# Novel Information Measures for Fermatean Fuzzy Sets and Their Applications to Pattern Recognition and Medical Diagnosis

Authors: Shahzaib Ashraf; Attaullah; Muhammad Naeem; Asghar Khan; Noor Rehman; M. K. Pandit
Journal: Computational Intelligence and Neuroscience
 (2023)

---

## Abstract

Fermatean fuzzy sets (FFSs) have piqued the interest of researchers in a wide range of domains. The striking framework of the FFS is keen to provide the larger preference domain for the modeling of ambiguous information deploying the degrees of membership and nonmembership. Furthermore, FFSs prevail over the theories of intuitionistic fuzzy sets and Pythagorean fuzzy sets owing to their broader space, adjustable parameter, flexible structure, and influential design. The information measures, being a significant part of the literature, are crucial and beneficial tools that are widely applied in decision-making, data mining, medical diagnosis, and pattern recognition. This paper aims to expand the literature on FFSs by proposing many innovative Fermatean fuzzy sets-based information measures, namely, distance measure, similarity measure, entropy measure, and inclusion measure. We investigate the relationship between distance, similarity, entropy, and inclusion measures for FFSs. Another achievement of this research is to establish a systematic transformation of information measures (distance measure, similarity measure, entropy measure, and inclusion measure) for the FFSs. To accomplish this aim, new formulae for information measures of FFSs have been presented. To demonstrate the validity of the measures, we employ them in pattern recognition, building materials, and medical diagnosis. Additionally, a comparison between traditional and novel similarity measures is described in terms of counter-intuitive cases. The findings demonstrate that the innovative information measures do not include any absurd cases.


---

## Body

## 1. Introduction

The idea of the fuzzy set (FS) was developed by Zadeh [1] in 1965, which addressed vagueness and ambiguity in real-world situations. In 1970, Bellman and Zadeh (1970) introduced the concept of decision-making (DM) problems with uncertainty. DM is a systematic procedure of selecting the most ideal choice from a collection of available alternatives. Therefore, the decision maker plays a crucial role in real world environments [2]. A smart decision may have a significant impact on the direction of someone’s lifestyle. Before making a final selection, a DM assesses the restrictions, advantages, and characteristics of each alternative. Since an FS is defined by a single parameter: membership degree. Several higher-order FSs have been described in recent decades by several scholars.Atanassov [3] established the notion of intuitionistic fuzzy sets (IFSs) capable of dealing with complexity and uncertainty and it has been extensively examined and utilized by several researchers in DM problems. An IFS is defined by three parameters: membership grade (MG), nonmembership grade (NMG), and hesitancy margin with the property that the sum of MG and NMG must be less than or equal to 1.In many situations, it is conceivable that the sum of the MG and NMG will be greater than 1. To overcome these challenges, Yager [4] introduced the Pythagorean fuzzy set (PyFS) as an extension of the IFS theory. PyFS is defined by an MG and NMG and satisfies the criterion that the square sum of its MG and NMG is less than or equal to 1. Therefore, PyFSs can more accurately express the fuzzy nature of information than IFS.In the field of PyFS, there are various approaches for solving real-life multiattribute decision-making (MADM) situations. A number of researchers have also suggested real-world applications in a Pythagorean fuzzy environment. However, if orthopair FSs as0.9,0.5, where 0.9 is the MG of specific criteria of a parameter and 0.5 is the NMG, it does not fulfill the IFS and PFS requirements. However, the cubic sum of the MG and NMG is equal to or less than one. In this context, Senapati and Yager [5] recently introduced the Fermatean fuzzy set (FFS). They also demonstrated that FFSs have larger degrees of uncertainty than IFSs and PyFSs, are capable of sustaining higher levels of uncertainty, and can solve MCDM challenges. Information measures are an essential notion for dealing with MADM challenges in a variety of domains, including pattern recognition, clinical diagnosis, and personnel appointment. There are several types of information measures established such as distance, similarity, entropy, and inclusion measures.The MADM process are normally assisted by similarity measures, distance measures, inclusion measures, entropy measures, and, in certain situations, aggregation operators. The degree of similarity measures has garnered considerable interest in recent decades due to its importance in DM, data mining, pattern recognition, and medical diagnosis applications. Szmidt and Kacprzyk [6] performed the first investigation, extending well-known distance measures such as the Hamming distance and the Euclidian distance to the IFS environment and comparing them to approaches used for conventional fuzzy sets. However, Wang and Xin [7] suggested that Szmidt and Kacprzyk [6] distance measure was ineffective in certain situations. Therefore, several innovative pattern recognition distance measures were developed and implemented. Grzegorzewski [8] also extended Hamming, Euclidean, and their normalized versions to the IFS framework. Later on, Chen [9] demonstrated that several flaws occurred in Grzegorzewski [8] by providing counter-examples. Hung and Yang [10] described three similarity measures and extended the Hausdorff distance to IFSs. On the other side, rather than expanding well-established measures, various research established novel similarity measures for IFS.Yong et al. [11] developed a novel similarity measure for IFS based on MG and NMG. Mitchell [12] demonstrated that Yong et al.’s [11] similarity measure had certain counter-intuitive circumstances and improved it statistically. Additionally, Liang and Shi [13] provided examples to demonstrate that the similarity measure proposed by Yong et al. [11] was unsuitable for certain scenarios, and hence developed various additional similarity measures for IFSs.Xu [14] formulated a series of IFS-based similarity measures and applied them to the MADM problem employing IF information. Xu and Chen [15] presented a set of distance and similarity measures that are different combinations and extensions of the weighted Hamming, Euclidean, and Hausdorff distances. Xu and Yager [16] constructed a similarity measure between IFSs and used it to MAGDM utilizing IF preference relations.In addition to this research, several researchers investigated the relationships between IFSs’ distance, similarity, and entropy measures. Zeng and Guo [17] analyzed the relationship between normalized distance, similarity, inclusion, and entropy of interval-valued fuzzy collections. Additionally, it was demonstrated that the similarity, inclusion, and entropy of interval-valued fuzzy sets may be induced using the normalized distance of their axiomatic definitions. Wei et al. [18] proposed a generalized entropy measure for IFSs and PyFSs. Additionally, a technique was developed for constructing similarity measures for IFS and PyFSs using entropy measures. Numerous researchers investigated information measures (distance measure, similarity measure, entropy measure, and inclusion measure) for IFSs and PyFSs and their transformations relationship. Dengfeng and Chuntian [19] investigated the similarity between IFSs and used their findings to pattern recognition. Huang and Yang [10] presented the Hausdorff distance as a similarity measure between IFSs and utilized it to assess the degree of similarity between IFSs. Ashraf et al. [20] gave the idea of a spherical fuzzy set then they implicated this concept also in decision-making [21].Nguyen et al. [22] developed a novel knowledge-based similarity measure for IFSs and demonstrated its application to pattern recognition. Zhang [23] pioneered a unique strategy for PyFSs MADM based on similarity measures. Zhang et al. [24] explored the use of the application of a scoring function on IFSs with double parameters for pattern recognition and medical diagnosis. Ejegwa established distance [25] and similarity measures [26] for PyFSs.Ye [27] designed and implemented a cosine similarity measure for IFSs (CIFS). In addition, Ye [28] introduced the cosine similarity measure for interval-valued IFSs (CIVIFSs) and described its use in solving MADM problems. Liu et al. [29] investigated the cosine similarity measure between hybrid IFSs and their application for diagnostic purposes. In recent years, several scholars have conducted research on PyFS information measures (distance measure, similarity measure, entropy measure, and inclusion measure). Wei and Wei [30] introduced a set of ten cosine-based PFS similarity measures relying on the MG, NMG, and hesitation of PyFSs in order to improve the capacity to cope with the two optimization challenges related to pattern recognition and medical diagnosis procedures. Peng [31] established a PyFS similarity measure based on the parameters Lp norm and levels of ambiguity, which were examined in detail in relation to the PyFS similarity measure. Peng et al. [32] developed the fundamental definitions of PyFS information measures, along with the similarity measure, as well as discussed the transformation principles for the established information measures.The advantages of existing information measures are as follows:An examination of the existing literature on FS, IFS, and PyFS exposes a number of weaknesses that spur us to create a more potent class of novel information measures (distance measure, similarity measure, entropy measure, and inclusion measure).The disadvantages of existing information measures are as follows:(i)
Some of them cannot help but be caught in pointless circumstances (i.e., dividing by zero).(ii)
Many of them struggle to avoid examples that seem to go against logic.(iii)
Many of them are unable to categorize the results, and some of them provide irrational results. We describe a class of useful FFS information measures (distance measure, similarity measure, entropy measure, and inclusion measure), offer associated information measure formulations, and examine their transformation connections to address the flaw in the prior research.The important contributions of the current manuscript are listed.


### 1.1. Important Contributions of the Manuscript

(i)
Development of axiomatic FFS information measures (distance measure, similarity measure, entropy, and inclusion measure).(ii)
We develop several equations for FFS information metrics and thoroughly examine the relevant transformation relationships.(iii)
To determine the necessary distances for pattern recognition and medical diagnostics in order to show that they are effective, we compared the suggested FFS distance measurements to those that are previously documented in the literature. Moreover, we demonstrate the viability and efficiency of using distance measurements between FFS data.(iv)
A number of illogical examples of traditional similarity measures are used to show the effectiveness of the novel measures.(v)
We use them for pattern recognition, building supplies, and medical diagnosis.(vi)
We come to the conclusion that the proposed similarity measures outperform current similarity measures for pattern recognition problems.(vii)
To show the effectiveness of the suggested information measures, which are also contrasted with current similarity measures, numerical examples are offered. For issues with medical diagnosis, a comparison between the proposed similarity measures and traditional similarity measurements is carried out.(viii)
Additionally, we used an example to demonstrate how the suggested FFS inclusion measures could be used for pattern identification. The results show that new illustration approaches are both feasible and successful.The manuscript is organized as follows: Section2 discusses the definitions and fundamental ideas of FS, HFS, IFS, and FFS, as well as the corresponding operational rules of FFS. Section 3 introduces a new type of information measures, provides related information measure formulations, and investigates their transformation relationships for FFSs. In Sections 4 to 7, we demonstrated the application of the novel information measures between FFSs to pattern recognition. Moreover, a comparative study has been presented between the proposed similarity measure and conventional similarity measures. Section 7 concludes the paper by outlining the future area of research.


## 1.1. Important Contributions of the Manuscript

(i)
Development of axiomatic FFS information measures (distance measure, similarity measure, entropy, and inclusion measure).(ii)
We develop several equations for FFS information metrics and thoroughly examine the relevant transformation relationships.(iii)
To determine the necessary distances for pattern recognition and medical diagnostics in order to show that they are effective, we compared the suggested FFS distance measurements to those that are previously documented in the literature. Moreover, we demonstrate the viability and efficiency of using distance measurements between FFS data.(iv)
A number of illogical examples of traditional similarity measures are used to show the effectiveness of the novel measures.(v)
We use them for pattern recognition, building supplies, and medical diagnosis.(vi)
We come to the conclusion that the proposed similarity measures outperform current similarity measures for pattern recognition problems.(vii)
To show the effectiveness of the suggested information measures, which are also contrasted with current similarity measures, numerical examples are offered. For issues with medical diagnosis, a comparison between the proposed similarity measures and traditional similarity measurements is carried out.(viii)
Additionally, we used an example to demonstrate how the suggested FFS inclusion measures could be used for pattern identification. The results show that new illustration approaches are both feasible and successful.The manuscript is organized as follows: Section2 discusses the definitions and fundamental ideas of FS, HFS, IFS, and FFS, as well as the corresponding operational rules of FFS. Section 3 introduces a new type of information measures, provides related information measure formulations, and investigates their transformation relationships for FFSs. In Sections 4 to 7, we demonstrated the application of the novel information measures between FFSs to pattern recognition. Moreover, a comparative study has been presented between the proposed similarity measure and conventional similarity measures. Section 7 concludes the paper by outlining the future area of research.


