Einstein Hybrid Structure of q-Rung Orthopair Fuzzy Soft Set and Its Application for Diagnosis of Waterborne Infectious Disease

1  Introduction

Flood is a natural disaster when water overflows its natural or artificial boundaries, such as rivers, lakes, or coastal areas. Floods can cause significant damage to human settlements, infrastructure, and agriculture and can also result in loss of life. Floods can be caused by many factors, including heavy rainfall, melting snow, hurricanes, and storm surges. Flooding can have far-reaching impacts on communities and the environment, including displacement of people, destruction of homes and businesses, loss of crops and livestock, contamination of drinking water, and damage to infrastructure, such as roads, bridges, and utilities. Floods can also lead to long-term economic, social, and environmental impacts, including reduced property values, loss of tourism, and degradation of ecosystems. It is important to be prepared for a flood and take measures to minimize its impacts, such as building flood-resistant structures, having evacuation plans, and being aware of flood warnings and alerts. It is also important to reduce the risk of flooding, such as preserving natural floodplains and wetlands, managing land use, and implementing effective drainage and stormwater management systems. Every nation has systems and infrastructure to direct rainfall into the appropriate basins and reservoirs. When it rains heavily, the water does not drain as rapidly as it should because those systems get overloaded, and water periodically surges into homes, resulting in a flood. Deforestation, lake loss, climate change, and sea level rise are the main aspects of how humans constantly damage the environment, increasing flood intensity and frequency.

Floods can increase the risk of water-borne infections and diseases. Floodwaters containing contaminated water from sewage, agricultural runoff, or hazardous waste sites can pose a significant health risk to those who come into contact with it. Some common water-borne infections and diseases associated with floods include Gastrointestinal infections: Floodwaters can contain bacteria such as Escherichia coli and Salmonella, which can cause stomach cramps, diarrhea, and other gastrointestinal symptoms. Skin infections: Floodwaters can contain bacteria such as Staphylococcus aureus and Streptococcus pyogenes, which can cause skin infections such as impetigo and cellulitis. Vector-borne diseases: Floods can increase the populations of mosquitoes, ticks, and other vectors that can transmit diseases such as the West Nile virus, dengue fever, and Lyme disease. Water-borne diseases: Floodwaters can contain parasites and bacteria such as Giardia lamblia and Cryptosporidium, which can cause water-borne diseases such as giardiasis and cryptosporidiosis. It is important to take precautions to reduce the risk of infection and disease during and after a flood. This includes avoiding contact with floodwaters whenever possible, washing hands frequently with soap and water, and wearing protective clothing and footwear when working in contaminated areas. If you suspect you have been infected with a water-borne disease, seeking medical attention as soon as possible is important. Ahmad et al. [1] described flood-associated risks with the agriculture department in flood-affected Punjab, Pakistan. Ochani et al. [2] illustrated the destruction in different areas of Pakistan after a severe attack flood in 2022 and elaborated on the health-related challenges caused by a flood. Ochani et al. [3] discussed the consequences caused by the flood in Pakistan in 2022 and argued about the effects of the flood on the feed of newborn children and their survival in a humid region.

Multi-criteria group decision-making (MCGDM) can be applied to flood-related infections and diseases to make informed decisions about the severity of certain conditions in patients affected by floods. For example, a group of experts could be assembled to evaluate a patient’s disease severity based on multiple criteria, such as the patient’s symptoms, medical history, and test results. The experts would provide their evaluations of the patient based on these criteria. The results would then be combined using an aggregation operator, such as the q-rung orthopair fuzzy soft Einstein hybrid weighted average (q-ROFSEHWA) operator, to arrive at a final decision about the severity of the disease. In this context, MCGDM can be a useful tool for improving the accuracy and reliability of disease severity assessments. By combining the opinions of multiple experts, MCGDM can account for uncertainty and imprecision in the evaluations and provide a more robust and comprehensive assessment of the disease severity. Furthermore, the use of MCGDM can improve the transparency and fairness of disease severity assessments, as the results of the assessments are based on a systematic and documented process that considers the perspectives and expertise of multiple experts. This can help ensure that decisions are based on the best available information and are made relatively and impartial.

