Research ArticleOn Structural Properties of ξ-Complex Fuzzy Sets andTheir Applications

Aneeza Imtiaz1 Umer Shuaib 1 Hanan Alolaiyan 2 Abdul Razaq3

and Muhammad Gulistan 4

1Department of Mathematics Government College University Faisalabad 38000 Pakistan2Department of Mathematics King Saud University Riyadh Saudi Arabia3Department of Mathematics Division of Science and Technology University of Education Lahore 54000 Pakistan4Department of Mathematics and Statistics Hazara University Mansehra Mansehra KPK Pakistan

Correspondence should be addressed to Hanan Alolaiyan holayanksuedusa

Received 30 March 2020 Revised 6 June 2020 Accepted 9 July 2020 Published 3 December 2020

Academic Editor Yan-Ling Wei

Copyright copy 2020 Aneeza Imtiaz et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Complex fuzzy sets are the novel extension of Zadehrsquos fuzzy sets In this paper we comprise the introduction to the concept ofξ-complex fuzzy sets and proofs of their various set theoretical properties We define the notion of (α δ)-cut sets of ξ-complexfuzzy sets and justify the representation of an ξ-complex fuzzy set as a union of nested intervals of these cut sets We also apply thisnewly defined concept to a physical situation in which one may judge the performance of the participants in a given task Inaddition we innovate the phenomena of ξ-complex fuzzy subgroups and investigate some of their fundamental algebraic at-tributes Moreover we utilize this notion to define level subgroups of these groups and prove the necessary and sufficientcondition under which an ξ-complex fuzzy set is ξ-complex fuzzy subgroup Furthermore we extend the idea of ξ-complex fuzzynormal subgroup to define the quotient group of a group G by this particular ξ-complex fuzzy normal subgroup and establish anisomorphism between this quotient group and a quotient group of G by a specific normal subgroup GAξ

1 Introduction

e competency of fuzzy logic to articulate steady adapta-tions from membership to nonmembership and the otherway around has played an effective role to solve manyphysical problems It provides us not only the powerful andmeaningful representations of measuring uncertainty butalso a useful approach to view the vague concepts expressedin the natural language Despite all of the advantages of thislogic we still face immense complications to counter variousphysical situations based on a real-valued membershipfunction It is therefore quite necessary to propose an ad-ditional development of fuzzy set theory on account of set ofcomplex numbers which is indeed an existing augmentationof real numbers e complex fuzzy logic is linear aug-mentation of traditional fuzzy logic which also allows anatural development of the difficulty based on the fuzzy logicthat is impracticable to solve with superficial membership

function is particular set has a paramount part in nu-merous executions in particular advanced control systemsand predicting of the periodic events where multiple fuzzyvariables are interrelated in a complex manner that cannotbe effectively characterized by simple fuzzy operations Inaddition these sets are also used to solve several problemsespecially the numerous periodic aspects and forecastproblems One of the far-reaching significances of studyingthe CFS is that they illustrate the data with uncertainty andperiodicity in a much effective way

Considering the inaccuracy in decision-making Zadeh[1] popularized the concept of fuzzy sets for the first time in1965 Roenfeld [2] used Zadehrsquos work to invent the approachof fuzzy subgroups in 1971 Mukherjee and Bhattacharya [3]initiated the study of the fuzzy cosets along with fuzzynormal subgroups in 1984 Mashour et al [4] studied manyimportant features of normal fuzzy subgroups For moredetails about the recent development of fuzzy subgroups we

HindawiComplexityVolume 2020 Article ID 2038724 13 pageshttpsdoiorg10115520202038724

