Complex Decision Modeling Framework with Fairly Operators and Quaternion Numbers under Intuitionistic Fuzzy Rough Context

1  Introduction

The main notion of the multi-attribute decision-making (MADM) method, which is a modified form of the simple decision-making scenario, is a fantastic and dominant strategy for illuminating the valuable opinion from a collection of preferences. We analyze or encounter several decision-making issues in daily life, and one of the most important things we can do is acquire knowledge of how to make good, excellent, or outstanding decisions [1,2]. Typically, the expert presents conventional knowledge without assessing its level of ambiguity and uncertainty. While a logical method uses data and facts to reach conclusions that are in line with science. Although emotional intelligence is an excellent technique to make choices, it frequently functions best when the option is straightforward, private, or urgent [3,4]. A more formal, systematic approach that involves both intuition and logical reasoning is fuzzy set theory, which is often required for more complex judgments. The theory of fuzzy set (FS) processing was introduced by Zadeh [5] in 1965 as a solution to such issues. For resolving difficult and ambiguous information in many real-life scenarios, fuzzy information is one of the most useful and realistic variations of classical data. Each component of a set is given a membership degree (MD) in the real standard [0 1]. Atanassov [6] presented intuitionistic fuzzy set (IFS), which expands on the concept of fuzzy sets and manages some key fuzzy data in multi-criteria decision making (MCDM) [7,8]. For each part of a set, IFS determines the levels of MD and NMD. Pawlak was the one who first put forth the idea of rough set (RS) theory [9,10]. An expansion of crisp set theory for the study of intelligent systems with imperfect, ambiguous, or insufficient knowledge is RS theory [11,12]. RS theory has drawn a lot of attention and enthusiasm from scholars over the past few years. There are many concepts that integrate the idea of RSs, including fuzzy rough sets (FRSs) [13], generalised fuzzy rough sets [14], ambiguous rough sets, rough grey sets, and intuitionistic fuzzy rough sets (IFRSs) [15,16]. Recent studies have demonstrated how these theories may be brought together to create a framework that is more adaptable and expressive for modeling and processing missing information in data systems. In the meantime, fuzzy sets have been expanded in an appealing way by intuitionistic fuzzy sets, which give them additional properties to reflect uncertainty (on top of vagueness) [17]. Similar to rough set theory, fuzzy set theory deals with the issue of handling incomplete knowledge. Regrettably, the multiple adhoc definitions of the term ”intuitionistic fuzzy rough set” that have since been proposed are a far cry from the original goals of rough set theory [18].

Researchers have also developed a number of novel theory that focuses on IFSs, including similarity measures and aggregation operators (AOs). On the basis of IFSs, many authors [19] developed multiple geometric AOs. Additionally, Naz et al. [20] introduced several average AOs built on IFSs. Double domination on intuitionistic fuzzy (IF) graphs was first proposed by Nagoorgani et al. [21]. Xu was the pioneer in introducing weighted arithmetic (WA) operators based on intuitionistic fuzzy numbers (IFNs) [22]. Subsequently, Xu and Yager [23,24] proposed the extended Bonferroni mean (BM) operators and weighted geometric (WG) operators, both based on IFNs. Qin and Liu [25] developed the weighted Maclaurin symmetric mean operator using IFNs. Additionally, Liu and Liu [26] presented partitioned BM operators for intuitionistic uncertain language variables. Furthermore, the extended MADM (Multiple Attribute Decision Making) method has been extensively discussed with detailed illustrations of useful approaches such as the TOPSIS method for IFNs [27,28], the triangular intuitionistic fuzzy-TODIM method [29,30], and the intuitionistic fuzzy PROMETHE II method [31]. However, the restriction that the sum of membership degree (MD) and non-membership degree (NMD) cannot exceed one limit of the applicability of IFS (Intuitionistic Fuzzy Sets). Decision-makers and experts prefer more complex decision-making situations, making this condition less valid. Due to its effectiveness as a general formal framework, IFS has received significant attention from academics. Nguyen et al. [32] developed the exponentially similarity matrix for Pythagorean fuzzy sets (PFSs) and demonstrated their application in pattern identification and decision-making. However, PFSs can only be implemented when the sum of the squares representing MD and NMD falls within the range [0,1]. To address this challenge, it becomes necessary to define information expression differently when the sum falls outside this range.