MCGDM is a technique used to determine the best alternative to a decision-making (DM) problem by considering the opinions of a team of experts. This process is based on multiple attributes of the alternatives being considered. The uncertainty and imprecision of information can present challenges in evaluating DM problems. A traditional crisp set, which assigns binary membership values of 0 or 1 to elements, is unsuitable for dealing with such uncertainty. To address this issue, Zadeh [4] introduced the concept of fuzzy sets (FS), which assigns membership values in the range of [0,1] to elements based on their properties. Ulucay [5] developed a novel similarity measure for trapezoidal fuzzy multi-numbers to resolve multi-criteria DM complications. FS allows for considering values between 0 and 1, providing a more flexible and comprehensive approach to evaluating DM problems that involve ambiguous information. There were limitations to the original concept of FS, as they only described elements’ membership degree (MD). To address these limitations, Atanassov [6] proposed the intuitionistic fuzzy set (IFS), which considers both MD and non-membership degrees (NMD). The linear relationship between MD and NMD ensures that the sum of the two does not exceed 1, and both belong to the range of [0,1]. Xu [7] introduced aggregation operators (AOs) for IFS and explored their fundamental properties. Wang et al. [8] extended this work by applying Einstein operational laws to IFS and developing Einstein AOs and weighted AOs for IFS. Similarly, Xu et al. [9] further expanded upon this by introducing geometric AOs for IFS and using them in evaluating DM problems. Zhao et al. [10] used the Einstein operational laws to develop Einstein hybrid AOs and applied them to multi-attribute decision-making (MADM) problems. Zeng et al. [11] applied these hybrid AOs in the digitalization of the manufacturing industry.

Pythagorean fuzzy sets (PFS) were introduced by Yager [12] to address the limitations of both FS and IFS. PFS is a generalization of IFS and can capture more complex and nuanced relationships between MD and NMD. One of the key differences between PFS and IFS is that PFS allows for a non-zero value for the square root of the product of the MD and NMD. This results in a more flexible and nuanced representation of uncertainty, simultaneously allowing elements to have uncertain degrees of membership and non-membership. In addition, PFS can also effectively handle conflicting information and attributes, making them well-suited for DM problems with multiple inconsistent criteria. Therefore, introducing PFS has expanded the scope of FS in DM and allowed for more accurate and robust representations of uncertainty and complexity. Ejegwa [13] explored the use of necessity and possibility operators in PFS, and Joshi [14] derived generalized average AOs for use in a PFS environment. Rahman et al. [15] introduced the Pythagorean fuzzy weighted averaging operator, which was applied to MCGDM problems. The Max-average approach in Fermatean fuzzy composite relations was implemented by Ejegwa et al. [16] to improve the validity of Fermatean fuzzy sets in machine learning using a soft computing technique. Zeng et al. [17] presented a multi-criteria decision-making (MCDM) model based on social networking to evaluate the efficiency of robotic vehicles. Rahman et al. [18] introduced a weighted ordered aggregation operator for PFS and a weighted geometric aggregation operator. Garg [19] proposed the Einstein operational laws for PFS, developed AOs, and ordered AOs for PFS. Rahman et al. [20] demonstrated the applicability of the Einstein-weighted geometric aggregation operator for solving DM problems. Yang et al. [21] presented a DM model constructed on the Fermatean fuzzy setting that employed a combination of weighted distance and the TOPSIS method to evaluate green low-carbon ports. Yager [22] proposed a new type of FS called the q-rung orthopair fuzzy set (q-ROFS) that encompasses FS, IFS, and PFS, where the sum of the qth power of the MD and NMD must belong to [0,1] where q≥1. If q=1, the q-ROFS becomes an IFS, and if q=2, it is known as a PFS. The q-ROFS provides a more comprehensive solution for addressing fuzziness and modelling vagueness in DM than IFS and PFS. Liu et al. [23] established weighted AOs for the q-ROFS to reduce uncertainty in DM. Riaz et al. [24] built upon this concept and investigated geometric AOs for the q-ROFS to address limitations in current methods and demonstrated their practical application in real-world problems. Limboo et al. [25] designed an approach for q-rung evidence sets to cope with dispute obstacles caused by q-rung fuzzy numbers. The method described here has been chosen because to provides numerous advantages over other fuzzy or discrete number systems. Khan et al. [26] introduced the Archimedean AOs in a T-spherical fuzzy setting to resolve MADM complications.