refer to [5ndash7] In addition another important aspect of fuzzysets is the level sets of this notion ese level sets have anincredible significance as they set up a connection betweenconcepts of crisp and fuzzy setsis phenomenon is utilizedto generalize various ideas and techniques for crisp sethypothesis as well as fuzzy sets In the succession of fuz-zification of subgroups the idea of level subgroup wasinitiated by Das [8] Yuan and Li [9] studied the impact ofthese level sets in the field of intuitionistic fuzzy sets To seemore on the level sets and level subgroups we refer to[10ndash12] Buckley [13] proposed the study of complex fuzzynumbers in 1989 e author [14] applied the theory ofcomplex fuzzy numbers to set up new techniques of dif-ferentiation Moreover some fundamental properties offuzzy counter integral in the complex plane were interpretedby the same author in [15] Zhang [16] established manyimportant properties of complex fuzzy numbers in 1992Ascia et al [17] designed a competent specific fuzzy pro-cessor which effectively deals with the complex fuzzy in-ference system In [18] Ramote et al launched the study ofCFS in 2002 e two novel operations namely reflectionand rotations of these sets were introduced in [19] Zhanget al [20] formulated the δ-equalities of CFS In 2011 Chenet al [21] developed the adaptive neuro complex fuzzy in-ference system In [22] Fu and Shen utilized CFS to design anew approach of linguistic evaluation of classifier perfor-mance Tamir and Kandel [23] provided granted bases forfirst order estimation complex class theory and complexfuzzy logic in the same year Ma et al [24] defined productsum accruement operator for these particular sets in 2012 Liet al [25] presented a new self-learning complex neuro fuzzysystem using the same idea In 2014 Alkouri and Salleh [26]illustrated many important multiple distance measuresdefined on CFS In 2015 Tamir et al [27] reintroduced theconcept of CFS and complex fuzzy logic in a more logicalway Al-Husban and Salleh [28] interpreted the study ofcomplex fuzzy hyper structure over complex fuzzy space in2016 e phenomenon of CFSG over complex fuzzy spacewas discussed in [29] e operator theory has an importantrole in modern mathematics because it is extensively used infuzzy theory for approximate reasoning e techniques oft-norms and t-co-norms with respect to complex fuzzy setswere presented by Nagarajan et al [30] For more on fuzzyset theory we suggest reading of [31ndash35]e competency ofthe complex fuzzy set has played an effective role to solvemany physical problems It provides us the meaningfulrepresentations of measuring uncertainty and periodicityDespite all of these advantages we still face vast compli-cations to counter various physical situations based on acomplex-valued membership function is motivates us todefine the notion of ξ-complex fuzzy set (ξ-CFS) throughwhich one can have multiple options to investigate a specificreal-world situation in much efficient way by choosingappropriate value of the parameter ξ

In this article we propose the idea of ξ-CFS as a powerfulextension of classical fuzzy set and use this idea to define thenotion of the cut sets of these particular sets We explore theimportance of these cut sets by proving the decompositiontheorems for a ξ-CFS Moreover we introduce the

phenomenon of ξ-complex fuzzy subgroup (ξ-CFSG) overan ξ-complex fuzzy set and investigate some of their fun-damental algebraic attributes

After a brief discussion about the historical backgroundand significance of CFS the rest of the article is organized asfollows Section 2 contains some definitions of the basicintroduction of the concept of ξ-CFS and the study of(α δ)-cut sets and strong (α δ)-cut sets of this newly definednotion In addition we establish the importance of definingthese ideas by viewing an ξ-CFS as a union of nested se-quence of both (α δ)-cut sets and strong (α δ)-cut setsrespectively Moreover we apply ξ-CFS to a physical situ-ation in which one may select the most suitable performanceof a participant In Section 4 we utilize the concept of ξ-CFSto propose the study of the idea of ξ-CFSG and prove thateach CFSG is ξ-CFSG Moreover we extend the importanceof these fuzzy subgroups by introducing the notions ofξ-complex fuzzy cosets and ξ-complex fuzzy normal sub-group (ξ-CFNSG) and investigate their many importantalgebraic aspects

2 Preliminaries

is section contains a brief review of the notion of CFS andrelated ideas which are quite essential to understand thenovelty of this article

Definition 1 (see [19]) A CFS A defined on a universe ofdiscourse U is characterized by a membership functionμA(m) that allocates each element of Uto a unit circle Clowast incomplex plane and is written as rA(m)eiωA(m) where rA(m)

denotes the real-valued function from U to the closed unitinterval and eiωA(m) is a periodic function whose periodic lawand principal period are 2π and 0lt argA(m)le 2πrespectively

Note that ωA(m) argA(m) + 2kπ k isin Z and argA(m)

is the principal argument

Definition 2 (see [8]) Let 0le αle 1 and 0le δ le 2π en the(α δ)-cut set of CFS A is denoted by A(αδ) and is defined asA(αδ) m isin U rA(m)ge αωA(m)ge δ1113864 1113865