Smarandache first proposed the neutrosophic set (NS), an analytical framework and quantitative tool for comprehending the origin, nature, and extent of neutralities [33]. It is a spiritual practice that focuses on neutralities’ origins, nature, and scope as well as how they interact with other ideational spectrums. The NS generalizes the ideas behind the classical set, fuzzy set, interval valued fuzzy set, interval-valued IFS, paraconsistent set, dialetheist set, paradoxist set, and tautological set. A NS is characterized by a truth membership function, indeterminacy membership function, and falsity membership function where all functions are real standard or nonstandard elements from [0−,1+]. It will be difficult to apply NS in actual scientific and engineering contexts, despite the fact that it philosophically generalizes the ideas of FS, IFS, and all existing structures. This idea is crucial in many situations, such as information fusion, which integrates data from several sensors. In recent years, engineering and other industries have mainly used neutrosophic sets to make decisions. A single-valued neutrosophic set (SVNS), which can deal with inaccurate, ambiguous, and incompatible data issues, was proposed by Wang et al. [34]. An SVNS, on the other hand, is an NS that enables us to depict ambiguity, imprecision, incompleteness, and inconsistent behavior in the real world [35]. Contrarily, SVNSs can be used in technical and scientific applications since SVNS theory is effective at modeling ambiguous, imperfect, and inconsistent data [36–38]. The SVNS can easily capture the ambiguous nature of subjective assessments, making it excellent for gathering vague, ambiguous, and inconsistent data in multi criteria decision-making analysis [39]. All the above sets can handle the uncertainty in real world issues but any set cannot treat the membership and non membership equally.

Previous AOs [40–44] have been proposed to handle MADM problems under fuzzy environments, but few have addressed the unbiased treatment of MD and NMD. Consolidated values reported in the literature [45–48] cannot be differentiated when a decision-maker compares both MD and NMD, indicating favoritism in the verdict. Therefore, new neutral procedures for IFRSs (Intuitionistic Fuzzy Rough Sets) are required to ensure equal treatment of information about members and non-members. Through the idea of ”intuitionistic” membership functions, IFRN expands the traditional fuzzy-rough framework. The IFRN contains two membership functions, one for a level of belongingness and a separate function for the degree of non-belongingness, in contrast to standard fuzzy sets. A more accurate portrayal of the ambiguity in real-world data is made possible by this dual perspective. To achieve this, the concept of proportional distribution rules of MD and NMD is used to establish two neutral or fair operations, enabling the evaluation of MD and NMD with true satisfaction in the decision-making process. A thorough understanding of the human decision-making process is essential to build a realistic model. This calls for the incorporation of fairness operators and a hybrid idea, such as intuitionistic fuzzy rough set, into the decision-making paradigm. Normally, decision-making involves weighing a range of options and standards. By placing a strong emphasis on the relationship between these options and the specified criteria, we provide a fresh strategy in this work. We use the quaternion function, that’s a key component of our technique, to clarify and codify the relationship between options and criteria. This novel strategy provides a more organized and thorough framework for decision-making, improving our capacity to make wise and effective decisions. Complex numbers and non-commutative four-dimensional algebra are both generalized by quaternion numbers. The concept of quaternion numbers and its applications to rotations were first published in 1840 by Olinde Rodrigues et al. [49]. However, an Irish mathematician by the name of Sir William Rowan Hamilton [50] independently made the discovery in 1843 and utilized it to research three-dimensional mechanics. A quaternion can be used to store each rotation in a 4-D coordinate system. The union of three complex components and one real element is known as a quaternion. It also applies to a lot more than just rotations. Rotations are better performed with quaternions than Euler angles, and the gamble lock problem is resolved. In computer animation, quaternions are frequently used to represent changes in the orientation of graphical elements. They offer a creative fix for issues like convenient interpolation, gimbal lock, and instability that plagued early animated systems. In order to express 4-dimensional complex information, we apply the intuitionistic fuzzy information to establish the relationship between intuitionistic fuzzy numbers and quaternion numbers. To assess the applicability of the suggested strategy, an example of a medical diagnosis based on the suggested representations is given. We can establish a connection between attributes and alternatives through this relationship.