Molodtsov [27] introduced the theory of soft sets (SS), which provides a way to describe the values of alternatives using parameters. Maji et al. [28] extended the concept by creating a hybrid structure called fuzzy soft set (FSS), which combines SS and FS to address inaccuracies in DM techniques. Maji et al. [29] developed an intuitionistic fuzzy soft set (IFSS), an SS and IFS hybrid with the same constraint as IFS. IFSS offers a broader range of options for dealing with vague information about alternatives in DM problems. Arora et al. [30] discussed the AOs for IFSS and highlighted their different properties. Peng et al. [31] introduced the Pythagorean fuzzy soft set (PFSS) by extending the idea of SS to PFS and defined various operations such as addition, multiplication, union, intersection, complement, etc. Zulqarnain et al. [32] introduced the AOs for PFSS and proposed a DM methodology, which they applied to green supplier chain management. Zulqarnain et al. [33] presented interaction AOs in the context of PFSS and demonstrated a DM approach using the developed operators and their properties. Athira et al. [34] introduced a DM technique that uses entropy and distance measures in the PFSS environment, highlighting the limitations of existing DM models. Athira et al. [35] defined new entropy measures for PFSS and showed their superiority over the existing entropy measures of IFSS. Zulqarnain et al. [36] presented the TOPSIS technique based on correlation coefficients and weighted correlation coefficients for PFSS, demonstrating its effectiveness in DM problems for green supply chain management. Siddique et al. [37] explained the basic operations in the PFSS environment and highlighted the superiority and significance of the proposed model. Riaz et al. [38] discussed cosine similarity measures, weighted cosine similarity measures, and Frobenius inner product spaces of matrices for PFSS and applied these models to evaluate DM problems. Naeem et al. [39] proposed extended models of TOPSIS and VIKOR for PFS and demonstrated their application in stock exchange investment through comparative analysis with existing methods. Zulqarnain et al. [40–42] introduced the operational laws of Einstein and proposed Pythagorean fuzzy soft Einstein weighted average and geometric operators. They further extended this idea and formulated Einstein-ordered weighted AOs for PFSS, which they used to address multiple attribute group decision-making (MAGDM) problems.

Hussain et al. [43] figured out the hybrid formation of SS and q-ROFS by demonstrating a q-rung orthopair fuzzy soft set (q-ROFSS), a more efficient and competent model to deal with uncertainties in a broader format than IFSS and PFSS. Chinram et al. [44] carried the AOs concept in q-ROFSS, established AOs for q-ROFSS, and presented algorithms based on these operators to evaluate DM problems. They discussed their importance and rationality by comparative analysis. Zulqarnain et al. [45] carried the idea of AOs in q-ROFSS, introduced different interactive AOs, and argued their characteristics, such as idempotency, boundedness, and homogeneity. There is still a need to explore more work under q-ROFSS as there is no contribution for hybrid AOs, especially based on Einstein operational laws.

