ}} ⟩$	$⟨ a e^{j (2 π - θ_{B}^{φ})}, a^{'} e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, a^{'} e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ 0 e^{j (2 π - θ_{A}^{φ})}, 1 e^{j (2 π - θ_{A}^{ϑ})} ⟩$	$⟨ 1 e^{j (2 π - θ_{B}^{φ})}, 0 e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ 0 e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, 1 e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ 0 e^{j (2 π - θ_{A}^{φ})}, 1 e^{j (2 π - θ_{A}^{ϑ})} ⟩$	$⟨ 0 e^{j (2 π - θ_{B}^{φ})}, 1 e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ 0 e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, 1 e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ 0 e^{j (2 π - θ_{A}^{φ})}, 1 e^{j (2 π - θ_{A}^{ϑ})} ⟩$	$⟨ a e^{j (2 π - θ_{B}^{φ})}, a^{'} e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ 0 e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, 1 e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ a e^{j (2 π - θ_{A}^{φ})}, a^{^{'}} e^{j (2 π - θ_{A}^{ϑ})} ⟩$	$⟨ 1 e^{j (2 π - θ_{B}^{φ})}, 0 e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, a^{'} e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ a e^{j (2 π - θ_{A}^{φ})}, a^{^{'}} e^{j (2 π - θ_{A}^{ϑ})} ⟩$	$⟨ 0 e^{j (2 π - θ_{B}^{φ})}, 1 e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ 0 e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, 1 e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$
$⟨ a e^{j (2 π - θ_{A}^{φ})}, a^{^{'}} e^{j (2 π - θ_{A}^{ϑ})} ⟩$	$⟨ a e^{j (2 π - θ_{B}^{φ})}, a^{'} e^{j (2 π - θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})}, a^{'} e^{j m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ})} ⟩$

Table 10.

LHS = ${C I S H S}_{A} (U) \hat{\cup} (C {I S H S}_{B} (U) \hat{\cap} C {I S H S}_{C} (U))$ .

$C {I S H S}_{A} (U)$	$C {I S H S}_{B} (U)$	$C {I S H S}_{C} (U)$		${C I S H S}_{A} (U) \hat{\cup} (C {I S H S}_{B} (U)$ $\hat{\cap} C {I S H S}_{C} (U))$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{C}^{φ}}, 1 e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ 1 e^{j m a x ((θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))}, \\ 1 e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j θ_{C}^{φ}}, a^{'} e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ a^{'} e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 0 e^{j θ_{A}^{φ}}, 1 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{C}^{φ}}, 0 e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ a^{'} e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{C}^{φ}}, 1 e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ 1 e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j θ_{C}^{φ}}, a^{'} e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m i n ((θ_{B}^{φ}, θ_{C}^{φ})}, \\ a^{'} e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))}, \\ 0 e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j θ_{A}^{φ}}, 0 e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{C}^{φ}}, 0 e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ a^{'} e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))} \\ 0 e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{B}^{φ}}, 1 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 0 e^{j θ_{C}^{φ}}, 1 e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ 1 e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))} \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{B}^{φ}}, 0 e^{j θ_{B}^{ϑ}} ⟩$	$⟨ a e^{j θ_{C}^{φ}}, a^{'} e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ a^{'} e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ a e^{j θ_{A}^{φ}}, a^{^{'}} e^{j θ_{A}^{ϑ}} ⟩$	$⟨ a e^{j θ_{B}^{φ}}, a^{'} e^{j θ_{B}^{ϑ}} ⟩$	$⟨ 1 e^{j θ_{C}^{φ}}, 0 e^{j θ_{C}^{ϑ}} ⟩$	$⟨ \begin{matrix} a e^{j m i n (θ_{B}^{φ}, θ_{C}^{φ})}, \\ a^{'} e^{j m a x (θ_{B}^{ϑ}, θ_{C}^{ϑ})} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j max (θ_{A}^{φ}, min (θ_{B}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$

Table 11.

RHS = $(C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U)) \hat{\cap} (C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{C} (U))$ .

$C {I S H S}_{A} (U) \hat{\cup} {C I S H S}_{B} (U)$	C ${I S H S}_{A} (U) \hat{\cup} {C I S H S}_{C} (U)$	(C ${I S H S}_{A} (U) \hat{\cup} {I S H S}_{B} (U)) \hat{\cap}$ C ${I S H S}_{A} (U) \hat{\cup} {C I S H S}_{C} (U)$
$⟨ 0 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 1 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 0 e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, 1 e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} 0 e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ 1 e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} a e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, a^{^{'}} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} a e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} 1 e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ 0 e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} 1 e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ 0 e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} 1 e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ 0 e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, a^{^{'}} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} a e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, a^{'} e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} a e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$
$⟨ a e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, a^{^{'}} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩$	$⟨ 1 e^{j m a x (θ_{A}^{φ}, θ_{C}^{φ})}, 0 e^{j m i n (θ_{A}^{ϑ}, θ_{C}^{ϑ})} ⟩$	$⟨ \begin{matrix} a e^{j m i n (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ}))}, \\ a^{'} e^{j m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ}))} \end{matrix} ⟩$