## 2. Basic Terminologies

In this section, we provide some relevant fundamental information, such as FS, HFS, IFS, FFSs, and some related operational laws, which are listed. These core concepts will assist readers in comprehending the proposed framework.Definition 1.
(see [1]). Let Z=ϰ1,ϰ2,ϰ3,…,ϰn be a finite set. The FS F on Z is defined as follows:(1)F=ϰi,IFϰi|ϰi∈Z,where Fϰi∈0,1 known to be MG.Definition 2.
(see [3]). Let Z=ϰ1,ϰ2,ϰ3,…,ϰn be a finite set. An IFS Ϝ over ϰ is follows:(2)Ϝ=ϰi,IϜϰi,ðϜϰi|ϰi∈Z,for each ϰi∈Z the functions IϜ:Z⟶0,1, 1 and ðϜ:ϰ⟶0,1 denotes the MG and NMG, respectively, which must satisfy the property 0≤IϜϰi+ðϜϰi≤1.Definition 3.
(see [33]). Let ϰ=ϰ1,ϰ2,ϰ3,…,ϰn be a finite set. The HFS H on ϰ is defined as follows:(3)H=ϰi,IhHϰi|ϰi∈Z,where hϰi is a set of values contain in (0,1), which shows the MG of ϰi∈H. The element of hϰi is known as the HF element.Definition 4.
(see [34]). Let Z=ϰ1,ϰ2,ϰ3,…,ϰn be a finite set, a Fermatean fuzzy sets (FFSs) F over Z is defined as follows:(4)F=ϰi,IFϰi,ðFϰi|ϰi∈ϰ,for each ϰi∈ϰ the functions IF:ϰ⟶0,1 and ðF:ϰ⟶0,1 denote the MG and NMG, respectively, which must satisfy ðFϰi3+IFϰi3≤1. The degree of indeterminacy is given as follows:(5)πFϰi=1−IFϰi3+ðFϰi33.Definition 5.
(see [5]). If Y and P be two FFSs, then the operations can be defined as follows:(i)
Addition:Y⊕P=ϰi∈Z,IY3ϰi+IP3ϰi−IY3ϰiIP3ϰi3,ðYϰiðPϰi|ϰi∈Z(ii)
Multiplication:Y⊗P=ϰi∈Z,IYϰiIPϰi,ðY3ϰi+ðP3ϰi−ðY3ϰiðP3ϰi3|ϰi∈Z(iii)
Scalar multiplication:λ⊙Y=ϰi∈Z,1−1−IY3ϰiλ,ðY3ϰiλ3ϰi∈Zandλ0(iv)
Exponent:Yλ=ϰi∈Z,IY3ϰi,1−1−ðP3ϰiλ3ϰi∈ZDefinition 6.
(see [5]). If Y and P be two FFSs, then the operations can be defined as follows:(i)
Complement:Yc=ðYϰi,IYϰi|ϰi∈Z;(ii)
Equality:Y=P iff for all ϰi∈Z,IYϰi=IPϰiandðYϰi=ðPϰi;(iii)
Intersection:Y∩P=minIYϰi,IPϰi,maxðYϰi,ðPϰi|ϰi∈Z;(iv)
Union:Y∪P=maxIYϰi,IPϰi,minðYϰi,ðPϰi|ϰi∈Z.Definition 7.
(see [5]). If Y, P,andM be three FFSs, then the following characteristics are held:(i)
Y∪P=P∪Y;(ii)
Y∩P=P∩Y;(iii)
Y∪P∪M=Y∪P∪M;(iv)
Y∩P∩M=Y∩P∩M;(v)
α∪Y∪P=αY∪αP;(vi)
Y∪Pα=Yα∪Pα.


## 3. Some New Types of Information Measures between FFSs

This section explains the axiomatic framework of FFSs information measures (distance, similarity, entropy, and inclusion), as well as their related formulations. Simultaneously, their transformation relationships are thoroughly examined.


### 3.1. Distance Measures for FFSs

This section introduces the idea of a distance measures for FFSs. A number that is assigned to a pair of points in a space which indicates how far those points are from one another. A distance measure is called a metric if it is always positive and also it is always symmetric.Definition 8.
LetY, P, and M be three FFSs on Z. A distance measure DY,P is a mapping D: FFS Z× FFS Z⟶0,1, carrying the following features:(1)
0≤DY,P≤1;(2)
DY,P=DP,Y;(3)
DY,P=0 iff Y=P;(4)
DY,Yc=1 iff Y is a crisp set;(5)
IfY⊆P⊆M, then DY,P≤DY,M and DP,M≤DY,M.Theorem 1.
LetY and P be two FFS s, then DiY,Pi=1,2,…,13 is a distance measure.(1)
DY,P=1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi;(2)
D2Y,P=1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi;(3)
D3Y,P=1/4X∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi+∑ϰi∈ZIY3ϰi−ðY3ϰi−IP3ϰi−ðP3ϰi(4)
D4Y,P=1/Z∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(5)
D5Y,P=2/Z∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(6)
D6Y,P=2∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/∑ϰi∈Z1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(7)
D7Y,P=1−α∑ϰi∈ZIY3ϰi∧IP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi−β∑ϰi∈ZðY3ϰi∧ðP3ϰi/∑ϰi∈ZðY3ϰi∨ðP3ϰi,α+β=1, α,β∈0,1;(8)
D8Y,P=1−α/Z∑ϰi∈ZIY3ϰi∧IP3ϰi/IY3ϰi∨IP3ϰi−β/Z∑ϰi∈ZðY3ϰi∧ðP3ϰi/ðY3ϰi∨ðP3ϰi,α+β=1, α,β∈0,1;(9)
D9Y,P=1−1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/IY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(10)
D10Y,P=1−∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(11)
D11Y,P=1−1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/IY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(12)
D12Y,P=1−∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(13)
D13Y,P=1/2Zl1+1t∑ϰi∈Zl1IY3ϰi−IP3ϰi−ðY3ϰi−ðP3ϰit+1/2Zl2+1t∑ϰi∈Zl2ðY3ϰi−ðP3ϰi−IY3ϰi−IP3ϰitt


### 3.2. Similarity Measure for FFSs

This section introduces the idea of similarity measures for FFSs. Similarity functions take a pair of points and return a large similarity value for nearby points, and a small similarity value for distant points. One way to transform between a distance function and a similarity measure is to take the reciprocal.Definition 9.
LetY, P, and M be three FFSs on Z. A similarity measure SY and P is a mapping S FFS (Z) × FFS (Z) ⟶0,1, possessing the following properties:(i)
0≤S (Y and P) ≤ 1;(ii)
S (Y and P) = S (P and Y);(iii)
S (Y and P) = 1 iff Y = P;(iv)
S (Y and Yc) = 0 iff Y is a crisp set;(v)
IfY⊆P⊆M, then S (Y and P) ≤S (Y and M) and S (P and M) ≤S (Y and M).Theorem 2.
LetY and P be two FFSs, then SiY,Pi=1,2,…,13 is a distance measure.(1)
S1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi(2)
S2Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi−ðY3ϰi−ðP3ϰi;(3)
S3Y,P=1−1/4X∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi+∑ϰi∈ZIY3ϰi−ðY3ϰi+IP3ϰi−ðP3ϰi(4)
S4Y,P=1−1/Z∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(5)
S5Y,P=1/Z∑ϰi∈Z1−IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(6)
S6Y,P=∑ϰi∈Z1−IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/∑ϰi∈Z1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(7)
S7Y,P=α∑ϰi∈ZIY3ϰi∧IP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+β∑ϰi∈ZðY3ϰi∧ðP3ϰi/∑ϰi∈ZðY3ϰi∨ðP3ϰi, α+β=1, α,β∈0,1;(8)
S8Y,P=α/Z∑ϰi∈ZIY3ϰi∧IP3ϰi/IY3ϰi∨IP3ϰi+β/Z∑ϰi∈ZðY3ϰi∧ðP3ϰi/ðY3ϰi∨ðP3ϰi,α+β=1, α,β∈0,1;(9)
S9Y,P=1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/IY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(10)
S10Y,P=∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(11)
S11Y,P=1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/IY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(12)
S12Y,P=∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(13)
S13Y,P=1/2Zl1+1t∑ϰi∈Zl1IY3ϰi−IP3ϰi−ðY3ϰi−ðP3ϰit+1/2Zl2+1t∑ϰi∈Zl2ðY3ϰi−ðP3ϰi−IY3ϰi−IP3ϰittTheorem 3.
Fori=1,2,3,…,13, if α=β=1/2, then we have(i)
SiY,Pc=SiPc,Y;(ii)
SiY,P=SiY∩P,Y∪P;(iii)
SiY,Y∩P=SiY,Y∪P;(iv)
SiY,Y∪P=SiY,Y∩P.Theorem 4.
Fori=1,2,…,6, we have(i)
SiY,Y⊗P=SiP,Y⊕P;(ii)
SiY,Y⊕P=SiP,Y⊗P;Theorem 5.
Fori=1,4,5,6 and for all ϰi∈Z, IY3ϰi+IP3ϰi=1, and ðY3ϰi+ðP3ϰi=1, we have(i)
SiY,Y⊘P=SiP,Y⊖P,IY3ϰi≤IP3ϰi,andðY3ϰi≥ðP3ϰi(ii)
SiY,Y⊖P=SiP,Y⊘P,IY3ϰi≥IP3ϰi,andðY3ϰi≤ðP3ϰi


### 3.3. Entropy for FFSs

LetY and P two FFSs on Z. An entropy measure EY is a mapping E: FFS Z⟶0,1, carrying the following features:(i)
0≤EY≤1;(ii)
EY=0 iff Y is a crisp set;(iii)
EY=1 iff IY3ϰi=ðYtϰi;(iv)
EY=EYc;(v)
IfEY≤EP if Y is less fuzzy than P, that is(6)IY3ϰi≤IP3ϰi≤ðP3ϰi≤ðY3ϰi,ðY3ϰi≤ðP3ϰi≤IP3ϰi≤IY3ϰi.Theorem 6.
LetYbe two FFSs, thenEiY,Pi=1,2,…,12is an entropy.