1.1 Motivation and Drawback of Existing Approaches

The hybrid weighted aggregation operator represents a sophisticated and all-encompassing evolution of the traditional weighted average and ordered weighted operators. While the conventional weighted average operator solely encapsulates experts’ viewpoints, the ordered weighted operator focuses solely on the prioritized position of each expert’s opinion based on a scoring mechanism. In stark contrast, the hybrid variant seamlessly merges the substance of experts’ arguments and their relative ranking, yielding a remarkably adaptable framework for grappling with the intricate intricacies of modelling ambiguous and uncertain data within decision-making problems. There is an assumption that this study might answer some of the following research questions: How is the simulation of unpredictability in the q-ROFSS context enhanced by combining the weighted average operator and ordered weighted average operator using Einstein’s operational laws? How are the developed hybrid model’s comparative advantages over existing aggregation operators in solving MCGDM concerns? How well does the suggested model handle confusing and inaccurate information in the decision-making process when diagnosing waterborne infectious diseases brought on by floods? Compared to other well-established methodologies, which is the suggested method to assess the extent of diseases in flood-affected patients employing the q-ROFSS structure? How can the advocated hybrid model handle the drawbacks of current approaches for addressing uncertainty and conflicts in MCGDM challenges? The model presented in [43,45] utilizes an aggregation and interactive aggregation framework to handle MCGDM problems within the q-ROFSS environment, where MD and NMD are represented as q-rung orthopair fuzzy soft numbers (q-ROFSNs), reflecting the experts’ perspectives on specific alternatives based on specific criteria. If we construct a matrix containing q-ROFSNs for evaluating the MCGDM problem where I={I1,I2} is a set of experts with weight vectors j=(0.4,0.6) who are going to provide their estimation in the form of q-ROFSNs as given in the following matrix based on attributes ℷ={ℷ1,ℷ2} having vector ℊ=(0.2,0.8). about the alternatives ε={ε1} whose associated weight vectors are τ′={0.7,0.3} and τ″={0.4,0.7}, κ=[(0.5,0.6)(0.3,0.4)(0.4,0.0)(0.2,0.7)]. The aggregated value obtained by applying the model mentioned above is (0.324529, 0), indicating that aggregated NMD is zero, affecting the arguments of all other experts. If any expert assigns NMD zero to any alternative, the above model cannot evaluate the DM problem.

1.2 Contribution

Using Einstein’s operational laws in AOs has become increasingly important in dealing with uncertainty and vague data in DM problems. These operators consider all experts’ opinions, even if some score zero. Developing Einstein hybridized models is crucial in this context, as they offer a wider range of solutions than traditional models. The article is distributed on the following objectives:

•   We are presenting Einstein norms-based operational laws within the q-ROFSS environment.

•   It introduces the q-ROFSEHWA operator, an extended version of the AO based on the Einstein operational laws and its basic principles.

•   An algorithm is presented using a deduced operator to cope with fuzziness and inappropriate data while evaluating DM problems.

•   MCGDM approach is based on a protracted algorithm to determine the nature of water-borne infections in different patients affected by the flood.

•   A comparative analysis of the latest MCGDM technique is conducted over prevalent MCGDM techniques for its validation and significance.

The article has been structured comprehensively to present the various aspects of the developed model. Section 1 discusses the basic definitions and concepts to understand the work’s background clearly. The concept of AOs based on Einstein’s operational laws is explained in detail in Section 2, where the properties and examples of the proposed model are also provided. The algorithm of the proposed model and its application to different DM problems are discussed in Section 4. The comparative study of the proposed model with existing methods is presented in a separate section, and the conclusion of the work is given in the final section. This structure provides a clear picture of the development and implementation of the proposed model and its comparison with existing methods.

2  Preliminaries

This manuscript comprises fundamental definitions of fuzzy set theory, including FS, IFS, PFS, q-ROFS, SS, FSS, IFSS, PFSS, and q-ROFSS. Familiarity with these concepts can aid readers in comprehending the paper’s framework.

2.1 Definition [4]

FS ℱ over a universal set χ′ is defined as:

ℱ={X,fℐ(X)⋮fℐ(X)∈[0,1],X∈𝒟⊆χ′}

where fℐ:𝒟⟶[0,1] and fℐ(X) is called MD of X, and the pairs (fℐ(X),dℐ(X)) are termed fuzzy numbers (FNs).

2.2 Definition [6]

IFS I over a universal set χ′ is defined as:

I={X,fℐ(X),dℐ(X)⋮fℐ(X),dℐ(X)∈[0,1],X∈𝒟⊆χ′}

where fℐ:𝒟⟶[0,1], dℐ:𝒟⟶[0,1] and fℐ(X), dℐ(X) are called MD and NMD of X with the constraint that fℐ(X) + dℐ(X)≤1 and the pairs (fℐ(X),dℐ(X)) are termed intuitionistic fuzzy numbers (IFNs).