Definition 3.1

Let $U$ represents the set of objects called the universe of discourse, and $μ_{A} : U \to φ_{A} (x) e^{j θ_{A}^{φ} (x)}$ , $v_{A} : U \to ϑ_{A} (x) e^{j θ_{A}^{ϑ} (x)}$ constructed the CIFS $A = f (x)$ over $U$ . Given a pair of thresholds $(w, p)$ ; with $0 \leq w \leq 1$ and $0 \leq p \leq 1$ . The $C {I S H S}_{A} (U) = {⟨ x, α_{(w, p)} f (x) ⟩ | x \in U}$ , where

$α_{(w, p)} : U \to {⟨ 1 e^{{j θ}^{φ}}, 0 e^{j θ^{ϑ}} ⟩$ , $⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩, ⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩}$ given by

α_{(w, p)} (f (x)) = ⟨ \hat{φ} e^{j θ^{φ}}, \hat{ϑ} e^{j θ^{ϑ}} ⟩ = {\begin{cases} ⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩, i f φ_{A} (x) \geq w a n d ϑ_{A} (x) < p \\ ⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩, i f φ_{A} (x) < w a n d ϑ_{A} (x) \geq p \\ ⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩, o t h e r w i s e, \end{cases}

where

a, a^{'} \in [0, 1]

. Also, each element in the range of

α_{(w, p)} (f (x))

satisfied the CIFS constrains, ((i.e.,

1 + 0, 0 + 1, and a + a^{'}

are less than or equal

1

, and

θ^{φ}

θ^{ϑ} \in [0, 2 π]

and

0 \leq θ^{φ} + θ^{ϑ} \leq 2 π

Now, $\forall x \in U$ , the CISHS $α$ is a mapping defined in $U$ into three regions in the unit disk, where the grade $⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩$ characterizes objects $‘ ‘ U^{″}$ that are full amplitude terms of members and zero amplitude terms of non-members in phase terms $θ^{φ}$ and $θ^{ϑ}$ , respectively. So, $U$ maps to $⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩$ usually have an amplitude term of membership with a large value, and amplitude term of non-membership with a small value $(φ_{A} (x) \geq w and ϑ_{A} (x) < p)$ in phase terms $θ^{φ}$ and $θ^{ϑ}$ , respectively, i.e. $⟨ μ_{A} (x), v_{A} (x) ⟩$ is close to $⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩$ . So, they accept those objects in $U$ . This region is shown in Figure 1 as the largest ring $R_{1}$ in membership shape and smallest disk $R_{3}$ in the non-membership shape, where $θ$ moving from $0$ to $2 π .$

The objects $U$ maps to $⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩$ usually have an amplitude term of membership with a large value and amplitude term of non-membership with a large value ( $φ_{A} (x) \geq w and ϑ_{A} (x) \geq p$ ) in phase terms $θ^{φ}$ and $θ^{ϑ}$ , respectively. Or an amplitude term of membership with a small value and amplitude term of non-membership with a small value( $ϑ_{A} (x) < w and φ_{A} (x) <$ $p)$ in phase terms $θ^{φ}$ and $θ^{ϑ}$ , respectively. So, they shadowed zone those objects in $U .$ This region is shown in Figure 1 as the ring $R_{2}$ in both membership and non-membership shapes, where $θ$ moving from $0$ to $2 π .$

The grade $⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩$ characterizes objects $" U "$ that are full amplitude terms of non-members and zero amplitude terms of members in phase terms $θ^{φ}$ and $θ^{ϑ}$ , respectively. So, $U$ maps to $⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩$ usually have an amplitude term of membership with a small value, and amplitude term of non-membership with a large value $(φ_{A} (x) < w and ϑ_{A} (x) \geq p)$ in phase terms $θ^{φ}$ and $θ^{ϑ}$ , respectively, i.e. $⟨ μ_{A} (x), v_{A} (x) ⟩$ is close to $⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩$ . So, they reject those objects in $U$ . This region is shown in Figure 1 as the internal disk $R_{3}$ in membership shape and largest ring $R_{1}$ in non-membership shape, where $θ$ moving from $0$ to $2 π .$ Therefore, in Figure 1, the red, yellow, and white colors represent the rejected, accepted, and shadowed regions, respectively, within both the membership and non-membership unit disks. Notably, the red region corresponds simultaneously to the innermost disk (R $_{3}$ ) in the membership function and the outermost ring (R $_{1}$ ) in the non-membership function for the same set of objects. Similarly, the yellow region represents the outermost ring (R $_{1}$ ) in the membership function and the innermost disk (R $_{3}$ ) in the non-membership function. Finally, the white region corresponds to the intermediate ring (R $_{2}$ ) in both the membership and non-membership functions for the same objects.

Definition 3.2

Let $C {I S H S}_{A} (U)$ and $C {I S H S}_{B} (U)$ be two complex intuitionistic shadowed sets over common universe $U$ . Then, $C {I S H S}_{A} (U)$ is a subset of $C {I S H S}_{B} (U)$ , denoted as $C {I S H S}_{A} (U) \hat{\subseteq} C {I S H S}_{B} (U)$ , if and only if $\forall x \in U,$ ${\hat{φ}}_{A} (x)$ $\leq$ ${\hat{φ}}_{B} (x)$ and ${\hat{ϑ}}_{A} (x) \geq {\hat{ϑ}}_{B} (x)$ . Also, $θ^{φ_{A}} \leq θ^{φ_{B}}$ , and $θ^{ϑ_{A}} \geq θ^{ϑ_{B}}$ . In the case of equality holds, then $C {I S H S}_{A} (U) \hat{=} C {I S H S}_{B} (U)$ .