Definition 3 (see [31]) Let A and B be any two CFS of auniverse U en

(1) A is homogeneous CFS if rA(m)le rA(n) impliesωA(m)leωA(n) and vice versa forallm n isin U

(2) A is homogeneous CFS with B if rA(m)le rB(n)

implies ωA(m)leωB(n) and vice versa forallm n isin U

Definition 4 (see [31]) A homogeneous CFS A of G is said tobe a CFSG if μA(mn)gemin μA(m) μA(n)1113864 1113865 andμA(mminus 1)ge μA(m) forallm n isin G

Definition 5 (see [5]) Let A be a CFSG (G) and m isin Genthe complex fuzzy left coset of A in G is represented bymAand is given by mA(g) μA(mminus1g) g isin G1113864 1113865

Similarly one can define complex fuzzy right coset ofAin G

2 Complexity

Definition 6 (see [5]) A CFSG A of a group Gis CFNSG (G)if mA Amforallm isin G

3 Decomposition Theorems of ξ-ComplexFuzzy Sets

In this section we initiate the idea of ξ-CFS as a powerfulextension of classical fuzzy sets We also define the conceptsof (α δ)-cut sets and strong (α δ)-cut sets of ξ-CFS andestablish fundamental properties of these phenomena Wealso prove three decomposition theorems of ξ-CFS

Definition 7 Let A be a CFS of a universe U and ξ αeiδ bean element of a unit circle with 0le αle 1 and 0le δ le 2πenthe CFS Aξ is called the ξ-complex fuzzy set (ξ-CFS) withrespect to CFS A and is expressed as μAξ(m) minμA(m) ξ1113864 1113865 forallm isin U

e family of all ξ-CFS defined on the universe U isdenoted by Fξ(U)

Definition 8 For any Aξ and Βξ isin Fξ(U)

(1) e union of ξ-CFS Aξ and Βξ is denoted by Aξ cupΒξand is defined as follows

μAξ cupΒξ(m) rAξ cupΒξ(m)eiω

Aξ cupΒξ (m)

max rAξ(m) rΒξ(m)( 1113857eimax ω

Aξ (m)ωBξ (m)( )

forallm isin U

(1)

(2) e intersection of ξ-CFS Aξ and Bξ is denoted byAξ capBξ and is defined as follows

μAξ capBξ(m) rAξ capBξ(m)eiω

Aξ capBξ (m)

min rAξ(m) rBξ(m)( 1113857eimin ω

Aξ (m)ωBξ (m)( )

forallm isin U

(2)

(3) e complement of ξ-CFS Aξ is denoted by Aξrsquo and isdescribed as follows

μAξrsquo (m) r

Aξrsquo (m)eiω

Aξrsquo (m)

1 minus rAξ(m)( 1113857ei 2πminusω

Aξ (m)( ) forallm isin U(3)

(4) e product of ξ-CFS Aξ and Bξ is represented byAξ ∘Bξ and is expressed as follows

μAξ ∘Bξ(m) rAξ ∘Bξ(m)eiω

Aξ ∘Bξ (m)

rAξ(m)rBξ(m)( 1113857ei2π ω

Aξ (m)2π( ) ωBξ (m)2π( )( )

forallm isin U

(4)

(5) Let Aξn n isin N be ξ-CFS of a universe U en

the Cartesian product of ξ-CFS Aξn is represented by

Aξ1 times A

ξ2 times middot middot middot times Aξ

n and is defined in the followingway

μAξ1timesA

ξ2timestimesA

ξn(m) r

Aξ1timesA

ξ2timestimesA

ξn(m)e

iωAξ1timesA

ξ2timestimesA

ξn

(m)

min rAξ1

m1( 1113857 rAξ2

m2( 1113857 rAξn

mn( 11138571113874 1113875eimin ω

Aξ1

m1( )ωAξ2

m2( )ωAξn

mn( )1113874 1113875

(5)

where m (m1 m2 middot middot middot mn) mi isin Ui i isin N

Definition 9 Let Aξ and Bξ be any two ξ-CFS of a universeU en

(1) Aξ is homogeneous ξ-CFS if rAξ(m)le rAξ(n) impliesωAξ(m)leωAξ(n) and vice versa forallm n isin U