2.3 Definition [12]

PFS 𝒫 over a universal set χ′ is defined as:

𝒫={X,fℐ(X),dℐ(X)⋮fℐ(X),dℐ(X)∈[0,1],X∈𝒟⊆χ′}

where fℐ:𝒟⟶[0,1], dℐ:𝒟⟶[0,1] and fℐ(X), dℐ(X) are called MD and NMD of X with the constraint that fℐ(X)2 + dℐ(X)2≤1 and the pairs (fℐ(X),dℐ(X)) are termed Pythagorean fuzzy numbers (PFNs).

2.4 Definition [22]

q-ROFS over a universal set χ′ is defined as:

Q={X,fℐ(X),dℐ(X)⋮fℐ(X),dℐ(X)∈[0,1],X∈D⊆χ′}

where fℐ:𝒟⟶[0,1], dℐ:𝒟⟶[0,1] and fℐ(X), dℐ(X) are called MD and NMD of X with the constraint that fℐ(X)q + dℐ(X)q≤1 for q≥1 and the pairs (fℐ(X),dℐ(X)) are termed q-rung orthopair fuzzy numbers (q-ROFNs).

2.5 Definition [27]

Let χ′ be a universal set and 𝔈′ be the set containing parameterized values. A mapping generates the SS fℐ:E′→T by assigning attributes to elements of set 𝒯⊆χ′. Also, it can be defined as:

(fℐ,E' )={ (ck,§k): ck∈  E,§k∈ ⊆ χ' }.

2.6 Definition [28]

Let χ′ be a universal set containing FNs. and 𝔈′ be the set containing parametrized values. Then, FSS is denoted as (fℐ,E') is generated through a mapping fℐ:E′→T by assigning attributes to FNs. Also, it can be defined as:

(fℐ,E')={ (ck,fℐ(Xk)):ck∈E,Xk∈𝒫⊆χ' }

where fℐ(Xk)∈[0,1] is MD of Xk.

2.7 Definition [29]

Let χ′ be a universal set containing IFNs. and 𝔈′ be the set containing parametrized values. Then, IFSS denoted as (fℐ,E') is generated through a mapping fℐ:E′→T by assigning attributes to IFNs. Also, it can be defined as:

(fℐ,E')={ (ck,fℐ(Xk)​,dℐ(X)):ck∈E,Xk∈𝒫⊆χ' }

where fℐ(Xk),dℐ(X)∈[0,1] are called MD and NMD of Xk, respectively, following the condition that fℐ(X) + dℐ(X)≤1 and the pairs (fℐ(X),dℐ(X)) are called intuitionistic fuzzy soft numbers (IFSNs).

2.8 Definition [31]

Let χ′ be a universal set containing PFNs. And 𝔈′ be the set containing parametrized values. Then PFSS is denoted as (fℐ,E') is generated through a mapping fℐ:E′→T by assigning attributes to PFNs. Also, it can be defined as:

(fℐ,E')={ (ck,fℐ(Xk)​,dℐ(X)):ck∈E,Xk∈𝒫⊆χ' }

where fℐ(Xk),dℐ(X)∈[0,1] are called MD and NMD of Xk, respectively, following the condition that fℐ(X)2 + dℐ(X)2≤1 and the pairs (fℐ(X),dℐ(X)) are called Pythagorean fuzzy soft numbers (PFSNs).

2.9 Definition [43]

Let χ′ be a universal set containing q-ROFNs. and 𝔈′ be the set containing parametrized values. Then q-ROFSS is denoted as (fℐ,E') is generated through a mapping fℐ:E′→T by assigning attributes to q-ROFNs. Also, it can be defined as:

(fℐ,E')={ (ck,fℐ(Xk)​,dℐ(X)):ck∈E,Xk∈𝒫⊆χ' }

where fℐ(Xk),dℐ(X)∈[0,1] are called MD and NMD of Xk, respectively, following the condition that fℐ(X)q + dℐ(X)q≤1 for q≥1 and the pairs (fℐ(X),dℐ(X)) are called q-rung orthopair fuzzy soft numbers (q-ROFSNs).