The following definitions may elaborate some special cases of the CIFHS sets. Definition 3.3, illustrate the Null CISHS characterizes all values/objects in the Universe that are full amplitude terms of non-members and zero amplitude terms of members in phase terms $θ^{φ}$ and $θ^{ϑ}$ . While Definition 3.4 is the absolute CISHS that characterizes all values/objects in the Universe that are full amplitude terms of non-members and zero amplitude terms of members in phase terms $θ^{φ}$ and $θ^{ϑ}$ . And Definition 3.5 called a completely CISHS characterizes all values/objects in the Universe that has amplitude term of membership and non-membership with a large value, or amplitude term of membership and non-membership with a small value.

Definition 3.3

A null complex intuitionistic shadowed set $C {I S H S}_{\emptyset} (U)$ over a universe $U$ is a complex intuitionistic shadowed set with $C {I S H S}_{\emptyset} (U) = 0 e^{j {(θ}^{φ} = 0)}, 1 e^{j (θ^{ϑ} = 2 π)},$ $\forall x \in U$ , where $j = \sqrt{- 1}$ .

Definition 3.4

An absolute complex intuitionistic shadowed set ${C I S H S}_{ψ} (U)$ over a universe $U$ is a complex intuitionistic shadowed set with ${C I S H S}_{ψ} (U) = 1 e^{j {(θ}^{φ} = 2 π)}, 0 e^{j (θ^{ϑ} = 0)},$ $\forall x \in U$ , where $j = \sqrt{- 1}$ and $θ^{φ}, θ^{ϑ} \in [0, 2 π]$ .

Definition 3.5

A completely complex intuitionistic shadowed set ${C I S H S}_{Ω} (U)$ over a universe $U$ is a complex intuitionistic shadowed set with ${C I S H S}_{Ω} (U) = a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}}$ , where $j = \sqrt{- 1}$ and $θ^{φ}, θ^{ϑ} \in [0, 2 π]$ .

Definition 3.6

Let $U$ be universe of discourse and $C {I S H S}_{A} (U)$ is the complex intuitionistic shadowed set and defined as $C {I S H S}_{A} : U \to {⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩, ⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩, ⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩}$ . Then, the complement of $C {I S H S}_{A} (U)$ , denoted $\bar{{C I S H S}_{A} (U)}$ , is defined by: $\bar{{C I S H S}_{A} (U)} = {⟨ 0 e^{j \bar{θ^{φ}}}, 1 e^{j \bar{θ^{ϑ}}} ⟩, ⟨ 1 e^{j \bar{θ^{φ}}}, 0 e^{j \bar{θ^{ϑ}}} ⟩, ⟨ a e^{j \bar{θ^{φ}}}, a^{'} e^{j \bar{θ^{ϑ}}} ⟩}$ , where $θ \in [0, 2 π]$ and the complement $C I S H S$ presented in Table 2.

Example 3.1

Let $U = {x_{1}, x_{2}$ , $x_{3}$ , $x_{4}}$ be a universe of discourse. Then, a CIFS $A$ is: $A = {\frac{x_{1}}{< 0.5 e^{j 0.1 π}, 0.4 e^{j 0.6 π} >}, \frac{x_{2}}{< 0.6 e^{j 0.5 π}, 0.3 e^{j π} >}, \frac{x_{3}}{< 0.4 e^{j 0.9 π}, 0.1 e^{j 0.2 π} >}, \frac{x_{4}}{< 0.7 e^{j 0.2 π}, 0.1 e^{j 0.9 π} >}}$ . Let $w = 0.6$ , and $p = 0.2$ be the threshold values. Then,

\begin{aligned} C {I S H S}_{A} (U) & = {\frac{x_{1}}{< 0 e^{j 0.1 π}, 1 e^{j 0.6 π} >}, \frac{x_{2}}{,} \frac{x_{3}}{}, \frac{x_{4}}{< 1 e^{j 0.2 π}, 0 e^{j 0.9 π} >}} . \\ \bar{C {I S H S}_{A} (U)} & = {\frac{x_{1}}{< 1 e^{j 1.9 π}, 0 e^{j 1.4 π} >}, \frac{x_{2}}{,} \frac{x_{3}}{}, \frac{x_{4}}{< 0 e^{j 1.8 π}, 1 e^{j 1.1 π} >}} \end{aligned}

Definition 3.7

The union of two complex intuitionistic shadowed sets ${C I S H S}_{A} (U)$ and $C {I S H S}_{B} (U)$ over $U$ is denoted as $C {I S H S}_{A} (U) \cup C {I S H S}_{B} (U)$ , where $C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U) = {⟨ x, \max {{\hat{φ}}_{A} (x), {\hat{φ}}_{B} (x)} e^{j m a x (θ_{A}^{φ}, θ_{B}^{φ})}, \min {({\hat{ϑ}}_{A} (x), {\hat{ϑ}}_{B} (x)} e^{j m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ}} ⟩ | \forall x \in U}$ , and presented in Table 3.