Following the discussion above, we described the objectives of this study as follows:

•   Although IFS and RS are merged, IFRS anticipates giving decision-makers additional space.

•   IFS lacks the positive and negative approximation spaces that IFRS utilizes.

•   The advance requirement that the sum (MD, NMD) must belong to [0,1] is used by IFRS.

•   To establish a few new neutral or fair functions that handle membership and non-membership functions equally using the connectivity coefficient.

•   Understanding the characteristics of weighted aggregation operators like intuitionistic fuzzy rough fairly weighted aggregation (IFRFWA) and intuitionistic fuzzy rough fairly ordered weighted aggregation (IFRFOWA). The level of expertise with evaluated substances that IFRFWA and IFRFOWA AOs can incorporate for first evaluation is a quality that is lacking in IFWA and IFWG aggregation operators.

•   To design a decision-making process that uses the weighted aggregation operators stated above to address multi-attribute decision-making issues with IFR data.

•   We also use intuitionistic fuzzy data to determine how quaternion numbers and intuitionistic fuzzy numbers relate to one another.

•   An example of a medical diagnosis based on the suggested representations is provided to demonstrate the strategy’s usefulness. Through this relationship, we can construct a link between attributes and possibilities.

According to the following, this paper is structured:

Section 2 focuses exclusively on the evaluation of a few preliminary documents. In Section 3, we describe the operators of the IF context and elucidate their fundamental characteristics. Section 4 provides a definition of the IFRFWA and IFRFOWA operators. Section 5 is intended to facilitate multi-attribute group decision-making (MAGDM). In accordance with the suggested methodology, which depicts the rating values of options on the characteristic in terms of IFR numbers, the best/most acceptable option is determined by ranking the alternatives based on their IFR score values. A case study with numerical examples for decision making is explained in Section 6. Relationship Model between Alternatives and Attributes is given in Section 7. In Section 8, Discussion and Comparison Analysis and positive aspects of the model are given. The conclusion can be found in Section 9.

2  Basic Concepts

We have discussed some of the key ideas related to IFSs in this section of the article. We will also go over the fundamental concepts of FSs, IFSs, scoring functions (SFs), accuracy functions (AFs), quaternion numbers (QNs), and IFQNs.

Definition 2.1. (See [5]) A fuzzy set (FS) Z in δ is mathematically represented as:

Z={⟨Γ,TZ(Γ)⟩|Γ∈δ},(1)

for each Γ∈δ, the MD TZ:δ→δ specifies the degree to which the element Γ∈Z, where TZ∈[0,1].

Definition 2.2. (See [6]) An intuitionistic fuzzy set (IFS) A in δ is mathematically represented as:

A={⟨♭,T(♭),F(♭)⟩|♭∈δ},(2)

here ♭∈δ,T(♭) is known as MD and F(♭) is NMD for IFS A, where (T(♭),F(♭))∈[0,1], satisfying 0≤(T(♭)+F(♭))≤1.