2.10 Definition [43]

The score function signifies the comparison of two q-ROFSNs, which help to decide the optimal outcome from aggregated values. If 𝒪=(f,d) be q-ROFSN; then the score function is defined as:

S(𝒪)=fq−dq+(efq−dqefq−dq+1−12)πq

where π=1−(fq+dq)q is termed as the degree of hesitancy for q-ROFSN and S(T)∈[−1,1]. If 𝒪11=⟨f11,d11⟩ and 𝒪12=⟨f12,d12⟩ be any q-ROFSNs and π11,π12 be their degrees of hesitancy, respectively. Then the following condition holds:

a) If S(𝒪11)>S(𝒪12) then 𝒪11>𝒪12

b) S(𝒪11)<S(𝒪12) then 𝒪11<𝒪12

c) If S(𝒪11)=S(𝒪12) then

i) If π11>π12 then 𝒪11<𝒪12

ii) If π11=π12 then 𝒪11=𝒪12

3  q-Rung Orthopair Fuzzy Soft Einstein Hybrid Weighted Average Operator

The hybrid weighted average AO based on Einstein operational laws, named the q-ROFSEHWA operator, is introduced in this section. Moreover, the properties of the proposed operator are also discussed.

3.1 Definition

Let α=(h,f), α1=(h1,f1) and α2=(h2,f2) be the q-ROFSNs and 𝒹>0, then Einstein-based operations on q-ROFSNs are defined as:

1.    α1⊕ℰα2=⟨h1q+h2q1+h1q.h2qq,f1.f21+(1−f1q).(1−f2q)q⟩

2.    α1⊗ℰα1=⟨h1.h21+(1−h1q).(1−h2q)q,f1q+f2q1+f1q.f2qq⟩

3.    𝒹.α=⟨(1+(h)q)𝒹−(1−(h)q)𝒹(1+(h)q)𝒹+(1−(h)q)𝒹q,2q(f)𝒹(2−(f)q)𝒹+((f)q)𝒹q⟩

4.    α𝒹=⟨2q(h)𝒹(2−(h)q)𝒹+((h)q)𝒹q,(1+(f)q)𝒹−(1−(f)q)𝒹(1+(f)q)𝒹+(1−(f)𝒹)𝒹q⟩

These operational laws will be used to develop the q-ROFSEHWA operator.

3.2 Definition

Let αij=(hij,fij) be the collection of q-ROFSNs and with its weight vectors μi=(μ1,μ2,…,μn) and νj=(ν1,ν2,…,νm) and suppose ℏi=(ℏ1,ℏ2,…,ℏn) and ℓj=(ℓ1,ℓ2,…,ℓm) represents weight vector for experts Γi and parameters ϵj′ s respectively such that ∑i=1nμi=1=∑j=1mνj=∑j=1mℏi=∑j=1m𝓁j and μi,νj,ℏi,𝓁j∈[0,1]∀1≤i≤nand1≤j≤m and k denote the total number of elements in ith row and jth column. Then, the q-ROFSEHWA is defined as:

q−ROFSEHWA(α11,α12,…,αnm)=⊕j=1m𝓁j(⊕i=1nℏi)(1)

where α∼ϕij=(h∼ϕij,f∼ϕij) signifies the permutation of the main value of ith row and jth column (α∼ϕ(i−1)j≥α∼ϕij and α∼ϕi(j−1)≥α∼ϕij∀i,j) from the collection of ordered pairs (i×j) of q-ROFSNs α∼ij=(h∼ij,f∼ij) and α∼ϕij=kμiνj.αϕij.

3.3 Theorem

Let q−ROFSEHWA(α11,α12,…,αnm)=⊕j=1m𝓁j(⊕i=1nℏiα∼ϕij) be a q-ROFSEHWA operator, where each αij represents q-ROFSNs, then

q−ROFSEHWA(α11,α12,…,αnm)=⊕j=1m𝓁j(⊕i=1nℏiα∼ϕij)