Example 3.2

Let $A$ and $B$ be two CIFSs over $U = {x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}}$ defined as follows:

\begin{aligned} A & = {\frac{x_{1}}{< 0.5 e^{j 0.7 π}, 0.3 e^{j 0.2 π} >}, \frac{x_{2}}{< 0 e^{j 0.5 π}, 0.2 e^{j 0.2 π} >}, \frac{x_{3}}{< 0.1 e^{j 0.5 π}, 0.2 e^{j 0.2 π} >}, \frac{x_{4}}{< 0 {.8 e}^{j π}, 0.2 e^{j 0.6 π} >}}, \\ B & = {\frac{x_{1}}{< 0.2 e^{j 0.5 π}, 0.6 e^{j 0.2 π} >}, \frac{x_{2}}{< 0.7 e^{j 0.2 π}, 0.3 e^{j 0.1 π} >}, \frac{x_{3}}{< 0.6 e^{j 0.5 π}, 0.1 e^{j 0.3 π} >}, \frac{x_{4}}{< 0.8 e^{j 0.1 π}, 0.1 e^{j 0.9 π} >}} . \end{aligned}

Let

w = 0.6

, and

p = 0.3

be the threshold values. Then, the union of two CISHSs

A

and

B

is:

\begin{aligned} C {I S H S}_{A} (U) & = {\frac{x_{1}}{,} \frac{x_{2}}{,} \frac{x_{3}}{}, \frac{x_{4}}{< 1 e^{j π}, 0 e^{j 0.6 π} >}}, \\ C {I S H S}_{B} (U) & = {\frac{x_{1}}{< 0 e^{j 0.5 π}, 1 e^{j 0.2 π} >}, \frac{x_{2}}{,} \frac{x_{3}}{< 1 e^{j 0.5 π}, 0 e^{j 0.3 π} >}, \frac{x_{4}}{< 1 e^{j 0.1 π}, 0 e^{j 0.9 π} >}} . \\ C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U) & = {\frac{x_{1}}{,} \frac{x_{2}}{,} \frac{x_{3}}{< 1 e^{j 0.5 π}, 0 e^{j 0.2 π} >}, \frac{x_{4}}{< 1 e^{j π}, 0 e^{j 0.6 π} >}} . \end{aligned}

Proposition 3.1

Let ${C I S H S}_{A} (U)$ , ${C I S H S}_{B} (U)$ , and ${C I S H S}_{C} (U)$ be any three CISHSs with a common universe $U$ . Then, the following results hold: a.

$C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U) = {C I S H S}_{B} (U) \hat{\cup} C {I S H S}_{A} (U)$ ,

${C I S H S}_{A} (U) \hat{\cup} (C {I S H S}_{B} (U) \hat{\cup} {C I S H S}_{C} (U)) =$

(C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U)) \hat{\cup} C {I S H S}_{C},

$C {I S H S}_{A} (U) \hat{\cup} {C I S H S}_{\emptyset} (U) = C {I S H S}_{A} (U)$ ,

$C {I S H S}_{A} (U) \hat{\cup} {C I S H S}_{ψ} (U) = {C I S H S}_{ψ} (U)$ ,

$C {I S H S}_{A} (U) \hat{\cup} {CISHS}_{Ω} (U) \hat{\subseteq} {C I S H S}_{ψ} (U)$ .

Proof. For part (a), we prove it by using Definition 3.7 as in (Table 4).

Clearly, the third column is equal to the fourth column in Table 4.

For parts (b), (c), (d), and (e) the proof is straightforward based on Definitions 3.2, 3.3, 3.4, 3.5, and 3.7. Definition 3.8

The intersection of two complex intuitionistic shadowed sets $C {I S H S}_{A} (U)$ and $C {I S H S}_{B} (U)$ over $U$ denoted as $C {I S H S}_{A} (U) \hat{\cap} C {I S H S}_{B} (U)$ , where $C {I S H S}_{A} (U) \hat{\cap} C {I S H S}_{B} (U) = {⟨ x, \min {{\hat{φ}}_{A} (x), {\hat{φ}}_{B} (x)} e^{j m i n (θ_{A}^{φ}, θ_{B}^{φ})}, m a x {{\hat{ϑ}}_{A} (x), {\hat{ϑ}}_{B} (x)} e^{j m a x (θ_{A}^{ϑ}, θ_{B}^{ϑ})} ⟩ | \forall x \in U},$ and presented in Table 5.