### 3.4. Inclusion Measure for FFSs

LetY, P, and M be three FFSs on Z. An inclusion measure IY,P is a mapping I: FFS Z× FFS Z⟶0,1, carrying the following features:(i)
0≤IY,P≤1;(ii)
IY,P=1 iff Y⊆P;(iii)
IY,P=0 iff Y=Φ and P=∅;(iv)
IfY⊆P⊆M, then IY,P≤IY,M and IP,M≤IY,M.Theorem 7.
LetY and P be two FFS s, then IiY,Pi=1,2,…,7 is an inclusion measure.(1)
I1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IY3ϰi∧IP3ϰi+ðY3ϰi−ðY3ϰi∨ðP3ϰi;(2)
I2Y,P=1,Y=∅∑ϰi∈Z1+IY3ϰi∧IP3ϰi−ðY3ϰi∨ðP3ϰi/∑ϰi∈Z1+IY3ϰi−ðY3ϰi,Y≠∅;(3)
I3Y,P=1,Y=P=∅∑ϰi∈Z1+IY3ϰi−ðY3ϰi/∑ϰi∈Z1+IY3ϰi∨IP3ϰi−ðY3ϰi∧ðP3ϰi,others;(4)
I4Y,P=1,Y=P=∅∑ϰi∈Z1+IY3ϰi−ðY3ϰi/∑ϰi∈Z1+ðY3ϰi∧ðP3ϰi−IY3ϰi∨IP3ϰi,others;(5)
I5Y,P=1,Y=P=∅1/Z∑ϰi∈Z1+IY3ϰi∧IP3ϰi−ðY3ϰi∨ðP3ϰi/1+IY3ϰi−ðY3ϰi,others(6)
I6Y,P=1,Y=P=∅1/Z∑ϰi∈Z1+IY3ϰi−ðY3ϰi/1+IY3ϰi∨IP3ϰi−ðY3ϰi∧ðP3ϰi,others(7)
I7Y,P=1,Y=P=∅1/Z∑ϰi∈Z1+IY3ϰi−ðY3ϰi/1+ðY3ϰi∧ðP3ϰi−IY3ϰi∨IP3ϰi,others


### 3.5. The Relations between These Measures

In this section, we study the relations between inclusion, entropy, similarity measure, and distance measure of Fermatean fuzzy sets. First, according to the definitions of similarity measure and distance measure of Fermatean fuzzy sets, one should note that they are all used for estimating the degree of similarity between two Fermatean fuzzy sets. The main difference is as follows: for the similarity measure, a greater value means that the two Fermatean fuzzy sets are more similar than are a pair with a lower value. The situation for the distance measure is just the opposite, that is, the smaller the value is, the more similar these two Fermatean fuzzy sets are. So, we can obtain the following theorem.


### 3.6. Transformation Relationships among Information Measures for FFSs

Theorem 8.
SupposeD be the Fermatean fuzzy distance measure for Y,P∈ FFS s, then SY,P=1−DY,P is the similarity measure of FFS sY and P. The proof is straightforward.Theorem 9.
ForY,P∈FFSs, and we order D1Y,P=1/Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi, then we haveSY,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi=S1Y,P.
Also,SiY,P=1−DiY,Pi=1,2,…,13.Theorem 10.
LetD and S be the distance and similarity measures of FFSs, for Y∈ FFSs, then EY=1−SY,Yc=1−SY,Yc is the entropy of FFSs.Proof.
(i)
(E1) It is straightforward.(ii)
(E2) If Y is a crisp set, then Y=∅ or Y=Φ, we have SY,Yc=0. Therefore, EY=0.(iii)
(E3) EY=1⇔SY,Yc=SYc,Y⇔IY3ϰi=IY3ϰi=ðY3ϰi for ϰi∈Z.(iv)
(E4) EY=SY,Yc=SYc,Y=EYc.(v)
(E5) Since IY3ϰi≤IP3ϰi≤ðP3ϰi≤ðY3ϰi implies Y⊆P⊆Pc⊆Yc. Therefore, according the definition of similarity measure of FFS s, we have SY,Yc≤SP,Yc≤EY,Yc, that is, EY≤EP. Similarly, if ðY3ϰi≤ðP3ϰi≤IP3ϰi≤IY3ϰi, then we have SY,Yc≤SP,Yc≤SP,Pc, that is, EY≤EP. This completes the proof.Theorem 11.
ForY,P∈FFSs, and we orderSY,P=S1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi, then we haveEY=S1Y,Yc=1−1/Z∑ϰi∈ZIY3ϰi−ðY3ϰi|=E4YAlso,E4Y=S2Y,Yc=S3Y,Yc=S4Y,Yc=S13Y,YcE5Y=S7Y,Yc=S10Y,Yc,E6Y=S8Y,Yc=S9Y,Yc,E7Y=S12Y,YcE8Y=S11Y,Yc.Definition 10.
LetY be an FFS, mY,nY∈ FFSs, ∀ϰi∈Z, mYϰi=ImYϰi,ðmYϰi, and nYϰi=InYϰi,ðnYϰi their membership and nonmembership functions are defined as follows:ImYϰi=1+IY3ϰi−ðY3ϰi33,ðmYϰi=1−IY3ϰi−ðY3ϰi3,InYϰi=1−IY3ϰi−ðY3ϰi33,ðnYϰi=1+IY3ϰi−ðY3ϰi3.Theorem 12.
LetD be the distance measure and S be the similarity measure of FFSs, for Y in FFSs, then EY=SmY,nY=1−DY,Yc=1−SmY,nY is the entropy of FFS Y.Proof.
(i)
(E1) It is straightforward.(ii)
(E2) If Y is a crisp set, then ∀ϰi∈Z, we have IYϰi=1, ðYϰi=0, or IYϰi=0, ðYϰi=1. Therefore, we can achieve(7)ImYϰi=1,ðmYϰi=0,ImYϰi=0,ðmYϰi=1.It implies thatmY=Φ,nY=∅, consequently, SmY,nY=0.(iii)
(E3) EY=1⇔SmY,nY=1⇔mY=nY⇔ImYϰi=InYϰi,ðmYϰi=ðnYϰi.(iv)
(E4 Using the definitions of mY and nY, we have mY=mYc,nY=nYc, hence SmY,nY=SmYc,nYc.(v)
(E5 Since IYϰi≤IPϰi≤ðYϰi≤ðPϰi implies Y⊆P⊆Pc⊆Yc. Therefore, we have IY3ϰi−ðY3ϰi≥IP3ϰi−ðP3ϰi. It means that n(A) ≤nP≤mP≤mY, so we have SmY,nY≤SmP,nY≤SmP,nP that is, EY≤EP.
Similarly, ifðYϰi≤ðPϰi≤IPϰi≤IYϰi, then we have(8)SmY,nY≤SmY,nP≤SmP,nP,thatis,EY≤EP.
This completes the proof.Theorem 13.
SupposeD and S be the distance measure and similarity measures of FFSs, respectively, for Y,P∈ FFSs, then IY,P=SY,Y∩P=1−DY,Y∩P is the inclusion measure of FFSs Y and P.Proof.
(i)
(I1) It is straightforward.(ii)
(I2) If Y⊆P, then SY,Y∩P=SY,Y=1=SY,P.(iii)
(I3) IY,P=0⇔SY,Y∩P=0⇔Y=Φ, P=∅.(iv)
(I4) If Y⊆P⊆O, then IO,Y=SO,O∩Y=SO,Y and IP,Y=SP,P∩Y=SP,Y. Known by the similarity measure of FFSs, we have IO,M≤IP,Y. Similarly, IO,Y≤IO,P. This completes the proof.Theorem 14.
SupposeD and S be the distance measure and similarity measures of FFSs, respectively, for Y,P in FFSs, then IY,P=SP,Y∪P=1−DP,Y∪P is the inclusion measure of FFSs Y and P.Definition 11.
LetY and P be two FFS s, then we define g(A,B) ∈ FFS s,∀ϰi∈Z,IgY,Pϰi=1+minIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi/2,ðgY,Pϰi=1−maxIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi/2.Theorem 15.
SupposeE be the entropy measure of FFSs, for Y, P∈ FFSs, then EgY,P is the similarity measure of FFSs Y and P.Proof.
(i)
(S1− ( S2 are straightforward.(ii)
(S3) Known by the definition of entropy of FFS s, EgY,P=1⇔IgY,Pϰi=ðgY,Pϰi⇔IY3ϰi−IPtϰi=0, ðY3ϰi−ðP3ϰi=0⇔IYϰi=IPϰi, ðYϰi=ðPϰi⇔Y=P.(iii)
(S4) If Y is a crisp set, then IYϰi=1, ðYϰi=0 or IYϰi=0, ðYϰi=1. Hence, IgY,Ycϰi=1,IYY,Pcϰi=0,ðYY,Pcϰi=0, it implies gY,Pc = Φ, so EgY,Pc=0.(iv)
(S5) Since Y⊆P⊆O, then ∀ϰi∈Z, we have IYϰi≤IPϰi≤IOϰi,ðOϰi≤ðPϰi≤ðYϰi. Therefore, we have IY3ϰi−IO3ϰi|≥IY3ϰi−IP3ϰi and ðY3ϰi−ðO3ϰi≥ðY3ϰi−ðP3ϰi.
Furthermore, we have(9)minIY3ϰi−IO3ϰi,ðY3ϰi−ðO3ϰi,≥minIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi,maxIY3ϰi−IO3ϰi,ðY3ϰi−ðO3ϰi,≥maxIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi.
Also, we can know(10)IgY,Oϰi≥IgY,Pϰi,ðYY,Oϰi≤ðYY,Pϰi,IgY,Pϰi≥ðYY,Pϰi,that is(11)ðgY,Oϰi≤ðgY,Pϰi≤IgY,Pϰi≤IgY,Oϰi,and known by the definition, EgY,O≤EgY,P.
Similarly, we can prove thatEgY,O≤EgP,O. This completes the proof. □Theorem 16.
SupposeI be the inclusion measure of FFS s, for Y∈ FFS s, then EY=IY∪Yc,Y∩Yc is the entropy of FFS sY.Proof.
(i)
(E1 It is straightforward.(ii)
(E2 If Y is a crisp set, then Y=Φ or Y=∅, we have IY∪Yc, Y∩Yc=IΦ,∅=0. Therefore, EY=0.(iii)
(E3EY = 1 ⇔IY∪Yc, Y∩Yc=1⇔Y∪Yc⊆Y∩Yc⇔Y∪Yc=Y∩Yc⇔IYϰi=IYϰi=ðYϰi(iv)
(E4EY=IY∪Yc, Y∩Yc=IYc∪Y,Yc∩Y=EYc.(v)
(E5) Since IYϰi≤IPϰi≤IPϰi≤IYϰi≤ðPϰi≤ðYϰi implies Y⊆P⊆Pc⊆Yc. Furthermore, Y∩Yc⊆P∩Pc⊆P∪Pc⊆Y∪Yc.
According to the definition of inclusion measure, we haveIY∪Yc, Y∩Yc≤IY∪Yc, P∩Pc≤IP∪Pc, P∩Pc, so EY≤EP.
Similarly, ifðYϰi≤ðPϰi≤IPϰi≤IYϰi, then we can have IY∪Yc, Y∩Yc≤IP∪Pc, Y∩Yc≤IP∪Pc, P∩Pc, that is E(A) ≤EP. This completes the proof.Example 1.
ForY, P∈ FFS s, and we order IY,P=I1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IY3ϰi∧IP3ϰi+ðY3ϰi−ðY3ϰi, then we have EY=I1Y∪Yc,Y∩Yc=1−1/2Z∑ϰi∈ZIY3ϰi−ðY3ϰi=E4YTheorem 17.
SIYϰi,ðYϰiis the entropy of FFSY.Proof.
(i)
(E1) It is straightforward.(ii)
(E2) If Y is a crisp set, then we have IYϰi=Φ, and ðYϰi=∅ or IYϰi=∅, and ðYϰi=Φ. Therefore, SIYϰi,ðYϰi=0.(iii)
(E3) Known by the definition of similarity measure of FFSs, we have(12)SIYϰi,ðYϰi=1⇔IYϰi=ðYϰi⇔EY=1.(iv)
(E4) EY=SIYϰi,ðYϰi=SðYϰi,IYϰi=EYc.(v)
(E5) Since IYϰi≤IPϰi≤ðPϰi≤ðYϰi implies Y⊆P⊆Pc⊆Yc. Namely, IY⊆IP⊆ðP⊆ðY. According to the definition of similarity measure, we can have SIYϰi,ðYϰi≤SIPϰi,ðPϰi, that is, EY≤EP.
Similarly, ifðYϰi≤ðPϰi≤IPϰi≤IYϰi, then we can have EY≤EP. This completes the proof.