(2) Aξ is homogeneous ξ-CFS with Bξ if rAξ(m)le rBξ(n)

implies ωAξ(m)leωBξ(n) and vice versa forallm n isin U

e next result illustrates that the intersection of any twoξ-CFS is also ξ-CFS

Proposition 1 For any two ξ-CFS Aξand Bξ (AcapB)ξ

Aξ capBξ

Proof Consider μ(AcapB)ξ

(m) min μAcapB(m) ξ1113864 1113865 m isin U

By applying Definition 8 (2) we have μ(AcapB)ξ

(m)

min(μAξ(m) μBξ(m)) μAξ capBξ(m)

Remark 1 e union of any two ξ-CFS is also ξ-CFS

Definition 10 e (α δ)-cut set of ξ-CFS Aξ is representedby A

ξ(αδ) and is defined as follows

Aξ(αδ) m isin U rAξ(m)ge αωAξ(m)ge δ 0le αle 1 0le δ le 2π1113864 1113865

(6)

Definition 11 For any Aξ isin Fξ(U) strong (α δ)-cut set ofAξ is defined as A

ξ+(αδ) m isin U rAξ(m)gt αωAξ1113864

(m)gt δ 0le αle 1 0le δ le 2π

Complexity 3

Definition 12 e level set ΩAξ of Aξ can be described asΩAξ m isin U rAξ(m) αωAξ(m) δ1113864 1113865 where α isin [0 1]

and δ isin [0 2π]

Theorem 1 Let Aξ and Bξ be any two ξ-CFS lten thefollowing attributes hold for any α αprime isin [0 1] andβ βprime isin [0 2π]

(1) Aξ+(αδ) subeA

ξ(αδ)

(2) αle αprime δ le δprime implies Aξ(αrsquoδrsquo) subeA

ξ(αδ)

(3) (Aξ capBξ)(αδ) Aξ(αδ) capB

ξ(αδ)

(4) (Aξ cupBξ)(αδ) Aξ(αδ) cupB

ξ(αδ)

(5) Aξrsquo(αδ) (Aξrsquo )+(1minusα2πminusδ)

Proof

(1) In view of Definition 10 for any element m isin UrAξ(m)gt α and ωAξ(m)gt δ It means that rAξ(m)ge αand ωAξ(m)ge δ us A

ξ+(αδ)sube A

ξ(αδ)

(2) Let m isin U and by applying Definition 10 we haverAξ(m)ge αωAξ(m)ge δ erefore A

ξ(αprime δprime)subeA

ξ(αδ)

(3) For any m isin (Aξ capBξ)(αδ) we have rAξ capBξ(m)ge αand ωAξ capBξ(m)ge δ is implies that min rAξ(m)1113864

rBξ(m)ge α and min ωAξ(m)ωBξ1113864 (m)ge δ enclearly rAξ(m)ge α rBξ(m)ge α and ωAξ(m)ge δωBξ

(m)ge δ erefore m isin Aξ(αδ) capB

ξ(αδ) and ulti-

mately we obtain

Aξ capB

ξ1113872 1113873

(αδ)subeAξ

(αδ) capBξ(αδ) (7)

Let m isin Aξ(αδ) capB

ξ(αδ)

By applying Definition 10 in the above relations weobtain

rAξ(m)ge αωAξ(m)ge δ rBξ(m)ge αωBξ(m)ge δ (8)

is shows that min rAξ(m) rBξ(m)1113864 1113865ge α and min ωAξ1113864

(m)ωBξ(m)ge δConsequently

Aξ(αδ) capB

ξ(αδ) sube A

ξ capBξ

1113872 1113873(αδ)

(9)

From (7) and (9) the required equality holds(4) For any m isin (Aξ cupBξ)(αδ) we obtain rAξ cupBξ(m)ge α

and ωAξ cupBξ(m)ge δ erefore max rAξ(m) rBξ1113864

(m)ge α and max ωAξ(m)ωBξ(m)1113864 1113865ge δ is meansthat rAξ(m)ge α rBξ(m)ge α and ωAξ(m)ge δ

ωBξ(m)ge δConsequently

Aξ cupB

ξ1113872 1113873

(αδ)subeAξ

(αδ) cupBξ(αδ) (10)