Example 3.3

Let $A$ and $B$ be two CIFSs over $U = {x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}}$ defined as follows:

\begin{aligned} A & = {\frac{x_{1}}{< 0.7 e^{j 0.4 π}, 0.1 e^{j 0.9 π} >}, \frac{x_{2}}{< 0 e^{j 1.5 π}, 0.1 e^{j 0.5 π} >}, \frac{x_{3}}{< 0.5 e^{j 0.2 π}, 0.1 e^{j π} >}, \frac{x_{4}}{< 0.6 e^{j π}, 0.3 e^{j 0.1 π} >}}, \\ B & = {\frac{x_{1}}{< 0.5 e^{j 0.5 π}, 0.3 e^{j 1.2 π} >}, \frac{x_{2}}{< 0.8 e^{j 0.7 π}, 0.2 e^{j 0.6 π} >}, \frac{x_{3}}{< 0.4 e^{j 0.1 π}, 0.5 e^{j 0.3 π} >}, \frac{x_{4}}{< 0.8 e^{j π}, 0.1 e^{j 0.4 π} >}} \end{aligned}

Let

w = 0.7

, and

p = 0.2

be the threshold values. Then the intersection of the two complex intuitionistic shadowed sets

A

and

B

is:

\begin{aligned} C {I S H S}_{A \hat{\cap} B} (U) & = {\frac{x_{1}}{< 1 e^{j 0.4 π}, 0 e^{j 0.9 π} >}, \frac{x_{2}}{,} \frac{x_{3}}{}, \frac{x_{4}}{< 0 e^{j π}, 1 e^{j 0.1 π} >}}, \\ C {I S H S}_{B} (U) & = {\frac{x_{1}}{< 0 e^{j 0.5 π}, 1 e^{j 1.2 π} >}, \frac{x_{2}}{,} \frac{x_{3}}{< 0 e^{j 0.1 π}, 1 e^{j 0.3 π} >}, \frac{x_{4}}{< 1 e^{j π}, 0 e^{j 0.4 π} >}} \\ C {I S H S}_{A} (U) \hat{\cap} C {I S H S}_{B} (U) & = {\frac{x_{1}}{< 0 e^{j 0.4 π}, 1 e^{j 1.2 π} >}, \frac{x_{2}}{,} \frac{x_{3}}{< 0 e^{j 0.1 π}, 1 e^{j π} >}, \frac{x_{4}}{< 0 e^{j π}, 1 e^{j 0.4 π} >}} . \end{aligned}

Proposition 3.2

Let $C {I S H S}_{A} (U)$ , $C {I S H S}_{B} (U)$ , and ${C I S H S}_{C} (U)$ be any three Complex intuitionistic shadowed sets with common universe $U$ . Then, the following results hold. a.

$C {I S H S}_{A} (U) \hat{\cap} {C I S H S}_{B} (U) = {C I S H S}_{B} (U) \hat{\cap} C {I S H S}_{A} (U)$ ,

$C {I S H S}_{A} (U) \hat{\cap} (C {I S H S}_{B} (U) \hat{\cap} C {I S H S}_{C} (U)) =$

(C {I S H S}_{A} (U) \hat{\cap} C {I S H S}_{B} (U)) \hat{\cap} C {I S H S}_{C} (U),

$C {I S H S}_{A} (U) \hat{\cap} {C I S H S}_{\emptyset} (U) = ({C I S H S}_{\emptyset} (U))$ ,

$C {I S H S}_{A} (U) \hat{\cap} {C I S H S}_{ψ} (U) = C {I S H S}_{A} (U)$ ,

$C {I S H S}_{A} (U) \hat{\cap} {CISHS}_{Ω} (U) \hat{\subseteq} C {I S H S}_{A} (U)$ .

Proof. For part (c), we prove it by using Definition 3.3 as in (Table 6).

Clearly, the second column is equal to the third column in Table 6. For parts (a), (b), (d), and (e) the proof is straightforward based on Definitions 3.2, Definition 3.4, Definition 3.5, and Definition 3.8.

Proposition 3.3

Let $C {I S H S}_{A} (U)$ be a CISHS over $U$ . Then, the following property holds:

\bar{\bar{C {I S H S}_{A} (U)}} = C {I S H S}_{A} (U) .

Proof: The proof is straightforward based on Definition 3.1, see Table 7.

Clearly, the third column is equal to the first column in Table 7 that proved the property $\bar{\bar{C {I S H S}_{A} (U)}} = C {I S H S}_{A} (U)$ .

Proposition 3.4

Let $C {I S H S}_{A} (U)$ and $C {I S H S}_{B} (U)$ be two CISHSs. Then, the following Demorgan’s laws properties holds: a.

$\bar{C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U)} = \bar{C {I S H S}_{A} (U)} \hat{\cap} \bar{C {I S H S}_{B} (U)}$

$\bar{C {I S H S}_{A} (U) \hat{\cap} C {I S H S}_{B} (U)} = \bar{C {I S H S}_{A} (U)} \hat{\cup} \bar{C {I S H S}_{B} (U)}$

Proof. The proof of (a) for the left-hand side (LHS) is conducted in Table 8, and the right-hand side (RHS) in Table 9, and by a similar method, we can prove (b).

Note that $2 π - m a x (θ_{A}^{φ}, θ_{B}^{φ}) = m i n (\bar{θ_{A}^{φ}}, \bar{θ_{B}^{φ}}) = m i n (2 π - θ_{A}^{φ}, 2 π - θ_{B}^{φ})$ , and $2 π - m i n (θ_{A}^{ϑ}, θ_{B}^{ϑ}) = m a x (\bar{θ_{A}^{φ}}, \bar{θ_{B}^{φ}}) = m a x (2 π - θ_{A}^{ϑ}, 2 π - θ_{B}^{ϑ}) .$

Proposition 3.5

Let ${C I S H S}_{A} (U)$ , ${C I S H S}_{B} (U)$ , and ${C I S H S}_{C} (U)$ be three CISHSs. Then, the following results hold: a.