Definition 2.3. (See [51]) Assume that ξ is a (US) universal set and ℜ is relation on ξ. A set valued mapping is mathematically represented as:

ℜ∗:ξ→M(ξ) by ℜ∗(ð)={a∈ξ|(ð,a)∈ℜ},(3)

The element’s successor neighborhood with respect to the relation ℜ is denoted by ℜ∗(ð). The pair (ξ,ℜ) is commonly referred to as the “crisp space of resemblance”. For any set k⊆ξ, the lower approximation (LA) and upper approximation (UA) with respect to the space of resemblance (ξ,ℜ) for this set are mathematically represented as follows:

ℜ_(k)={ð∈ξ|ℜ∗(ð)⊆k};(4)

ℜ¯(k)={ð∈ξ|ℜ∗(ð)∩k≠ϕ}.(5)

The pair (ℜ_(k),ℜ¯(k)) is called fuzzy RS where both ℜ_(k),ℜ¯(k):M(ξ)→M(ξ) are LA and UA operators.

Definition 2.4. (See [52]) Assume that F={(T_,F_),(T¯,F¯)} is a IFRN. Then, SF and AF are describe as:

Sc=14{2+T_⋋+T¯ ⋋−F_⋋−F¯⋋},S∈[0,1](6)

Ac=14{2+T_⋋+T¯⋋+F_⋋+F¯⋋},A∈[0,1].(7)

Definition 2.5. Assume R1=⟨T1,F1⟩ and R2=⟨T2,F2⟩ are two IFNs and ℵ,ℵ1,ℵ2 ⋗0 be the real numbers, then we have,

•   R1⊕R2=R2⊕R1

•   R1⊗R2=R2⊗R1

•   ℵ(R1⊕R2)=(ℵR1)⊕(ℵR2)

•   (R1⊗R2)ℵ=R1ℵ⊗R2ℵ

•   (ℵ1+ℵ2)R1=(ℵ1R1)⊕(ℵ2R2)

•   R1ℵ1+ℵ2=R1ℵ1⊗R2ℵ2

If TR1=FR1 and TR2=FR2 then, based on description, we have,

TR1⊕R2≠FR1⊕R2, TR1⊗R2≠FR1⊗R2, TℵR1≠FℵR1,TR1ℵ≠FR1ℵ.

Consequently, none of the procedures

R1⊕R2,R1⊗R2,ℵR1,R1ℵ

deemed to be impartial or equitable in reality. Therefore, developing some fair operations between IFRNs must be the focus of our attention in the outset.

Example 1. For the Definition 2.5, we prove it by the following example. A generalization of conventional fuzzy sets, intuitionistic fuzzy sets (IFS) involve both membership and non-membership degrees. To merge two IFSs into only one IFS, utilize the bounded operations of an IFS.

Assume that R1={(0.7,0.2)} and R2={(0.6,0.3)} are two IFSs, then the proof of above results are as following:

•   R1⊕R2={(0.7,0.2)}⊕{(0.6,0.3)}={(0.82,0.44)}The bounded sum can be solved as, If R1=⟨T1,F1⟩ and R2=⟨T2,F2⟩ are two IFNs then R1⊕R2={(⟨T1+T2−T1∗T2⟩,⟨F1+F2−F1∗F2⟩)}

•   R2⊕R1={(0.6,0.3)}⊕{(0.7,0.2)}={(0.82,0.44)}Hence R1⊕R2=R2⊕R1

•   R1⊗R2={(0.7,0.2)}⊗{(0.6,0.3)}={(0.42,0.44)}The bounded product can be solved as, If R1=⟨T1,F1⟩ and R2=⟨T2,F2⟩ are two IFNs then R1⊗R2={(⟨T1∗T2⟩,⟨F1+F2−F1∗F2⟩)}

•   R2⊗R1={(0.6,0.3)}⊗{(0.7,0.2)}={(0.42,0.44)}Hence R1⊗R2=R2⊗R1

•   (R1⊗R2)ℵ=R1ℵ⊗R2ℵ Let us solve (R1⊗R2)ℵ, where ℵ is a positive integer and ⊗ denotes the bounded product. This example will involve the constrained product operation and raising the outcome to the power of 2 (ℵ=2)

•   R1={(0.8,0.1)} and R1={(0.6,0.3)} and ℵ=2Hence

R1⊗R2 ={(0.48,0.38)}

(R1⊗R2)ℵ ={(0.2304,0.1444)}

R1ℵ⊗R2ℵ ={(0.2304,0.1444)}

It can compute (R1⊗R2)ℵ by repeating this procedure for any positive integer ℵ. Similarly, other all results can be computed by the above process and all the results satisfied for IFNs.