## 3.1. Distance Measures for FFSs

This section introduces the idea of a distance measures for FFSs. A number that is assigned to a pair of points in a space which indicates how far those points are from one another. A distance measure is called a metric if it is always positive and also it is always symmetric.Definition 8.
LetY, P, and M be three FFSs on Z. A distance measure DY,P is a mapping D: FFS Z× FFS Z⟶0,1, carrying the following features:(1)
0≤DY,P≤1;(2)
DY,P=DP,Y;(3)
DY,P=0 iff Y=P;(4)
DY,Yc=1 iff Y is a crisp set;(5)
IfY⊆P⊆M, then DY,P≤DY,M and DP,M≤DY,M.Theorem 1.
LetY and P be two FFS s, then DiY,Pi=1,2,…,13 is a distance measure.(1)
DY,P=1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi;(2)
D2Y,P=1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi;(3)
D3Y,P=1/4X∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi+∑ϰi∈ZIY3ϰi−ðY3ϰi−IP3ϰi−ðP3ϰi(4)
D4Y,P=1/Z∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(5)
D5Y,P=2/Z∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(6)
D6Y,P=2∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/∑ϰi∈Z1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(7)
D7Y,P=1−α∑ϰi∈ZIY3ϰi∧IP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi−β∑ϰi∈ZðY3ϰi∧ðP3ϰi/∑ϰi∈ZðY3ϰi∨ðP3ϰi,α+β=1, α,β∈0,1;(8)
D8Y,P=1−α/Z∑ϰi∈ZIY3ϰi∧IP3ϰi/IY3ϰi∨IP3ϰi−β/Z∑ϰi∈ZðY3ϰi∧ðP3ϰi/ðY3ϰi∨ðP3ϰi,α+β=1, α,β∈0,1;(9)
D9Y,P=1−1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/IY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(10)
D10Y,P=1−∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(11)
D11Y,P=1−1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/IY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(12)
D12Y,P=1−∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(13)
D13Y,P=1/2Zl1+1t∑ϰi∈Zl1IY3ϰi−IP3ϰi−ðY3ϰi−ðP3ϰit+1/2Zl2+1t∑ϰi∈Zl2ðY3ϰi−ðP3ϰi−IY3ϰi−IP3ϰitt


## 3.2. Similarity Measure for FFSs

This section introduces the idea of similarity measures for FFSs. Similarity functions take a pair of points and return a large similarity value for nearby points, and a small similarity value for distant points. One way to transform between a distance function and a similarity measure is to take the reciprocal.Definition 9.
LetY, P, and M be three FFSs on Z. A similarity measure SY and P is a mapping S FFS (Z) × FFS (Z) ⟶0,1, possessing the following properties:(i)
0≤S (Y and P) ≤ 1;(ii)
S (Y and P) = S (P and Y);(iii)
S (Y and P) = 1 iff Y = P;(iv)
S (Y and Yc) = 0 iff Y is a crisp set;(v)
IfY⊆P⊆M, then S (Y and P) ≤S (Y and M) and S (P and M) ≤S (Y and M).Theorem 2.
LetY and P be two FFSs, then SiY,Pi=1,2,…,13 is a distance measure.(1)
S1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi(2)
S2Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi−ðY3ϰi−ðP3ϰi;(3)
S3Y,P=1−1/4X∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi+∑ϰi∈ZIY3ϰi−ðY3ϰi+IP3ϰi−ðP3ϰi(4)
S4Y,P=1−1/Z∑ϰi∈ZIY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(5)
S5Y,P=1/Z∑ϰi∈Z1−IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(6)
S6Y,P=∑ϰi∈Z1−IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi/∑ϰi∈Z1+IY3ϰi−IP3ϰi∨ðY3ϰi−ðP3ϰi;(7)
S7Y,P=α∑ϰi∈ZIY3ϰi∧IP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+β∑ϰi∈ZðY3ϰi∧ðP3ϰi/∑ϰi∈ZðY3ϰi∨ðP3ϰi, α+β=1, α,β∈0,1;(8)
S8Y,P=α/Z∑ϰi∈ZIY3ϰi∧IP3ϰi/IY3ϰi∨IP3ϰi+β/Z∑ϰi∈ZðY3ϰi∧ðP3ϰi/ðY3ϰi∨ðP3ϰi,α+β=1, α,β∈0,1;(9)
S9Y,P=1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/IY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(10)
S10Y,P=∑ϰi∈ZIY3ϰi∧IP3ϰi+ðY3ϰi∧ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+ðY3ϰi∨ðP3ϰi;(11)
S11Y,P=1/Z∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/IY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(12)
S12Y,P=∑ϰi∈ZIY3ϰi∧IP3ϰi+1−ðY3ϰi∧1−ðP3ϰi/∑ϰi∈ZIY3ϰi∨IP3ϰi+1−ðY3ϰi∨1−ðP3ϰi;(13)
S13Y,P=1/2Zl1+1t∑ϰi∈Zl1IY3ϰi−IP3ϰi−ðY3ϰi−ðP3ϰit+1/2Zl2+1t∑ϰi∈Zl2ðY3ϰi−ðP3ϰi−IY3ϰi−IP3ϰittTheorem 3.
Fori=1,2,3,…,13, if α=β=1/2, then we have(i)
SiY,Pc=SiPc,Y;(ii)
SiY,P=SiY∩P,Y∪P;(iii)
SiY,Y∩P=SiY,Y∪P;(iv)
SiY,Y∪P=SiY,Y∩P.Theorem 4.
Fori=1,2,…,6, we have(i)
SiY,Y⊗P=SiP,Y⊕P;(ii)
SiY,Y⊕P=SiP,Y⊗P;Theorem 5.
Fori=1,4,5,6 and for all ϰi∈Z, IY3ϰi+IP3ϰi=1, and ðY3ϰi+ðP3ϰi=1, we have(i)
SiY,Y⊘P=SiP,Y⊖P,IY3ϰi≤IP3ϰi,andðY3ϰi≥ðP3ϰi(ii)
SiY,Y⊖P=SiP,Y⊘P,IY3ϰi≥IP3ϰi,andðY3ϰi≤ðP3ϰi


## 3.3. Entropy for FFSs

LetY and P two FFSs on Z. An entropy measure EY is a mapping E: FFS Z⟶0,1, carrying the following features:(i)
0≤EY≤1;(ii)
EY=0 iff Y is a crisp set;(iii)
EY=1 iff IY3ϰi=ðYtϰi;(iv)
EY=EYc;(v)
IfEY≤EP if Y is less fuzzy than P, that is(6)IY3ϰi≤IP3ϰi≤ðP3ϰi≤ðY3ϰi,ðY3ϰi≤ðP3ϰi≤IP3ϰi≤IY3ϰi.Theorem 6.
LetYbe two FFSs, thenEiY,Pi=1,2,…,12is an entropy.


## 3.4. Inclusion Measure for FFSs

LetY, P, and M be three FFSs on Z. An inclusion measure IY,P is a mapping I: FFS Z× FFS Z⟶0,1, carrying the following features:(i)
0≤IY,P≤1;(ii)
IY,P=1 iff Y⊆P;(iii)
IY,P=0 iff Y=Φ and P=∅;(iv)
IfY⊆P⊆M, then IY,P≤IY,M and IP,M≤IY,M.Theorem 7.
LetY and P be two FFS s, then IiY,Pi=1,2,…,7 is an inclusion measure.(1)
I1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IY3ϰi∧IP3ϰi+ðY3ϰi−ðY3ϰi∨ðP3ϰi;(2)
I2Y,P=1,Y=∅∑ϰi∈Z1+IY3ϰi∧IP3ϰi−ðY3ϰi∨ðP3ϰi/∑ϰi∈Z1+IY3ϰi−ðY3ϰi,Y≠∅;(3)
I3Y,P=1,Y=P=∅∑ϰi∈Z1+IY3ϰi−ðY3ϰi/∑ϰi∈Z1+IY3ϰi∨IP3ϰi−ðY3ϰi∧ðP3ϰi,others;(4)
I4Y,P=1,Y=P=∅∑ϰi∈Z1+IY3ϰi−ðY3ϰi/∑ϰi∈Z1+ðY3ϰi∧ðP3ϰi−IY3ϰi∨IP3ϰi,others;(5)
I5Y,P=1,Y=P=∅1/Z∑ϰi∈Z1+IY3ϰi∧IP3ϰi−ðY3ϰi∨ðP3ϰi/1+IY3ϰi−ðY3ϰi,others(6)
I6Y,P=1,Y=P=∅1/Z∑ϰi∈Z1+IY3ϰi−ðY3ϰi/1+IY3ϰi∨IP3ϰi−ðY3ϰi∧ðP3ϰi,others(7)
I7Y,P=1,Y=P=∅1/Z∑ϰi∈Z1+IY3ϰi−ðY3ϰi/1+ðY3ϰi∧ðP3ϰi−IY3ϰi∨IP3ϰi,others


## 3.5. The Relations between These Measures

In this section, we study the relations between inclusion, entropy, similarity measure, and distance measure of Fermatean fuzzy sets. First, according to the definitions of similarity measure and distance measure of Fermatean fuzzy sets, one should note that they are all used for estimating the degree of similarity between two Fermatean fuzzy sets. The main difference is as follows: for the similarity measure, a greater value means that the two Fermatean fuzzy sets are more similar than are a pair with a lower value. The situation for the distance measure is just the opposite, that is, the smaller the value is, the more similar these two Fermatean fuzzy sets are. So, we can obtain the following theorem.