${C I S H S}_{A} (U) \hat{\cup} (C {I S H S}_{B} (U) \hat{\cap} C {I S H S}_{C} (U)) =$

(C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{B} (U)) \hat{\cap} (C {I S H S}_{A} (U) \hat{\cup} C {I S H S}_{C} (U)),

${C I S H S}_{A} (U) \hat{\cap} ({C I S H S}_{B} (U) \hat{\cup} {C I S H S}_{C} (U)) =$

({C I S H S}_{A} (U) \hat{\cap} {C I S H S}_{B} (U)) \hat{\cup} ({C I S H S}_{A} (U) \hat{\cap} {C I S H S}_{C} (U)) .

Proof. The proof of the left-hand side (LHS) of (a) is conducted in Table 10 and right-hand side (RHS) in Table 11, and by a similar method, we can prove part (b).

Note that $min (\max (θ_{A}^{φ}, θ_{B}^{φ}), \max (θ_{A}^{φ}, θ_{C}^{φ})) = max (θ_{A}^{φ}, \min (θ_{B}^{φ}, θ_{C}^{φ}))$ , and

$m a x (\min (θ_{A}^{ϑ}, θ_{B}^{ϑ}), \min (θ_{A}^{ϑ}, θ_{C}^{ϑ})) = m i n (θ_{A}^{ϑ}, \max (θ_{B}^{ϑ}, θ_{C}^{ϑ}))$ .

4. Comparison Analysis Between CSHS and CISHS

The innovation lies in the richness of data types and the level of detail they carry; thus, constructing a set capable of fully capturing and emphasizing this detailed information becomes essential for effective analysis. Also, The CIFS stands out from CFS by using two functions instead of one, allowing it to represent both the fuzziness and the periodic nature of data in more details and comprehensively. As known in the literature, the CIFS can address and solving certain real-world problems that CFS cannot. CISHS is considered as a generalization of CSHS, it incorporates additional complex non-membership functions that enhance its representational power and applicability. A more thorough examination of ambiguity and repetition is made possible by CISHS, which specifically offers a multi-dimensional viewpoint on complex membership, complex non-membership, and complex hesitation degrees. This extension enables researchers to tackle more intricate problems that cannot be adequately addressed by CSHS alone. Also, Table 12. Represent how many regions, number of functions that represent the information, the mathematical structures, universe of discourse, and graphs shape for CSHS and CISHS.

5. Application

In this section, a numerical example is studied to show the efficiency of the proposed algorithm. For this purpose, a selection model is used based on Reference (Garg & Rani, 2019). Moreover, CISHS is utilized to get the desired machine or the best alternative for a company to select a machine from five models with its production date simultaneously. Therefore, they are two-dimensional problems, i.e., the model and production date of the machine. As a result, the following steps are used to reach a decision:

Table 12.
Comparison Between CSHS and CISHS.

CSHS CISHS

The codomain function Three regions Three regions

Information Representation Information is represented by one complex membership function. Information is represented by both complex membership and complex non-membership functions.

Mathematical structure of three regions Accept: $⟨ 1 e^{j θ^{φ}} ⟩,$ Reject: $⟨ 0 e^{j θ^{φ}} ⟩,$ Shadow: $⟨ a e^{j θ^{φ}} ⟩$ Accept: $⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩,$ Reject: $⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩,$ Shadow: $⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩$

Universe of discourse All values lie in the unit disk of the complex plane All values lie in the unit disk of the complex plane

Graph shape Need one unit disk to represent three regions. Need two-unit disks to represent the three regions.

	CSHS	CISHS
The codomain function	Three regions	Three regions
Information Representation	Information is represented by one complex membership function.	Information is represented by both complex membership and complex non-membership functions.
Mathematical structure of three regions	Accept: $⟨ 1 e^{j θ^{φ}} ⟩,$ Reject: $⟨ 0 e^{j θ^{φ}} ⟩,$ Shadow: $⟨ a e^{j θ^{φ}} ⟩$	Accept: $⟨ 1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}} ⟩,$ Reject: $⟨ 0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}} ⟩,$ Shadow: $⟨ a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}} ⟩$
Universe of discourse	All values lie in the unit disk of the complex plane	All values lie in the unit disk of the complex plane
Graph shape	Need one unit disk to represent three regions.	Need two-unit disks to represent the three regions.

Table 13.

Information in the Form of CISHS.