Definition 2.6. (See [53]) Assume that, R is a Complex fuzzy set (CFS) over Z mathematically represented as:

R={(T,ℷR(T)):T∈Z},(8)

Here ℷR(T) is complex valued grade of MD of T in Z. By definition, the value ℷR(T) may receive all lie with in the unit circle in the complex plane, and are thus of the form ΓR(T).eȷ^ϖR(T), where ȷ^=−1, ΓR(T) and ΓR(T)∈[0,1]

Definition 2.7. (See [54]) Assume that, Γ is a Complex intuitionistic fuzzy set (CIFS) over Z is distinguished by a MD TZ(Γ) and NMD FZ(Γ), respectively, that assign a complex-valued grade to both MD and NMD in Z. The values of TZ(Γ) and FZ(Γ) all lie with in the unit circle in the complex plane and are of the form TZ(Γ)=TZ(Γ)eȷ^ϖZ(Γ) and FZ(Γ)=FZ(Γ)eȷ^ℷZ(Γ), where TZ(Γ)andFZ(Γ)∈[0,1] with ȷ^=−1 is denoted as:

Γ={(Γ,TZ(Γ),FZ(Γ):Γ∈Z},(9)

Similarly, the pure complex NMD was added to the concept of complex fuzzy class to create the concept of complex intuitionistic fuzzy class.

Definition 2.8. (See [49,50]) A quaternion q is a four dimensional complex number also called a hyper complex number, introduced by Hamilton in 1843. Let a^,b^,c^,d^ are real numbers and ı^,ȷ^,k^ are imaginary units then a quaternion is expressed as q~=a^+bi^+cj^+dk^. The imaginary units ı^,ȷ^,k^ are mutually orthogonal unit vectors. These imaginary units have the following properties:

•   i^2=j^2=k^2=ijk^=−1

•   ij^=−ji^=k^,jk^=−kj^=ı^,ki^=−ik^=ȷ^.

Definition 2.9. (See [49,50]) A fuzzy quaternion number is given by Q´:B→[0,1] such that

Q´(ϖ+Φı^+ℑȷ^+Θk^)=min{ℵ¯(ϖ),χ¯(Φ),F¯(ℑ),Z¯(Θ)}(10)

for some (ϖ,Φ,ℑ,Θ)∈BQ and ℵ¯,χ¯,F¯,Z¯∈ℜF, here ℜF fuzzy quaternion numbers.

Definition 2.10. A intuitionistic fuzzy quaternion set (IFQS) is given by the complex function Z=T+jF, where T,F:χ⟶[0,1] are the function of MD and NMD, respectively and mathematically represented as: as a set of ordered pairs, the IFS A¨ can be represented as:

Z={(ϰ,Z)|ϰ∈χ,Z=T(ϰ)+jF(ϰ)}.(11)

In other hands, we can defined as:

Definition 2.11. Assume δ is a space. Zq is the IFQS on δ defined on the quaternion function Q=ϖ+ı^T+ȷ^F+k^Λ, where ı^,ȷ^,k^ are the complex roots, ı^2=ȷ^2=k^2=−1. Here, ϖ,T,F and Λ are the functions of real MD, complex MD, real NMD and complex NMD, respectively. For all ϰ∈δ, the function ϖ,T,F and Λ satisfy the following conditions:

 ϖ(ϰ),T(ϰ),F(ϰ) and Λ(ϰ)∈[0,1],

{ϖ(ϰ)+T(ϰ)≤1,F(ϰ)+Λ(ϰ)≤1,ϖ(ϰ)+F(ϰ) ≤1,T(ϰ)+Λ(ϰ)≤1.}

mathematically represented as:

δ=(ϖ(ϰ)+ı^T(ϰ))+ȷ^(F(ϰ)−ı^Λ(ϰ)),(12)

the values ϖ(ϰ),T(ϰ),F(ϰ) and Λ(ϰ) are known as degree of real membership, imaginary membership, real non-membership and imaginary non-membership, respectively.