=⟨(∏j=1m∏i=1n(1+(h∼ϕij)q)ℏi)𝓁j−(∏j=1m∏i=1n(1−(h∼ϕij)q)ℏi)𝓁j(∏j=1m∏i=1n(1+(h∼ϕij)q)ℏi)𝓁j+(∏j=1m∏i=1n(1−(h∼ϕij)q)ℏi)𝓁jq,2q(∏j=1m∏i=1n(f∼ϕij)ℏi)𝓁j(∏j=1m∏i=1n(2−(f∼ϕij)q)ℏi)𝓁j+(∏j=1m∏i=1n((f∼ϕij)q)ℏi)𝓁jq⟩(2)

where α∼ϕij=(h∼ϕij,f∼ϕij) denotes the permutation of the biggest value of ith row and jth column α∼ϕ(i−1)j≥α∼ϕij and α∼ϕi(j−1)≥α∼ϕij∀i,j. From the group of ordered pairs (i×j) of q-ROFSNs α∼ij=(h∼ij,f∼ij) and α∼ϕij=kμiνj.αϕij.

Proof: Using mathematical induction

Let i=1, then ℏi=1.

From Eq. (2), we have

q−ROFSEHWA(α11,α12,…,α1m)=⊕j=1m𝓁j(⊕i=1nℏiα∼ϕij)=⊕j=1m𝓁jα∼ϕij=⟨∏j=1m(1+(h∼ϕ1j)q)𝓁j−∏j=1m(1−(h∼ϕ1j)q)𝓁j∏j=1m(1+(h∼ϕ1j)q)𝓁j+∏j=1m(1−(h∼ϕ1j)q)𝓁jq,2q∏j=1m(f∼ϕ1j)𝓁j∏j=1m(2−(f∼ϕ1j)q)𝓁j+q(∏j=1m(f∼ϕ1j)q)𝓁j⟩

=⟨∏j=1m(∏i=11(1+(h∼ϕij)q)ℏi)𝓁j−∏j=1m(∏i=11(1−(h∼ϕij)q)ℏi)𝓁j∏j=1m(∏i=11(1+(h∼ϕij)q)ℏi)𝓁j+∏j=1m(∏i=11(1−(h∼ϕij)q)ℏi)𝓁jq,2q∏j=1m(∏i=11((f∼ϕij)q)ℏi)𝓁j∏j=1m(∏i=11(2−(f∼ϕij)q)ℏi)𝓁j+∏j=1m(∏i=11((f∼ϕij)q)ℏi)𝓁jq⟩

=⊕j=1m𝓁j(⊕i=11ℏiα∼ϕij)

Let j=1, then 𝓁j=1.

From Eq. (2), we have

q−ROFSEHWA(α11,α12,…,αn1)=⊕j=11𝓁j(⊕i=1nℏiα∼ϕij)=⊕i=1nℏiα∼ϕij=⟨∏i=1n(1+(h∼ϕi1)q)ℏi−∏i=1n(1−(h∼ϕi1)q)ℏi∏i=1n(1+(h∼ϕi1)q)ℏi+∏j=1m(1−(h∼ϕi1)q)ℏiq,2q∏j=1m(f∼ϕi1)ℏi∏j=1m(2−(f∼ϕi1)q)ℏi+(∏j=1m(f∼ϕi1)q)ℏiq⟩

=⟨(∏j=11∏i=1n(1+(h∼ϕij)q)ℏi)𝓁j−(∏j=11∏i=1n(1−(h∼ϕij)q)ℏi)𝓁j(∏j=11∏i=1n(1+(h∼ϕij)q)ℏi)𝓁j+(∏j=11∏i=1n(1−(h∼ϕij)q)ℏi)𝓁jq,2q(∏j=11∏i=1n(f∼ϕij)ℏi)𝓁j(∏j=11∏i=1n(2−(f∼ϕij)q)ℏi)𝓁j+(∏j=11∏i=1n((f∼ϕij)q)ℏi)𝓁jq⟩

=⊕j=11𝓁j(⊕i=1nℏiα∼ϕij).

Hence, the Eq. (2) is true for n=1 and m=1.