	$V_{1}$	$V_{2}$	$V_{3}$	$V_{4}$	$V_{5}$
$B_{1}$	$⟨ \begin{matrix} 1 e^{j 2 π (0.9)}, \\ 0 e^{j 2 π (0.1)} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j 2 π (0.6)}, \\ a^{'} e^{j 2 π (0.3)} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π (0.4)}, \\ 1 e^{j 2 π (0.4)} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π (0.8)}, \\ 1 e^{j 2 π (0.1)} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π (0.7)}, \\ 0 e^{j 2 π (0.2)} \end{matrix} ⟩$
$B_{2}$	$⟨ \begin{matrix} 1 e^{j 2 π (0.5)}, \\ 0 e^{j 2 π (0.4)} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j 2 π (0.9)}, \\ a^{'} e^{j 2 π (0.1)} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j 2 π (0.6)}, \\ a^{'} e^{j 2 π (0.4)} \end{matrix} ⟩$	$⟨ \begin{matrix} a e^{j 2 π (0.3)}, \\ a^{'} e^{j 2 π (0.3)} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π (0.7)}, \\ 0 e^{j 2 π (0.1)} \end{matrix} ⟩$
$B_{3}$	$⟨ \begin{matrix} a e^{j 2 π (0.6)}, \\ a^{'} e^{j 2 π (0.2)} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π (0.7)}, \\ 0 e^{j 2 π (0.3)} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π (0.4)}, \\ 1 e^{j 2 π (0.6)} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π (0.5)}, \\ 0 e^{j 2 π (0.3)} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π (0.6)}, \\ 0 e^{j 2 π (0.2)} \end{matrix} ⟩$
$B_{4}$	$⟨ \begin{matrix} a e^{j 2 π (0.7)}, \\ a^{'} e^{j 2 π (0.2)} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π (0.6)}, \\ 1 e^{j 2 π (0.1)} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π (0.7)}, \\ 0 e^{j 2 π (0.1)} \end{matrix} ⟩$	$⟨ \begin{matrix} 0 e^{j 2 π (0.5)}, \\ 1 e^{j 2 π (0.4)} \end{matrix} ⟩$	$⟨ \begin{matrix} 1 e^{j 2 π (0.8)}, \\ 0 e^{j 2 π (0.1)} \end{matrix} ⟩$

Step 1: Provide $n$ candidates $V_{r}, r = 1, 2, 3, 4, \dots, n$ for employment using the CIFS form.

Step 2: Assume two parameters ( $w,\ p)$ as threshold values.

Step 3: Approximation $V_{r}$ from CIFS to CISHS using values in step 2 on a tabular form.

Step 4: Find the score of each element in $U$ as the value of $X_{A c c}$ , $X_{S h}$ , and $X_{R e j}$ , where -

$X_{A c c}$ = $\frac{n u m b e r o f r e b e t i t i o n o f (1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}}) \sum_{A c c} (θ^{φ} + θ^{ϑ})}{2 (m) (π)}$ , where $m$ is the number of elements, and $\sum_{A c c} (θ^{φ} + θ^{ϑ})$ is the summation of the sum of $θ^{φ},$ and $θ^{ϑ}$ , corresponds to the term $(1 e^{j θ^{φ}}, 0 e^{j θ^{ϑ}}) .$

$X_{S h}$ = $\frac{n u m b e r o f r e b e t i t i o n o f (a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}}) \sum_{S h} (θ^{φ} + θ^{ϑ})}{2 (m) (π)}$ , where $m$ is the number of element, and $\sum_{S h} (θ^{φ} + θ^{ϑ})$ is the summation of the sum of $θ^{φ},$ and $θ^{ϑ}$ , corresponds to the term $(a e^{j θ^{φ}}, a^{'} e^{j θ^{ϑ}})$ .

$X_{R e j}$ = $\frac{n u m b e r o f r e b e t i t i o n o f (0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}}) \sum_{R e j} (θ^{φ} + θ^{ϑ})}{2 (m) (π)}$ , where $m$ is the number of element, and $\sum_{R e j} (θ^{φ} + θ^{ϑ})$ is the summation of the sum of $θ^{φ},$ and $θ^{ϑ}$ , corresponds to the term $(0 e^{j θ^{φ}}, 1 e^{j θ^{ϑ}})$ .

Step 5: Rank the feasible alternatives, hence select the best alternative that has the largest number of $X_{A c c}$ .

Example 5.1

Assume a contractor wants to buy for his company a new machine from a machine maker. The machine maker offers information on five models of machine which form the set of alternatives, $U =$ ( $V_{1}, V_{2}, V_{3}, V_{4}, V_{5})$ with varying production dates. To select a machine, the contractor consider four criteria as a set of parameters, namely , $B =$ ( $B_{1}, B_{2}, B_{3}, B_{4}),$ where $B_{1} : Reliability$ , $B_{2}$ : Safety, $B_{3} :$ Flexibility, and $B_{4} :$ Productivity. The optimal alternative can now be obtained by implementing the steps of the proposed approach to CIF data, which is recalled from Example 2.2.6, as follows:

Step 1: Provide five candidates $V_{r}, r = 1, 2, 3, 4, 5$ for employment using the CIFS form