## 3.6. Transformation Relationships among Information Measures for FFSs

Theorem 8.
SupposeD be the Fermatean fuzzy distance measure for Y,P∈ FFS s, then SY,P=1−DY,P is the similarity measure of FFS sY and P. The proof is straightforward.Theorem 9.
ForY,P∈FFSs, and we order D1Y,P=1/Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi, then we haveSY,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi=S1Y,P.
Also,SiY,P=1−DiY,Pi=1,2,…,13.Theorem 10.
LetD and S be the distance and similarity measures of FFSs, for Y∈ FFSs, then EY=1−SY,Yc=1−SY,Yc is the entropy of FFSs.Proof.
(i)
(E1) It is straightforward.(ii)
(E2) If Y is a crisp set, then Y=∅ or Y=Φ, we have SY,Yc=0. Therefore, EY=0.(iii)
(E3) EY=1⇔SY,Yc=SYc,Y⇔IY3ϰi=IY3ϰi=ðY3ϰi for ϰi∈Z.(iv)
(E4) EY=SY,Yc=SYc,Y=EYc.(v)
(E5) Since IY3ϰi≤IP3ϰi≤ðP3ϰi≤ðY3ϰi implies Y⊆P⊆Pc⊆Yc. Therefore, according the definition of similarity measure of FFS s, we have SY,Yc≤SP,Yc≤EY,Yc, that is, EY≤EP. Similarly, if ðY3ϰi≤ðP3ϰi≤IP3ϰi≤IY3ϰi, then we have SY,Yc≤SP,Yc≤SP,Pc, that is, EY≤EP. This completes the proof.Theorem 11.
ForY,P∈FFSs, and we orderSY,P=S1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IP3ϰi+ðY3ϰi−ðP3ϰi+πY3ϰi−πP3ϰi, then we haveEY=S1Y,Yc=1−1/Z∑ϰi∈ZIY3ϰi−ðY3ϰi|=E4YAlso,E4Y=S2Y,Yc=S3Y,Yc=S4Y,Yc=S13Y,YcE5Y=S7Y,Yc=S10Y,Yc,E6Y=S8Y,Yc=S9Y,Yc,E7Y=S12Y,YcE8Y=S11Y,Yc.Definition 10.
LetY be an FFS, mY,nY∈ FFSs, ∀ϰi∈Z, mYϰi=ImYϰi,ðmYϰi, and nYϰi=InYϰi,ðnYϰi their membership and nonmembership functions are defined as follows:ImYϰi=1+IY3ϰi−ðY3ϰi33,ðmYϰi=1−IY3ϰi−ðY3ϰi3,InYϰi=1−IY3ϰi−ðY3ϰi33,ðnYϰi=1+IY3ϰi−ðY3ϰi3.Theorem 12.
LetD be the distance measure and S be the similarity measure of FFSs, for Y in FFSs, then EY=SmY,nY=1−DY,Yc=1−SmY,nY is the entropy of FFS Y.Proof.
(i)
(E1) It is straightforward.(ii)
(E2) If Y is a crisp set, then ∀ϰi∈Z, we have IYϰi=1, ðYϰi=0, or IYϰi=0, ðYϰi=1. Therefore, we can achieve(7)ImYϰi=1,ðmYϰi=0,ImYϰi=0,ðmYϰi=1.It implies thatmY=Φ,nY=∅, consequently, SmY,nY=0.(iii)
(E3) EY=1⇔SmY,nY=1⇔mY=nY⇔ImYϰi=InYϰi,ðmYϰi=ðnYϰi.(iv)
(E4 Using the definitions of mY and nY, we have mY=mYc,nY=nYc, hence SmY,nY=SmYc,nYc.(v)
(E5 Since IYϰi≤IPϰi≤ðYϰi≤ðPϰi implies Y⊆P⊆Pc⊆Yc. Therefore, we have IY3ϰi−ðY3ϰi≥IP3ϰi−ðP3ϰi. It means that n(A) ≤nP≤mP≤mY, so we have SmY,nY≤SmP,nY≤SmP,nP that is, EY≤EP.
Similarly, ifðYϰi≤ðPϰi≤IPϰi≤IYϰi, then we have(8)SmY,nY≤SmY,nP≤SmP,nP,thatis,EY≤EP.
This completes the proof.Theorem 13.
SupposeD and S be the distance measure and similarity measures of FFSs, respectively, for Y,P∈ FFSs, then IY,P=SY,Y∩P=1−DY,Y∩P is the inclusion measure of FFSs Y and P.Proof.
(i)
(I1) It is straightforward.(ii)
(I2) If Y⊆P, then SY,Y∩P=SY,Y=1=SY,P.(iii)
(I3) IY,P=0⇔SY,Y∩P=0⇔Y=Φ, P=∅.(iv)
(I4) If Y⊆P⊆O, then IO,Y=SO,O∩Y=SO,Y and IP,Y=SP,P∩Y=SP,Y. Known by the similarity measure of FFSs, we have IO,M≤IP,Y. Similarly, IO,Y≤IO,P. This completes the proof.Theorem 14.
SupposeD and S be the distance measure and similarity measures of FFSs, respectively, for Y,P in FFSs, then IY,P=SP,Y∪P=1−DP,Y∪P is the inclusion measure of FFSs Y and P.Definition 11.
LetY and P be two FFS s, then we define g(A,B) ∈ FFS s,∀ϰi∈Z,IgY,Pϰi=1+minIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi/2,ðgY,Pϰi=1−maxIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi/2.Theorem 15.
SupposeE be the entropy measure of FFSs, for Y, P∈ FFSs, then EgY,P is the similarity measure of FFSs Y and P.Proof.
(i)
(S1− ( S2 are straightforward.(ii)
(S3) Known by the definition of entropy of FFS s, EgY,P=1⇔IgY,Pϰi=ðgY,Pϰi⇔IY3ϰi−IPtϰi=0, ðY3ϰi−ðP3ϰi=0⇔IYϰi=IPϰi, ðYϰi=ðPϰi⇔Y=P.(iii)
(S4) If Y is a crisp set, then IYϰi=1, ðYϰi=0 or IYϰi=0, ðYϰi=1. Hence, IgY,Ycϰi=1,IYY,Pcϰi=0,ðYY,Pcϰi=0, it implies gY,Pc = Φ, so EgY,Pc=0.(iv)
(S5) Since Y⊆P⊆O, then ∀ϰi∈Z, we have IYϰi≤IPϰi≤IOϰi,ðOϰi≤ðPϰi≤ðYϰi. Therefore, we have IY3ϰi−IO3ϰi|≥IY3ϰi−IP3ϰi and ðY3ϰi−ðO3ϰi≥ðY3ϰi−ðP3ϰi.
Furthermore, we have(9)minIY3ϰi−IO3ϰi,ðY3ϰi−ðO3ϰi,≥minIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi,maxIY3ϰi−IO3ϰi,ðY3ϰi−ðO3ϰi,≥maxIY3ϰi−IP3ϰi,ðY3ϰi−ðP3ϰi.
Also, we can know(10)IgY,Oϰi≥IgY,Pϰi,ðYY,Oϰi≤ðYY,Pϰi,IgY,Pϰi≥ðYY,Pϰi,that is(11)ðgY,Oϰi≤ðgY,Pϰi≤IgY,Pϰi≤IgY,Oϰi,and known by the definition, EgY,O≤EgY,P.
Similarly, we can prove thatEgY,O≤EgP,O. This completes the proof. □Theorem 16.
SupposeI be the inclusion measure of FFS s, for Y∈ FFS s, then EY=IY∪Yc,Y∩Yc is the entropy of FFS sY.Proof.
(i)
(E1 It is straightforward.(ii)
(E2 If Y is a crisp set, then Y=Φ or Y=∅, we have IY∪Yc, Y∩Yc=IΦ,∅=0. Therefore, EY=0.(iii)
(E3EY = 1 ⇔IY∪Yc, Y∩Yc=1⇔Y∪Yc⊆Y∩Yc⇔Y∪Yc=Y∩Yc⇔IYϰi=IYϰi=ðYϰi(iv)
(E4EY=IY∪Yc, Y∩Yc=IYc∪Y,Yc∩Y=EYc.(v)
(E5) Since IYϰi≤IPϰi≤IPϰi≤IYϰi≤ðPϰi≤ðYϰi implies Y⊆P⊆Pc⊆Yc. Furthermore, Y∩Yc⊆P∩Pc⊆P∪Pc⊆Y∪Yc.
According to the definition of inclusion measure, we haveIY∪Yc, Y∩Yc≤IY∪Yc, P∩Pc≤IP∪Pc, P∩Pc, so EY≤EP.
Similarly, ifðYϰi≤ðPϰi≤IPϰi≤IYϰi, then we can have IY∪Yc, Y∩Yc≤IP∪Pc, Y∩Yc≤IP∪Pc, P∩Pc, that is E(A) ≤EP. This completes the proof.Example 1.
ForY, P∈ FFS s, and we order IY,P=I1Y,P=1−1/2Z∑ϰi∈ZIY3ϰi−IY3ϰi∧IP3ϰi+ðY3ϰi−ðY3ϰi, then we have EY=I1Y∪Yc,Y∩Yc=1−1/2Z∑ϰi∈ZIY3ϰi−ðY3ϰi=E4YTheorem 17.
SIYϰi,ðYϰiis the entropy of FFSY.Proof.
(i)
(E1) It is straightforward.(ii)
(E2) If Y is a crisp set, then we have IYϰi=Φ, and ðYϰi=∅ or IYϰi=∅, and ðYϰi=Φ. Therefore, SIYϰi,ðYϰi=0.(iii)
(E3) Known by the definition of similarity measure of FFSs, we have(12)SIYϰi,ðYϰi=1⇔IYϰi=ðYϰi⇔EY=1.(iv)
(E4) EY=SIYϰi,ðYϰi=SðYϰi,IYϰi=EYc.(v)
(E5) Since IYϰi≤IPϰi≤ðPϰi≤ðYϰi implies Y⊆P⊆Pc⊆Yc. Namely, IY⊆IP⊆ðP⊆ðY. According to the definition of similarity measure, we can have SIYϰi,ðYϰi≤SIPϰi,ðPϰi, that is, EY≤EP.
Similarly, ifðYϰi≤ðPϰi≤IPϰi≤IYϰi, then we can have EY≤EP. This completes the proof.


## 4. Apply the Distance Measure between FFSs to Pattern Recognition

In this part, we use numerical examples to show the feasibility and effectiveness of the innovative FFS distance measures to illustrate the applications of the established distance measures for specific FFSs in pattern recognition. Furthermore, we compare them to the existing distance measures.Example 2.
LetM1,M2,M3, and M4 be four known patterns that are illustrated by the following FFSs in X as follows:(13)M1=ϰ1,0.32,0.36,ϰ2,0.41,0.47,ϰ3,0.54,0.48,M2=ϰ1,0.41,0.43,ϰ2,0.52,0.51,ϰ3,0.60,0.32,M3=ϰ1,0.34,0.57,ϰ2,0.56,0.61,ϰ3,0.86,0.81,M4=ϰ1,0.23,0.11,ϰ2,0.31,0.30,ϰ3,0.45,0.35,L=ϰ1,0.25,0.22,ϰ2,0.46,0.34,ϰ3,0.47,0.54,where L is an unknown pattern. Its aimed is to determine the class to which L belongs. In order to do that, the distance between L and classes M1, M2, M3, and M4 are measured, and L is then allocated to the class Mg specified as follows:(14)g=argmaxgDMg,L.
For all the newly developed distance measuresD1−D13 for FFS, the distance between DM1,L, DM2,L, DM3,L, and DM4,L is determined and displayed in Table 1. It is observed in the Table 1, that an unknown pattern L belongs to a class M3 when D1 to D13 are used. It is clear that the cause for this difference is the first characteristic, i.e., ϰ1. The FFNs of ϰ1 are as follows:(15)0.32,0.36,0.41,0.43,0.34,0.57,0.23,0.11,0.25,0.22,for M1, M2, M3,M4, and L, respectively. It is predicted that the distance between 0.34,0.57 and 0.25,0.22 is larger than the distance between 0.41,0.43 and 0.25,0.22 is larger than the distance between 0.23,0.11 and 0.25,0.22 is larger than 0.32,0.36 and 0.25,0.22. As a conclusion, it appears that(16)DM3,L>DM2,L>DM4,L>DM4,L,is more acceptable. By routine calculations, we can find the aforementioned relation for D to D13 as shown in Table 1.Table 1
Distance measures for Example2withα=β=0.5,l1=l2=2,andt=1.
DM1,LDM2,LDM3,LDM4,LClassification resultsD10.08270.18290.43130.1137M1D20.03540.05040.06980.0272M1D30.05900.11670.25050.0704M1D40.05140.09560.29820.0638M1D50.09750.17380.43600.1169M1D60.09780.17460.45930.1200M1D70.41180.63340.76270.5214M1D80.79180.87450.92530.8486M1D90.79130.86980.92260.8360M1D100.41410.62330.76360.5184M1D110.69300.71670.80230.6902M1D120.07890.14830.37610.0699M1D130.03160.05990.11210.0212M1Example 3.
Assume that a doctor would like to diagnose the condition ofC:  (viral fever, malaria, typhoid, or chest problem) for patients P:  (Ragu, Mathi,Velu, and Karthi) with disease symptoms V: (headache, acidity, burning eyes and depression). The symptoms associated with the considered diagnosis are listed in Table 2–4, and the symptoms of the disease associated with each patient are listed in Table 2. Each table element is represented by a specific FFSs. For each patient, a precise diagnosis is necessary.The distance measuring methods mentioned here are used to assess the distance between each patient and each diagnosis. Each patient was then diagnosed using the concept of the shortest possible distance. To determine a condition of the patient, we may assess the distance measure between the symptoms associated with each illness and those associated with the patient. The diagnostic findings are provided in Table 2–4 using the distance measure formula D13. We may conclude that all the patients suffer from viral fever.Table 2
Symptomatic characteristics of the diagnosis under consideration.
HeadacheAcidityBurning eyesDepressionStress0.2,0.60.3,0.70.2,0.60.1,0.8Ulcer0.3,0.70.1,0.40.2,0.40.1,0.4Vision problem0.2,0.50.1,0.50.1,0.30.2,0.3Blood pressure0.1,0.40.2,0.50.2,0.40.1,0.3Table 3
Symptoms and features of the patients under consideration.
Blood pressureUlcerVision problemStressRagu0.2,0.40.1,0.30.1,0.40.2,0.3Mathi0.1,0.50.1,0.40.2,0.30.1,0.3Velu0.1,0.30.1,0.50.2,0.30.2,0.3Karthi0.1,0.40.2,0.40.1,0.50.1,0.3Table 4
The distance between the patient and the set of probable diagnoses usingD1.
Blood pressureUlcerVision problemStressRagu0.29610.15390.10470.0760Mathi0.26580.12360.04590.0899Velu0.28820.20620.06830.0522Karthi0.25360.16180.10930.0803