Suppose that the statement is true for n=ϵ2,m=ϵ1+1 and n=ϵ2+1,m=ϵ1

Then, we have

q−ROFSEHWA(α11,α12,…,αϵ2(ϵ1+1))=⊕j=1ϵ1+1𝓁j(⊕i=1ϵ2ℏiα∼ϕij)

=⟨(∏j=1ϵ1+1∏i=1ϵ2(1+(h∼ϕij)q)ℏi)𝓁j−(∏j=1ϵ1+1∏i=1ϵ2(1−(h∼ϕij)q)ℏi)𝓁j(∏j=1ϵ1+1∏i=1ϵ2(1+(h∼ϕij)q)ℏi)𝓁j+(∏j=1ϵ1+1∏i=1ϵ2(1−(h∼ϕij)q)ℏi)𝓁jq,2q(∏j=1ϵ1+1∏i=1ϵ2(f∼ϕij)ℏi)𝓁j(∏j=1ϵ1+1∏i=1ϵ2(2−(f∼ϕij)q)ℏi)𝓁j+(∏j=1ϵ1+1∏i=1ϵ2((f∼ϕij)q)ℏi)𝓁jq⟩

and

q−ROFSEHWA(α11,α12,…,α(ϵ2+1)ϵ1)=⊕j=1ϵ1𝓁j(⊕i=1ϵ2+1ℏiα∼ϕij)

=⟨(∏j=1ϵ1∏i=1ϵ2+1(1+(h∼ϕij)q)ℏi)𝓁j−(∏j=1ϵ1∏i=1ϵ2+1(1−(h∼ϕij)q)ℏi)𝓁j(∏j=1ϵ1∏i=1ϵ2+1(1+(h∼ϕij)q)ℏi)𝓁j+(∏j=1ϵ1∏i=1ϵ2+1(1−(h∼ϕij)q)ℏi)𝓁jq,2q(∏j=1ϵ1∏i=1ϵ2+1(f∼ϕij)ℏi)𝓁j(∏j=1ϵ1∏i=1ϵ2+1(2−(f∼ϕij)q)ℏi)𝓁j+(∏j=1ϵ1∏i=1ϵ2+1((f∼ϕij)q)ℏi)𝓁jq⟩

Need to prove Eq. (2) for m=ϵ1+1 and n=ϵ2+1

⊕j=1ϵ1+1𝓁j(⊕i=1ϵ2+1ℏiα∼ϕij)=⊕j=1ϵ1+1𝓁j(⊕i=1ϵ2ℏiα∼ϕij⊕ℏϵ2+1α∼ϕ(ϵ2+1)j)

=⊕j=1ϵ1+1𝓁j(⊕i=1ϵ2ℏiα∼ϕij)⊕(⊕j=1ϵ1+1𝓁jℏϵ2+1α∼ϕ(ϵ2+1)j)

=⟨(∏j=1ϵ1+1∏i=1ϵ2(1+(h∼ϕij)q)ℏi)𝓁j−(∏j=1ϵ1+1∏i=1ϵ2(1−(h∼ϕij)q)ℏi)𝓁j(∏j=1ϵ1+1∏i=1ϵ2(1+(h∼ϕij)q)ℏi)𝓁j+(∏j=1ϵ1+1∏i=1ϵ2(1−(h∼ϕij)q)ℏi)𝓁jq,2q(∏j=1ϵ1+1∏i=1ϵ2(f∼ϕij)ℏi)𝓁j(∏j=1ϵ1+1∏i=1ϵ2(2−(f∼ϕij)q)ℏi)𝓁j+(∏j=1ϵ1+1∏i=1ϵ2((f∼ϕij)q)ℏi)𝓁jq⟩

⊕⟨∏j=1ϵ1+1((1+(h∼ϕ(ϵ2+1)j)q)ℏϵ2+1)𝓁j−∏j=1ϵ1+1((1−(h∼ϕ(ϵ2+1)j)q)ℏϵ2+1)𝓁j∏j=1ϵ1+1((1+(h∼ϕ(ϵ2+1)j)q)ℏϵ2+1)𝓁j+∏j=1ϵ1+1((1−(h∼ϕ(ϵ2+1)j)q)ℏϵ2+1)𝓁jq,2q∏j=1ϵ1+1((f∼ϕ(ϵ2+1)j)ℏϵ2+1)𝓁j∏j=1ϵ1+1(2−((f∼ϕ(ϵ2+1)j)q)ℏϵ2+1)𝓁j+∏j=1ϵ1+1(((f∼ϕ(ϵ2+1)j)q)ℏϵ2+1)𝓁jq⟩