\begin{aligned} V_{1} & = {\begin{matrix} \frac{B_{1}}{⟨ 0.7 e^{j 2 π (0.9)}, 0.1 e^{j 2 π (0.1)} ⟩}, \frac{B_{2}}{⟨ 0.8 e^{j 2 π (0.5)}, 0.1 e^{j 2 π (0.4)} ⟩}, \frac{B_{3}}{⟨ 0.6 e^{j 2 π (0.6)}, 0.3 e^{j 2 π (0.2)} ⟩}, \\ \frac{B_{4}}{⟨ 0.7 e^{j 2 π (0.7)}, 0.3 e^{j 2 π (0.2)} ⟩} \end{matrix}}, \\ V_{2} & = {\begin{matrix} \frac{B_{1}}{⟨ 0.7 e^{j 2 π (0.6)}, 0.3 e^{j 2 π (0.3)} ⟩}, \frac{B_{2}}{⟨ 0.4 e^{j 2 π (0.9)}, 0.2 e^{j 2 π (0.1)} ⟩}, \frac{B_{3}}{⟨ 0.7 e^{j 2 π (0.7)}, 0.2 e^{j 2 π (0.3)} ⟩}, \\ \frac{B_{4}}{⟨ 0.4 e^{j 2 π (0.6)}, 0.3 e^{j 2 π (0.1)} ⟩} \end{matrix}}, \\ V_{3} & = {\begin{matrix} \frac{B_{1}}{⟨ 0.3 e^{j 2 π (0.4)}, 0.6 e^{j 2 π (0.4)} ⟩}, \frac{B_{2}}{⟨ 0.6 e^{j 2 π (0.6)}, 0.3 e^{j 2 π (0.4)} ⟩}, \frac{B_{3}}{⟨ 0.3 e^{j 2 π (0.4)}, 0.5 e^{j 2 π (0.6)} ⟩}, \\ \frac{B_{4}}{⟨ 0.7 e^{j 2 π (0.7)}, 0.1 e^{j 2 π (0.1)} ⟩} \end{matrix}}, \\ V_{4} & = {\begin{matrix} \frac{B_{1}}{⟨ 0.4 e^{j 2 π (0.8)}, 0.5 e^{j 2 π (0.1)} ⟩}, \frac{B_{2}}{⟨ 0.7 e^{j 2 π (0.3)}, 0.3 e^{j 2 π (0.3)} ⟩}, \frac{B_{3}}{⟨ 0.6 e^{j 2 π (0.5)}, 0.1 e^{j 2 π (0.3)} ⟩}, \\ \frac{B_{4}}{⟨ 0.5 e^{j 2 π (0.5)}, 0.3 e^{j 2 π (0.4)} ⟩} \end{matrix}}, \\ V_{5} & = {\begin{matrix} \frac{B_{1}}{⟨ 0.9 e^{j 2 π (0.7)}, 0.1 e^{j 2 π (0.2)} ⟩}, \frac{B_{2}}{⟨ 0.7 e^{j 2 π (0.7)}, 0.2 e^{j 2 π (0.1)} ⟩}, \frac{B_{3}}{⟨ 0.7 e^{j 2 π (0.6)}, 0.2 e^{j 2 π (0.2)} ⟩}, \\ \frac{B_{4}}{⟨ 0.8 e^{j 2 π (0.8)}, 0.1 e^{j 2 π (0.1)} ⟩} \end{matrix}} . \end{aligned}

Step 2: Assume two parameters (w, p) as threshold values. Suppose w = 0.6, p = 0.3 based on the entrepreneur‘s requirements.

Step 3: Approximate $V_{r}$ from CIFS to CISHS using threshold values in step 2 in a tabular form (Table 13).

Step 4: Find the score function as in Table 14

Step 5: The ranking of all the feasible alternatives ( = 1, 2, 3, 4, 5) according to Table 14, is given as 5 $>$ 1 $>$ 2 $>$ 3 $\geq$ 4, which concludes that alternatives 3 and 4 have the highest rejection scores, 2 has the highest waiting score, and 5 has the highest acceptance score. This means that 5 is the optimal alternative.

Table 14.

Score of $V_{r}$ .

Score $V_{r}$	$X_{A c c}$	$X_{S h}$	$X_{R e j}$
$V_{1}$	$0.95$	0.85	0
$V_{2}$	0.25	0.95	0.175
$V_{3}$	0.2	0.25	0.9
$V_{4}$	0.2	0.15	0.9
$V_{5}$	3.4	0	0

6. Conclusions

This research presented a new concept, namely, complex intuitionistic shadowed sets. The novel idea may be used to address the issue of sophisticated intuitionistic ambiguity and frequency at the same time, which requires a significant amount of computation and data. Therefore, the main advantage of CISHSs is its ability to divide CIFS values into three regions at the unit disk of the complex plane in a flexible and comprehensive manner comparable with CSHS. These regions were illustrated to simplify and present data in an easy-to-handle format. Furthermore, several operations on CISHSs were defined, i.e., subset, complement, union, and intersection, and their properties were proved. Finally, the CISHS decision-making algorithm and real-life example were construed to demonstrate the concept’s validity and effectiveness. As shown, the score function was used by the proposed algorithm to study the best alternative for the current research. CISHS introduces additional computational complexity and may require more intricate algorithms for practical applications as a limitation. Also, some information has more details comparable with CIFS, such as bipolarity or multi parameters which cannot be represented and approximated by current approach. Therefore, to cover this limitation, we suggest extending the presented approach to study and combining shadowed set and both complex bipolar fuzzy set and complex hesitant fuzzy set as future research.

Footnotes

ORCID iDs

Abd Ulazeez MJS Alkouri

Leen Frehat

Eman A AbuHijleh

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability

The author confirms that all data generated or analyzed during this study is included in this published article. Furthermore, primary and secondary sources and data supporting the findings of this study were all publicly available at the time of submission.

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