### 4.1. Comparison of the Distance Measure between FFSs in Medical Diagnosis

To illustrate the effectiveness of the novel distance measure for specific FFSs in pattern recognition, we present a numerical example and compare the novel findings to those reported in the literature.Example 4.
Consider a set of four patients, Al, Bob, Joe, and Ted, as represented byM=M1,M2,M3,M4 who have the symptoms temperature, headache, stomach pain, cough, and chest pain, which are represented by S=ϰ1,ϰ2,ϰ3,ϰ4,ϰ5. Let L=L1,L2,L3,L4,L5 be a list of possible diagnoses. Possible disease are defined as follows: L1: viral fever, L2: malaria, L3: typhoid, L4: stomach problem, and L5: chest problem. The FF relation M⟶S is illustrated by FFS, as can be seen in Table 5. The FF relation S⟶L is denoted by the FFS, as seen in Table 6. Each element in Table 6 is represented by FFS. The established distance measure methods are used to determine the distance between each patient and each diagnosis. Then, using the idea of minimal distance degree, each patient was diagnosed. We demonstrated the distance measure results of the patient Mjj=1,2,3,4 with regard to the diagnostic Lii=1,2,3,4,5 and the final diagnosis findings are given in Al has malaria Table 7, Bob has stomach problem Table 8, Joe has typhoid Table 9, and Ted has viral fever Table 10. We perform a comparison study with other methodologies to demonstrate the capability and validity of the presented distance measures, and the findings are provided in Table 11. Table 11 shows that the suggested distance measure approaches achieve the same result as in D [18], D [35], D [36], and D [37], demonstrating that using the proposed distance measure methods to solve the medical diagnosis problem is possible and beneficial. From the preceding practical implementation of the measures techniques, we may deduce that the proposed distance measures approaches are more effective and superior in handling real world challenges.Table 5
Symptomatic characteristics of the patient.
ϰ1ϰ2ϰ3ϰ4ϰ5M10.80,0.100.60,0.100.20,0.800.60,0.100.10,0.60M20.00,0.800.40,0.400.60,0.100.10,0.700.10,0.80M30.80,0.100.80,0.100.00,0.600.20,0.700.00,0.50M40.60,0.100.50,0.400.30,0.400.70,0.200.30,0.40Table 6
Symptomatic characteristics of the diagnosis.
ϰ1ϰ2ϰ3ϰ4ϰ5L10.40,0.000.30,0.500.10,0.700.40,0.300.10,0.70L20.70,0.000.20,0.600.00,0.900.70,0.000.10,0.80L30.30,0.300.60,0.100.20,0.700.20,0.600.10,0.90L40.10,0.700.20,0.400.80,0.000.20,0.700.20,0.70L50.10,0.800.00,0.800.20,0.800.20,0.800.80,0.10Table 7
Diagnostic results of the FFS distance measure for Al.
DM1,L1DM1,L2DM1,L3DM1,L4DM1,L5Classification resultsD10.36430.34170.46070.56900.6813L2D20.20450.20670.26930.46830.5323L1D30.28440.27420.36500.51870.6068L2D40.36170.34130.46070.56670.6813L2D50.57170.56060.65810.81100.9018L2D60.35660.34000.43310.50750.5804L2D70.63940.48720.67440.92290.8799L2D80.87030.80570.90470.97770.9591L2D90.87440.81990.89780.98640.9678L2D10.63390.48920.66280.92020.8568L2D110.74570.73340.76210.82740.8555L2D120.23050.23210.29980.45220.5370L2D130.21460.16790.24800.39120.4480L2Table 8
Diagnostic results of the FFS distance measure for Bob.
DM2,L1DM2,L2DM2,L3DM2,L4DM2,L5Classification resultsD10.46670.57830.44100.23230.5490L4D20.29650.44870.27130.11770.4027L4D30.38160.51350.35620.17500.4758L4D40.46670.57830.44100.23230.5467L4D50.69040.76480.66960.38500.7542L4D60.43750.51520.41850.24470.4940L4D70.84480.87000.73080.42780.8139L4D80.95280.96170.89800.80850.9388L4D90.93800.96100.87830.78800.9336L4D100.79280.82100.66210.34860.7230L4D110.78990.84710.78110.72010.8534L4D120.35770.49940.34560.15690.4980L4D130.28540.38700.24470.14320.3383L4Table 9
Diagnostic results of the FFS distance measure for Joe.
DM3,L1DM3,L2DM3,L3DM3,L4DM3,L5Classification resultsD10.53230.63930.54930.58430.6693L3D20.29520.41070.27600.39300.5230L3D30.41370.52500.41270.48870.5962L3D40.53130.63870.54630.58170.6667L1D50.77850.90210.77130.79390.9256L3D60.48350.55410.49370.51740.5714L1D70.79480.77380.68570.81500.8678L3D80.92800.91410.90850.93260.9549L3D90.94330.93370.90700.96850.9673L3D100.79150.77880.67390.82980.8467L3D110.77470.79540.76570.80890.8521L3D120.31060.40960.30540.39030.5265L3D130.26320.33020.25120.34330.4011L3Table 10
Diagnostic results of the FFS distance measure for Ted.
DM4,L1DM4,L2DM4,L3DM4,L4DM4,L5Classification resultsD10.36270.46670.51970.51930.7987L1D20.20300.26180.31680.36520.5255L1D30.28280.36430.41830.44230.6621L1D40.36230.46670.51970.51930.7987L1D50.58370.64090.74060.77020.0791L1D60.35720.43750.47540.47510.6479L1D70.77440.60620.84800.91950.9493L2D80.92950.85420.95510.97980.9825L2D90.92860.91730.96350.98240.9839L2D100.77360.67400.86100.92060.9454L2D110.73820.74720.77420.79010.8373L1D120.22030.27690.33400.36000.5182L1D130.16520.19030.23430.27740.3843L1Table 11
Diagnosis results of different methods in Example4 (summary of the comparison).
AlBobJoeTedWei et al. [18]MalSPTYPVFXiao and Ding [35]MalSPTYPVFZhou et al. [36]MalSPTYPVFEuclidean distanceMalSPTYPVFDeng and Wang [37]MalSPTYPVFDeng and Wang [37]MalSPTYPVFPresent————D1MalSPTYPVFD2VFSPTYPVFD3MalSPTYPVFD4MalSPVFVFD5MalSPTYPVFD6MalSPVFVFD7MalSPTYPVFD8MalSPTYPVFD9MalSPTYPVFD10MalSPTYPVFD11MalSPTYPVFD12MalSPTYPVFD13MalSPTYPVF