Now suppose that m isin Aξ(αδ) cupB

ξ(αδ) then m isin A

ξ(αδ)

or m isin Bξ(αδ) By applying Definition 10 in the above

relations we get rAξ(m)ge αωAξ(m)ge δ orrBξ(m)ge αωBξ(m)ge δ It further shows that

max rAξ(m) rBξ(m)1113864 1113865ge α and max ωAξ(m)ωBξ1113864 (m)

ge δConsequently

Aξ(αδ) cupB

ξ(αδ)sube A

ξ cupBξ

1113872 1113873(αδ)

(11)

From (10) and (11) the required equality is satisfied(5) Let m isin A

ξprime(αδ) then μ

Aξprime(m) 1 minus rAξ(m)

ei(2πminusωAξ (m)) implying that 1 minus rAξ(m)ge α 2π minus ωAξ

(m)ge δ

It follows that rAξ(m)le 1 minus α ωAξ(m)le 2π minus δ whichshows that m notin (Aξ)+(1minusα2πminusδ)

erefore m isin (Aξprime)+(1minusα2πminusδ) and hence

Aξprime(αδ)sube A

ξprime1113874 1113875

+(1minusα2πminusδ) (12)

Now suppose m isin (Aξprime)+(1minusα2πminusδ) then m notin (Aξ)+

(1 minus α 2π minus δ)is implies that 1 minus αge rAξ(m) ωAξ(m)le 2π minus δ

αle 1 minus rAξ(m) and δ le 2π minus ωAξ(m)It means that m isin A

ξprime(αδ) therefore

Aξprime

1113874 1113875+(1minusα2πminusδ)

subeAξprime(αδ) (13)

From (12) and (13) the required result is satisfied

Theorem 2 Let Aξ and Bξ be any two ξ-CFS lten thefollowing attributes hold for all α αprime isin [0 1] andδ δprime isin [0 2π]

(1) αle αprime δ le δprime implies Aξ+(αprime δprime)subeA

ξ+(αδ)

(2) (Aξ capBξ)+(αδ) Aξ+(αδ) capB

ξ+(αδ)

(3) (Aξ cupBξ)(αδ) Aξ(αδ) cupB

ξ(αδ)

Theorem 3 lte following properties hold for any family ofξ-CFS A

ξi i isin I

(1) cup iisinI(Aξi )(αδ)subecup iisinI(A

ξi )(αδ)

(2) cap iisinI(Aξi )(αδ) cap iisinI(A

ξi )(αδ)

Proof

(1) Let m isin cup iisinI(Aξi )(αδ) in view of Definition 10 we

get rAξio

(m)ge α and ωAξio

(m)ge δe above relation holds only if Supr

Aξi

(m)ge α andSupω

Aξi

(m)ge δ that is cup iisinI(rξi )(m)ge α and

cup iisinI(ωξi )(m)ge δ

It follows that m isin cup iisinI(Aξi )(αδ)

(2) Let m isin cap iisinI(Aξi )(αδ) by using Definition 10 we

obtain rAξio

(m)ge α ωAξio

(m)ge δ

e above inequality holds only if infrAξi

(m)ge αinfω

Aξi

(m)ge δ that is cap iisinI(rξi )(m)ge α and cap iisinI(ω

ξi )(m)

ge δ is implies that m isin cap iisinI(Aξi )(αδ) Hence cap iisinI

(Aξi )(αδ)subecap iisinI(A

ξi )(αδ) Now suppose that m isin cap iisinI

(Aξi )(αδ) Again by applying Definition 10 we get

4 Complexity

rAξi

(m)ge α ωAξi

(m)ge δ en obviously m isin cap iisinI(Aξi )(αδ)

Consequently cap iisinI(Aξi )(αδ)subecap iisinI(A

ξi )(αδ)

Theorem 4 Any family of ξ-CFS Aξi i isin I admits the fol-

lowing properties

(1) cup iisinI(Aξi )+(αδ) cup iisinI(A

ξi )+(αδ)

(2) cup iisinI(Aξi )+(αδ)subecup iisinI(A

ξi )+(αδ)

Theorem 5 Any two ξ-CFS Aξ and Bξ satisfy the followingrelations

(1) AξsubeBξ if and only if Aξ(αδ)subeB

ξ(αδ)

(2) Aξ Bξ if and only if Aξ(αδ) B

ξ(αδ)

Proof

(1) Suppose AξsubeBξ Assume that there exist αo isin [0 1]

and δo isin [0 2π] such that Aξ(αoδo)subeB

ξ(αoδo)