3  Devolpment Fairly Operators of IFR Context

In this part, we combined RSs with IFS to develop some aggregation operators and examine their fundamental characteristics.

Definition 3.1. Assume that (R_1,R¯1)={(T_R1,F_R1),(T¯R1,F¯R1)} and (R_2,R¯2)={(T_R2,F_R2),(T¯R2,F¯R2)} are two IFRSs, and then we determine.

{(R_1⊕R_2)R¯1⊕R¯2}=({(T_R1T_R2T_R1T_R2+F_R1F_R2)(1−(1−T_R1−F_R1)(1−T_R2−F_R2)),(T¯R1T¯R2T¯R1T¯R2+F¯R1F¯R2)(1−(1−T¯R1−F¯R1)(1−T¯R2−F¯R2))},{(F_R1F_R2T_R1T_R2+F_R1F_R2)(1−(1−T_R1−F_R1)(1−T_R2−F_R2)),(F¯R1F¯R2T¯R1T¯R2+F¯R1F¯R2)(1−(1−T¯R1−F¯R1)(1−T¯R2−F¯R2))}.)(13)

{(ℵ∗R_1),(ℵ∗R¯1)}=({((T_R1ℵT_R1ℵ+F_R1ℵ)(1−(1−T_R1−F_R1)ℵ))((T¯R1ℵT¯R1ℵ+F¯R1ℵ)(1−(1−T¯R1−F¯R1)ℵ))},{((F_R1ℵT_R1ℵ+F_R1ℵ)(1−(1−T_R1−F_R1)ℵ))((F¯R1ℵT¯R1ℵ+F¯R1ℵ)(1−(1−T¯R1−F¯R1)ℵ))}.)(14)

It is simple to validate that {(R_1⊕R_2),(R¯1⊕R¯2)},{(ℵ∗R_1),(ℵ∗R¯1)} where ℵ>0, are the IFRSs.

Theorem 3.1. Assume that (R_1,R¯1)={(T_R1,F_R1),(T¯R1,F¯R1)} and (R_2,R¯2)={(T_R2,F_R2),(T¯R2,F¯R2)} are the IFRSs. If (T_R1T¯R1)=(F_R1F¯R1), (T_R2T¯R2)=(F_R2F¯R2), then,

1.   {(T_R1⊕R2T¯R1⊕R2)=(F_R1⊕R2F¯R1⊕R2)}

2.   {(T_ℵ∗R1T¯ℵ∗R1)=(F_ℵ∗R1F¯ℵ∗R1)}

Proof. As given (T_R1T¯R1)=(F_R1F¯R1), (T_R2T¯R2)=(F_R2F¯R2), then

1.

{(T_R1⊕R2F_R1⊕R2)(T¯R1⊕R2F¯R1⊕R2)} ={((T_R1T_R2T_R1T_R2+F_R1F_R2)(1−(1−T_R1−F_R1)(1−T_R2−F_R2))),((T¯R1T¯R2T¯R1T¯R2+F¯R1F¯R2+Λ¯R1Λ¯R2)(1−(1−T¯R1−F¯R1)(1−T¯R2−F¯R2)))}{((F_R1F_R2T_R1T_R2+F_R1F_R2+Λ_R1Λ_R2)(1−(1−T_R1−F_R1)(1−T_R2−F_R2))),((F¯R1F¯R2T¯R1T¯R2+F¯R1F¯R2+Λ¯R1Λ¯R2)(1−(1−T¯R1−F¯R1)(1−T¯R2−F¯R2)))}(15)

  =1(16)

consequently, (T_R1⊕R2(T¯R1⊕R2))=(F_R1⊕R2F¯R1⊕R2), if (T_R1T¯R1)=(F_R1F¯R1), and (T_R2T¯R2)=(F_R2F¯R2)

2.