## 4.1. Comparison of the Distance Measure between FFSs in Medical Diagnosis

To illustrate the effectiveness of the novel distance measure for specific FFSs in pattern recognition, we present a numerical example and compare the novel findings to those reported in the literature.Example 4.
Consider a set of four patients, Al, Bob, Joe, and Ted, as represented byM=M1,M2,M3,M4 who have the symptoms temperature, headache, stomach pain, cough, and chest pain, which are represented by S=ϰ1,ϰ2,ϰ3,ϰ4,ϰ5. Let L=L1,L2,L3,L4,L5 be a list of possible diagnoses. Possible disease are defined as follows: L1: viral fever, L2: malaria, L3: typhoid, L4: stomach problem, and L5: chest problem. The FF relation M⟶S is illustrated by FFS, as can be seen in Table 5. The FF relation S⟶L is denoted by the FFS, as seen in Table 6. Each element in Table 6 is represented by FFS. The established distance measure methods are used to determine the distance between each patient and each diagnosis. Then, using the idea of minimal distance degree, each patient was diagnosed. We demonstrated the distance measure results of the patient Mjj=1,2,3,4 with regard to the diagnostic Lii=1,2,3,4,5 and the final diagnosis findings are given in Al has malaria Table 7, Bob has stomach problem Table 8, Joe has typhoid Table 9, and Ted has viral fever Table 10. We perform a comparison study with other methodologies to demonstrate the capability and validity of the presented distance measures, and the findings are provided in Table 11. Table 11 shows that the suggested distance measure approaches achieve the same result as in D [18], D [35], D [36], and D [37], demonstrating that using the proposed distance measure methods to solve the medical diagnosis problem is possible and beneficial. From the preceding practical implementation of the measures techniques, we may deduce that the proposed distance measures approaches are more effective and superior in handling real world challenges.Table 5
Symptomatic characteristics of the patient.
ϰ1ϰ2ϰ3ϰ4ϰ5M10.80,0.100.60,0.100.20,0.800.60,0.100.10,0.60M20.00,0.800.40,0.400.60,0.100.10,0.700.10,0.80M30.80,0.100.80,0.100.00,0.600.20,0.700.00,0.50M40.60,0.100.50,0.400.30,0.400.70,0.200.30,0.40Table 6
Symptomatic characteristics of the diagnosis.
ϰ1ϰ2ϰ3ϰ4ϰ5L10.40,0.000.30,0.500.10,0.700.40,0.300.10,0.70L20.70,0.000.20,0.600.00,0.900.70,0.000.10,0.80L30.30,0.300.60,0.100.20,0.700.20,0.600.10,0.90L40.10,0.700.20,0.400.80,0.000.20,0.700.20,0.70L50.10,0.800.00,0.800.20,0.800.20,0.800.80,0.10Table 7
Diagnostic results of the FFS distance measure for Al.
DM1,L1DM1,L2DM1,L3DM1,L4DM1,L5Classification resultsD10.36430.34170.46070.56900.6813L2D20.20450.20670.26930.46830.5323L1D30.28440.27420.36500.51870.6068L2D40.36170.34130.46070.56670.6813L2D50.57170.56060.65810.81100.9018L2D60.35660.34000.43310.50750.5804L2D70.63940.48720.67440.92290.8799L2D80.87030.80570.90470.97770.9591L2D90.87440.81990.89780.98640.9678L2D10.63390.48920.66280.92020.8568L2D110.74570.73340.76210.82740.8555L2D120.23050.23210.29980.45220.5370L2D130.21460.16790.24800.39120.4480L2Table 8
Diagnostic results of the FFS distance measure for Bob.
DM2,L1DM2,L2DM2,L3DM2,L4DM2,L5Classification resultsD10.46670.57830.44100.23230.5490L4D20.29650.44870.27130.11770.4027L4D30.38160.51350.35620.17500.4758L4D40.46670.57830.44100.23230.5467L4D50.69040.76480.66960.38500.7542L4D60.43750.51520.41850.24470.4940L4D70.84480.87000.73080.42780.8139L4D80.95280.96170.89800.80850.9388L4D90.93800.96100.87830.78800.9336L4D100.79280.82100.66210.34860.7230L4D110.78990.84710.78110.72010.8534L4D120.35770.49940.34560.15690.4980L4D130.28540.38700.24470.14320.3383L4Table 9
Diagnostic results of the FFS distance measure for Joe.
DM3,L1DM3,L2DM3,L3DM3,L4DM3,L5Classification resultsD10.53230.63930.54930.58430.6693L3D20.29520.41070.27600.39300.5230L3D30.41370.52500.41270.48870.5962L3D40.53130.63870.54630.58170.6667L1D50.77850.90210.77130.79390.9256L3D60.48350.55410.49370.51740.5714L1D70.79480.77380.68570.81500.8678L3D80.92800.91410.90850.93260.9549L3D90.94330.93370.90700.96850.9673L3D100.79150.77880.67390.82980.8467L3D110.77470.79540.76570.80890.8521L3D120.31060.40960.30540.39030.5265L3D130.26320.33020.25120.34330.4011L3Table 10
Diagnostic results of the FFS distance measure for Ted.
DM4,L1DM4,L2DM4,L3DM4,L4DM4,L5Classification resultsD10.36270.46670.51970.51930.7987L1D20.20300.26180.31680.36520.5255L1D30.28280.36430.41830.44230.6621L1D40.36230.46670.51970.51930.7987L1D50.58370.64090.74060.77020.0791L1D60.35720.43750.47540.47510.6479L1D70.77440.60620.84800.91950.9493L2D80.92950.85420.95510.97980.9825L2D90.92860.91730.96350.98240.9839L2D100.77360.67400.86100.92060.9454L2D110.73820.74720.77420.79010.8373L1D120.22030.27690.33400.36000.5182L1D130.16520.19030.23430.27740.3843L1Table 11
Diagnosis results of different methods in Example4 (summary of the comparison).
AlBobJoeTedWei et al. [18]MalSPTYPVFXiao and Ding [35]MalSPTYPVFZhou et al. [36]MalSPTYPVFEuclidean distanceMalSPTYPVFDeng and Wang [37]MalSPTYPVFDeng and Wang [37]MalSPTYPVFPresent————D1MalSPTYPVFD2VFSPTYPVFD3MalSPTYPVFD4MalSPVFVFD5MalSPTYPVFD6MalSPVFVFD7MalSPTYPVFD8MalSPTYPVFD9MalSPTYPVFD10MalSPTYPVFD11MalSPTYPVFD12MalSPTYPVFD13MalSPTYPVF


## 5. Apply the Similarity Measure between FFSs to Pattern Recognition

In this part, we describe some examples to show the use of the suggested similarity measures based on FFS to pattern recognition.Example 5.
Suppose the four classesM1,M2,M3, and M4 of known construction materials and L, an unknown construction material, are defined in the space X=ϰ1,ϰ2,ϰ3 and are represented by FFS is given. Its goal is to ascertain to which class L belongs to (see Table 12).
Here,L is a known building materials. Its objective is to determine the class to which L belongs. To do this, the degrees of similarity between L and classes M1, M2, M3, and M4 are measured, and L is then allocated to the class Mg specified as follows:(17)g=argmaxgSMg,L.
For all the established similarity measureS1−S13 for FFS, the degree of similarity between the four classes of known building materials SM1,L,SM2,L,SM3,L, and SM4,L are determined and displayed in Table 13. It is clearly observed in the Table 13, that an unknown building material L belongs to a class M1 when S1,S3,S7, to S10 are used and L belongs to a class M3 when S2, S4, to S6 and S11 to S13 are used. It is clear that the cause for this difference is the first feature, i.e., (ϰ1). The FFNs of ϰ1 are 0.21,0.33, 0.22,0.41, 0.11,0.11,0.33,0.11, and 0.25,0.22 for M1, M2, M3,M4, and L, respectively. It is predicted that the similarity degree between 0.21,0.33 and 0.25,0.22 is larger than the similarity degree between 0.11,0.11 and 0.25,0.22 is larger than the similarity degree between 0.33,0.11 and 0.25,0.22 is larger than 0.22,0.41 and 0.25,0.22. As a conclusion, it appears that SM1,L>SM3,L>S ( M4,L>SM2,L is more acceptable. Similarly, we can find the previously mentioned relations for S2 to S13.Table 12
Classes information with FFSs.
ϰ1ϰ2ϰ3M10.21,0.330.22,0.430.32,0.32M20.22,0.410.32,0.310.13,0.34M30.11,0.110.12,0.210.32,0.32M40.33,0.110.21,0.220.23,0.32L0.25,0.220.42,0.340.33,0.24Table 13
Similarity measure for Example5.
SM1,LSM2,LSM3,LSM4,LClassification resultsS10.96410.95240.95170.9541M1S20.97380.97430.98840.9819M3S30.96890.96330.97010.9680M1S40.97310.96670.97360.9728M3S50.16360.16290.16370.1636M3S60.49090.48860.49100.4908M3S70.42480.36540.29010.2829M1S80.12610.11550.08220.0888M1S90.12810.13120.08240.0816M2S100.42520.36560.28910.2749M1S110.31610.31460.31720.3155M3S120.94860.94360.95220.9469M3S130.97600.97760.99070.9848M3


## 6. A Comparison of the Proposed Similarity Measures between FFSs

To illustrate the effectiveness of the novel similarity measures for specific FFSs in pattern recognition, we present some examples and compare the novel findings to those reported in the literature.Example 6.
Comparison analysis of similarity measure for three known patternsM1,M2, and M3 that are presented by the following FFSs in X=ϰ1,ϰ2,ϰ3,ϰ4:M1=ϰ1,0.3,0.3,ϰ2,0.4,0.4,ϰ3,0.4,0.4,ϰ4,0.4,0.4,M2=ϰ1,0.5,0.5,ϰ2,0.1,0.1,ϰ3,0.5,0.5,ϰ4,0.1,0.1,M3=ϰ1,0.5,0.4,ϰ2,0.4,0.5,ϰ3,0.3,0.3,ϰ4,0.2,0.2.
The following is an unknown patternL: L=ϰ1,0.4,0.4,ϰ2,0.5,0.5,ϰ3,0.2,0.2,ϰ4,0.3,0.3.
Our objective is to ascertain the class to whichL belongs. The classification result of the suggested similarity measures S1−S13 displayed in Table 14 is contrasted to the classification result of the existing similarity measures S [38]-S [39]) depicted in Table 15. From Table 14, we observed that the developed similarity measures S1,S3−S13 addressing the shortcomings of conventional similarity measures S [19], S [27], S [30], S [44], S [32], and S [45].Table 14
Comparison for the proposed similarity measure for Example6.
SM1,LSM2,LSM3,LClassification resultS10.89730.79870.9467M3S21.00001.00000.9797Cannot be classifiedS30.94870.89930.9632M3S40.96150.92450.9648M3S50.16230.15790.1627M3S60.48680.47350.4880M3S70.39130.19470.7015M3S80.14330.06490.2470M3S90.14330.06490.2383M3S100.39130.19470.6755M3S110.30070.27200.3160M3S120.90230.81710.9481M3S130.98280.97210.9760M1Table 15
Comparison for similarity measures for Example6.
SM1,LSM2,LSM3,LClassification resultsS [38]0.86770.72610.9134M3S [40]1.00001.00000.9750Cannot be classifiedS [41]0.86790.74250.8923M3S [10]0.87500.75000.9000M3S [10]0.81410.65010.8495M3S [10]0.77780.60000.8182M3S [42]0.87500.75000.9250M3S [19]1.00001.00000.9750Cannot be classifiedS [43]0.93750.87500.9500M3S [13]0.87500.75000.9250M3S [13]0.93750.87500.9500M3S [13]0.91670.83330.9417M3S [12]0.87500.75000.9250M3S [27]1.00001.00000.9969Cannot be classifiedS [30]1.00001.00000.9884Cannot be classifiedS [44]0.50000.50000.5000Cannot be classifiedS [32]1.00001.00000.9775Cannot be classifiedS [32]0.50380.23780.6223M3S [32]0.87850.79120.9205M3S [45]0.95830.91670.9583Cannot be classifiedS [39]0.98410.97270.9816M1Example 7.
Assume that a doctor would like to diagnose the condition ofC:  (viral fever, malaria, or typhoid) for a set of patients P:  (Al, Bob, Joe, and Ted) having symptoms V:  (temperature, headache, and cough). The symptoms associated with the considered diagnosis are listed in Table 16, and the symptoms associated with each patient are listed in Table 17. Each table element is represented by a specific FFSs. Each patient requires proper diagnosis, which need be assessed. We will identify a diagnosis for each patient based on the similarity between the symptoms associated with each diagnosis and those associated with the patient. The diagnostic observations are described in Table 18 Al, Table 19 Bob, Table 20 Joe, and Table 21 Ted, respectively, using the novel similarity measures formula S1−S13. The patient Al is diagnosed with malaria (Mal.) in 12 of the 13 of the approaches; the remaining approach indicates that Al is diagnosed with viral fever (VF) as presented in Table 18. It is obvious that Bob has a stomach problem (SP), since all of the measures yield the same findings as shown in Table 19. Joe is diagnosed with typhoid in 12 of the 13 methods; the other approach represented that Joe is diagnosed with VF as shown in Table 20. Similarly, 9 of the 13 measures indicated that Ted has VF, whereas, the remaining methods imply that Ted has Mal as presented in Table 21. For patient Al, it could be observed from Table 18 and Table 22 that the established similarity measures S1,S3−S11 yield the same findings as those in S [41], S [10], S [49], S [13], S [12], S [44], S [32], S [48], and S [18], and the measures S2,S12,S13 provide the same results as in S [38], S [40], S [42], S [19], S [13], S [27], S [30], S [32], S [45], S [39], S [46], S [48], S [47], and S [50]. For patient Bob the novel similarity measures provided the same results as in the literature presented in Table 19 and Table 22. Similarly, for patient Joe the proposed similarity measure provided the same result as in the literature shown in Table 20 except S4. For patient Ted, the suggested similarity measures S1−S6,S11−S13 yield the same findings as in S [38], S [40], S [10], S [42], S [19], S [43], S [13], S [12], S [27], S [30], S [32], S [45], S [39], S [18], S [47], S [48], and S [50], and the measures S7–S10 provide the results as in S [41], S [44], S [32], S [46], and S [49] figure out in Tables 21 and 22. Table 23 shows the present summary of medical diagnosis.Table 16
Symptom features for the diagnosis under consideration.
TemperatureHeadacheStomach painCoughChest painViral fever0.4,0.00.3,0.50.1,0.70.4,0.30.1,0.7Malaria0.7,0.00.2,0.60.0,0.90.7,0.00.1,0.8Typhoid0.3,0.30.6,0.10.2,0.70.2,0.60.1,0.9Stomach problem0.1,0.70.2,0.40.8,0.00.2,0.70.2,0.7Chest problem0.1,0.80.0,0.80.2,0.80.2,0.80.8,0.1Table 17
Symptoms of the patients under examination.
TemperatureHeadacheStomach painCoughChest painAl0.8,0.10.6,0.10.2,0.80.6,0.10.1, 0.6Bob0.0,0.80.4,0.40.6,0.10.1,0.70.1, 0.8Joe0.8,0.10.8,0.10.0,0.60.2,0.70.0, 0.5Ted0.6,0.10.5,0.40.3,0.40.7,0.20.3, 0.4Table 18
Symptoms features of the patient (Al).
VFMalTypSPChest problemS10.64880.69840.72030.50140.4562S20.79550.79330.73070.53170.4677S30.72210.74590.72550.51650.4619S40.72880.74400.65450.57500.4890S50.14720.14780.14240.13430.1272S60.43910.44300.41980.39760.3715S70.36060.51280.32560.07710.1201S80.12970.19430.09530.02230.0409S90.12560.18010.10220.01360.0322S100.36610.51080.33720.07980.1432S110.25430.26660.23790.17260.1445S120.76950.76790.70020.54780.4630S130.84370.76230.75570.60000.5030Table 19
Symptoms features of the patient (Bob).
VFMalTypSPChest problemS10.55820.45090.71430.86660.5881S20.70350.55130.72870.88230.5973S30.63090.50110.72150.87450.5927S40.65000.56620.66930.82570.5900S50.14110.13640.14240.15470.1370S60.41860.39500.42380.46250.4019S70.15520.13000.26920.57220.1861S80.04720.03830.10200.19150.0612S90.06200.03900.12170.21200.0664S100.20720.17900.33790.65140.2770S110.21010.15290.21890.27990.1466S120.64230.50060.65440.84310.5020S130.77190.58220.75060.87110.5903Table 20
Symptoms features of the patient (Joe).
VFMalTypSPChest problemS10.56160.49170.70860.58130.4720S20.70480.58930.72400.60700.4770S30.63320.54050.71630.59420.4745S40.60150.52100.59020.56370.5000S50.13630.12840.13660.13460.1266S60.40520.38150.40200.39430.3750S70.20520.22620.31430.18500.1322S80.07200.08590.09150.06740.0451S90.05670.06630.09300.03150.0327S100.20850.22120.32610.17020.1533S110.22530.20460.23430.19110.1479S120.68940.59040.69460.60970.4735S130.76130.62630.77060.64930.5101Table 21
Symptoms features of the patient (Ted).
VFMalTypSPChest problemS10.65790.64470.67270.60800.4644S20.79700.73820.68320.63480.4745S30.72740.69140.67790.62140.4694S40.72830.65000.61030.61050.4010S50.14670.14330.13850.13690.1142S60.43900.41860.40770.40770.3425S70.22560.39380.15200.08050.0507S80.07050.14580.04490.02020.0175S90.07140.08270.03650.01760.0161S100.22640.32600.13900.07940.0546S110.26180.25280.22580.20990.1627S120.77970.72310.66600.64000.4818S130.82680.70690.70620.66880.5084Table 22
The summary of existing similarity measures in medical diagnosis.
AlBobJoeTedS [38]VFSPTYPVFS [40]VFSPTYPVFS [41]MalSPTYPMalS [10]MalSPTYPVFS [10]MalSPTYPVFS [10]MalSPTYPVFS [42]VFSPTYPVFS [19]VFSPTYPVFS [43]VF/MalSPTYPVFS [13]MalSPTYPVFS [13]VFSPTYPVFS [13]MalSPTYPVFS [12]MalSPTYPVFS [27]VFSPTYPVFS [30]VFSPTYPVFS [44]MalSPTYPMalS [32]MalSPTYPVFS [32]MalSPTYPMalS [32]VFSPTYPVFS [45]VFSPTYPVFS [39]VFSPTYPVFS [18]MalSPTYPVFS [46]VFSPTYPMalS [47]VFSPTYPVFS [48]VFSPTYPVFS [48]VFSPTYPVFS [48]MalSPSPVFS [49]MalSPTYPMalS [50]VFSPTYPVFTable 23
The summary of present similarity measures in medical diagnosis.
AlBobJoeTedS1TYPSPTYPTYPS2VFSPTYPVFS3MalSPTYPVFS4MalSPVFVFS5MalSPTYPVFS6MalSPVFVFS7MalSPTYPMalS8MalSPTYPMalS9MalSPTYPMalS10MalSPTYPMalS11MalSPTYPVFS12VFSPTYPVFS13VFSPTYPVF