{(T_ℵ∗R1F_ℵ∗R1)(T¯ℵ∗R1F¯ℵ∗R1)} ={((T_R1ℵT_R1ℵ+F_R1ℵ)(1−(1−T_R1−F_R1)ℵ))((T¯R1ℵT¯R1ℵ+F¯R1ℵ)(1−(1−T¯R1−F¯R1)ℵ))}{((F_R1ℵT_R1ℵ+F_R1ℵ)(1−(1−T_R1−F_R1)ℵ))((F¯R1ℵT¯R1ℵ+F¯R1ℵ)(1−(1−T¯R1−F¯R1)ℵ))}(17)

 =1(18)

consequently, {(T_ℵ∗R1T¯ℵ∗R1)=(F_ℵ∗R1F¯ℵ∗R1)}.

if (T_R1T¯R1)=(F_R1F¯R1), and (T_R2T¯R2)=(F_R2F¯R2)

The processes revealed by this theorem {(R1⊕R2R1⊕R2),(ℵ∗R1ℵ∗R1)}, when MD and NMD are originally equal, demonstrate to the DMs a fair or neutral nature. Because of this, we refer to the activities ⊗~,∗ as fairly operations. This is why we call the operations fairly operations.

Theorem 3.2. consider (R_1,R¯1)={(T_R1,F_R1)(T¯R1,F¯R1)} and (R_2,R¯2)={(T_R2,F_R2)(T¯R2,F¯R2)} are the IFRSs and If the numbers ℵ,ℵ1 and ℵ2 are any three real values, we have got,

1.   {(R_1⊕R_2R¯1⊕R¯2)=(R_2⊕R_1R¯2⊕R¯1)}

2.   {(ℵ∗(R_1⊕R_2)ℵ∗(R¯1⊕R¯2))=(ℵ∗R_1ℵ∗R¯1)⊕(ℵ∗R_2ℵ∗R¯2)}

3.   {((ℵ1+ℵ2)∗R_1(ℵ1+ℵ2)∗R¯1)=(ℵ1∗R_1ℵ1∗R¯1)⊕(ℵ2∗R_1ℵ2∗R¯1)}

1. This one is trivial.

2. ℵ∗{(R_1⊕R_2)(R¯1⊕R¯2)}

=({((((T_R1T_R2T_R1T_R2+F_R1F_R2)ℵ)((T_R1T_R2T_R1T_R2+F_R1F_R2)ℵ+(F_R1F_R2T_R1T_R2+F_R1F_R2)ℵ))(1−(1−(1−(1−T_R1−F_R)(1−T_R2−F_R2)))ℵ))((((T¯R1T¯R2T¯R1T¯R2+F¯R1F¯R2)ℵ)((T¯R1T¯R2T¯R1T¯R2+F¯R1F¯R2)ℵ+(F¯R1F¯R2T¯R1T¯R2+F¯R1F¯R2)ℵ))(1−(1−(1−(1−T¯R1−F¯R1)(1−T¯R2−F¯R2)))ℵ))}{((((F_R1F_R2T_R1T_R2+F_R1F_R2)ℵ)((T_R1T_R2T_R1T_R2+F_R1F_R2)ℵ+(F_R1F_R2T_R1T_R2+F_R1F_R2)ℵ))(1−(1−(1−(1−T_R1−F_R1)(1−T_R2−F_R2)))ℵ))((((F¯R1F¯R2T¯R1T¯R2+F¯R1F¯R2)ℵ)((T¯R1T¯R2T¯R1T¯R2+F¯R1F¯R2)ℵ+(F¯R1F¯R2T¯R1T¯R2+F¯R1F¯R2)ℵ))(1−(1−(1−(1−T¯R1−F¯R1)(1−T¯R2−F¯R2)))ℵ))}.)(19)