## 7. Application of the Inclusion Measure between FFSs and Pattern Recognition

This section illustrates the applicability of the suggested FFS inclusion measures to pattern recognition.Example 8.
Let,M1,M2,M3, and M4 are the known patterns illustrated by FFSs in X=ϰ1,ϰ2,ϰ3 described as follows:M1=ϰ1,0.21,0.19,ϰ2,0.22,0.23,ϰ3,0.32,0.27,M2=ϰ1,0.22,0.12,ϰ2,0.24,0.21,ϰ3,0.27,0.25,M3=ϰ1,0.12,0.21,ϰ2,0.14,0.30,ϰ3,0.21,0.32,M4=ϰ1,0.10,0.22,ϰ2,0.11,0.31,ϰ3,0.13,0.33.
The following is an unknown patternL: L=ϰ1,0.15,0.21,ϰ2,0.19,0.27,andϰ3,0.32,0.28. Its purpose is to determine which class L belongs to. To do this, the inclusion degrees between L and classes M1, M2, M3, and M4 are measured, and L is then allocated to the class Mg specified as follows:(18)g=argmaxgIMg,L.
For all the established inclusion measuresI1−I7 for FFS, the degree of inclusion between IM1,L, IM2,L, IM3,L, and IM4,L are determined and displayed in Table 24. It is clearly observed in the Table 24 that an unknown pattern L belongs to a class M4 when I1 to I3 and I5 to I6 are used, and L belongs to a class M3 and M2 when I4 and I7 are, respectively, used. It is clear that the cause for this difference is the first characteristic, i.e., ϰ1. The FFNs of ϰ1 are 0.21,0.19, 0.22,0.12, 0.12,0.21,0.10,0.22, and 0.15,0.21 for M1, M2, M3,M4, and L, respectively. It is predicted that the inclusion degree between 0.10,0.22 and 0.15,0.21 is larger than the inclusion degree between 0.12,0.21 and 0.15,0.21 is larger than the inclusion degree between 0.21,0.19 and 0.15,0.21 is larger than 0.22,0.12 and 0.15,0.21. As a conclusion, it appears that IM1,L>IM3,L>IM4,L>IM2,L is more acceptable. In a similar way, we can find the previously mentioned relations for I2 to I7.Table 24
Inclusion measure for Example8.
IM1,LIM2,LIM3,LIM4,LClassification resultsI10.99640.99360.99700.9997M4I20.99270.98720.99390.9995M4I30.99270.98730.99400.9995M4I40.97320.97870.99060.9852M3I50.33090.32910.33130.3332M4I60.33090.32910.33130.3332M4I70.33560.33690.33310.3321M2


## 8. Conclusions and Future Recommendations

### 8.1. Conclusions

The main findings of this research are emphasized and encapsulated as follows:(1)
We developed axiomatically FFSs information measures (distance measure, similarity measure, entropy, and inclusion measure).(2)
We constructed various formulae for FFSs information measures and analyzed the associated transformation relationships in detail.(3)
We used the established distance measures (D1-D13) to pattern recognition and medical diagnosis to demonstrate their efficacy. The applications substantiate the results and also illustrate the feasibility and effectiveness of the distance measures between FFSs information.(4)
We demonstrated the efficacy of the novel similarity measures (S1-S13); several counterintuitive examples of existing similarity measures are shown. We employed them to pattern recognition, construction materials, and medical diagnosis. For pattern recognition problems, we conclude that the proposed similarity measures dominate existing similarity measures. In some special situations, it has been shown that many conventional similarity measures are incapable of providing reasonable findings. However, in these specific cases, the proposed similarity measure is proficient of discriminating FFSs. A comparison of the proposed similarity measures with conventional similarity measures is performed for medical diagnosis problems. The applications emphasise the results and also illustrate the significance and reliability of the established similarity measures.(5)
Additionally, we illustrated the applicability of the suggested FFS inclusion measures to pattern recognition with an example. The findings demonstrate the feasibility and effectiveness of new inclusion measures.The experimental findings demonstrated that the proposed measures are more reliable and can avoid the counter-intuitive situation in dealing with practical applications based on Fermatean fuzzy environment. [24, 51, 52].


### 8.2. Limitations and Future Works

(1)
The FFSs are inappropriate to deal with situations where the cube sum of membership and nonmembership grades of exceeds 1(2)
A near future target is to unfold the application of the proposed information measures in scientific investigations for decision-making, pattern recognition, linguistic summarization, and data mining(3)
We have also a plan to apply the presented approach to procurement planning, water desalination station selection, wind power plant site selection, and many more domains of real world problems(4)
Additionally, we will be further interested to immerse them in a variety of fuzzy environments(5)
Furthermore, since this work presents an applicative analysis of the FFS information measures, we should develop an appropriate software to effectively apply the presented information measures in a realistic situation


## 8.1. Conclusions

The main findings of this research are emphasized and encapsulated as follows:(1)
We developed axiomatically FFSs information measures (distance measure, similarity measure, entropy, and inclusion measure).(2)
We constructed various formulae for FFSs information measures and analyzed the associated transformation relationships in detail.(3)
We used the established distance measures (D1-D13) to pattern recognition and medical diagnosis to demonstrate their efficacy. The applications substantiate the results and also illustrate the feasibility and effectiveness of the distance measures between FFSs information.(4)
We demonstrated the efficacy of the novel similarity measures (S1-S13); several counterintuitive examples of existing similarity measures are shown. We employed them to pattern recognition, construction materials, and medical diagnosis. For pattern recognition problems, we conclude that the proposed similarity measures dominate existing similarity measures. In some special situations, it has been shown that many conventional similarity measures are incapable of providing reasonable findings. However, in these specific cases, the proposed similarity measure is proficient of discriminating FFSs. A comparison of the proposed similarity measures with conventional similarity measures is performed for medical diagnosis problems. The applications emphasise the results and also illustrate the significance and reliability of the established similarity measures.(5)
Additionally, we illustrated the applicability of the suggested FFS inclusion measures to pattern recognition with an example. The findings demonstrate the feasibility and effectiveness of new inclusion measures.The experimental findings demonstrated that the proposed measures are more reliable and can avoid the counter-intuitive situation in dealing with practical applications based on Fermatean fuzzy environment. [24, 51, 52].


## 8.2. Limitations and Future Works

(1)
The FFSs are inappropriate to deal with situations where the cube sum of membership and nonmembership grades of exceeds 1(2)
A near future target is to unfold the application of the proposed information measures in scientific investigations for decision-making, pattern recognition, linguistic summarization, and data mining(3)
We have also a plan to apply the presented approach to procurement planning, water desalination station selection, wind power plant site selection, and many more domains of real world problems(4)
Additionally, we will be further interested to immerse them in a variety of fuzzy environments(5)
Furthermore, since this work presents an applicative analysis of the FFS information measures, we should develop an appropriate software to effectively apply the presented information measures in a realistic situation



---

*Source: 9273239-2023-03-08.xml*