It means that mo isin U such that mo isin Aξ(αoδo) but

mo notin Bξ(αoδo) en rAξ(mo)ge αo ωAξ(mo)ge δo rBξ

(mo)lt αo and ωBξ(mo)lt δo Hence rBξ(mo)ltrAξ(mo) and ωBξ(mo)ltωAξ(mo) which is contra-diction to our suppositionConversely let A

ξ(αδ)subeB

ξ(αδ) Consider AξsubeBξ im-

plying that mo isin U such thatrAξ(mo)gt rBξ(mo)ωAξ(mo)gtωBξ(mo) is further shows thatmo isin A

ξ(αδ) and mo notin B

ξ(αδ) which contradicts our

assumption(2) In a similarwaywe can obtain the required equality

Theorem 6 Any two ξ-CFS Aξ and Bξ satisfy the followingcharacteristics

(1) AξsubeBξ if and only if Aξ+(αδ)subeB

ξ+(αδ)

(2) Aξ Bξ if and only if Aξ+(αδ) B

ξ+(αδ)

Theorem 7 Every ξ-CFS Aξ satisfies the following relations

(1) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ(αprime δprime)

(2) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ(αprime δprime)

(3) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ+(αprimeδprime)

(4) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ+(αprime δprime)

Proof

(1) In view of eorem 1 (2)

Aξ(αδ) sube cap

αprimeltαδprimeltδ

Aξαprime δprime( )

(14)

for all αprime lt α and δprime lt δNext suppose m isin cap αprime lt α

δprime lt δA

ξ(αprime δprime) Again by applying

eorem 1 (2) we have rAξ(m)ge αrsquo and ωAξ(m)ge δrsquoe application of the given condition in the above

relation yields that rAξ(m)ge α and ωAξ(m)ge δ is impliesthat m isin A

ξ(αδ)

Hence

cap αprime lt αδprime lt δ

Aξαprime δprime( )subeAξ

(αδ) (15)

From (14) and (15) the required equality holds eremaining parts can be proved in a similar manner In thefollowing definitions we present a new approach to defineξ-CFS which is quite necessary to establish the proofs ofdecomposition theorems

Definition 13 Let Aξ(αδ) be a (α δ)-cut set of Aξ isin Fξ(U)

en the ξ-CFS Aξlowast(αδ) with respect to A

ξ(αδ) is defined as

follows

Aξlowast(αδ)(m)

αeiδ

0

if m isin Aξ(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Definition 14 Let Aξ+(αδ) be a strong (α δ) cut set of

Aξ isin Fξ(U) en the ξ-CFS Aξlowast+(αδ) with respect to A

ξ+(αδ)

can be described as follows

Aξlowast+(αδ)(m)

αeiδ

0

if m isin Aξ+(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

e subsequent result illustrates the decomposition of anξ-CFS as a union of ξ-CFS A

ξlowast(αδ)

Theorem 10 (first decomposition theorem) For everyAξ isin Fξ(U) Aξ cup

αisin[01]δisin[02π]

Aξlowast(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

Complexity 5

cupαisin[01]

δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup

αisin[01]

δisin[02π]

Aξlowast(αδ)(m)

max Supαisin[0u]

δisin[0v]

Aξlowast(αδ)(m) Sup

αisin(u1]

δisin(v2π]

Aξlowast(αδ)(m)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

Choose any α isin (u 1] and δ isin (v 2π] thenrAξ(m) ult α andωAξ(m) vlt δ erefore A

ξlowast(αδ)

(m)

0ei0 On the contrary for any choice of α isin [0 u] andβ isin [0 v] we have rAξ(m) uge α andωAξ(m) vge δ

erefore Aξlowast(αδ)(m) αeiδ which implies that

cupαisin[01]δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (m) Sup αisin[0 u]

δisin[0 v]

αeiδ ueiv μAξ (m)

e following result describes the decomposition of Aξ as aunion of A

ξlowast+(αδ)

Theorem 11 (second decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisin[0 1]

δisin[0 2π]

Aξlowast+(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

cupαisin[01]

δisin[02π]

Aξlowast+(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup αisin[01]δisin[02π]

Aξlowast+(αδ)(m)

max Sup αisin[0u]δisin[0v]

Aξlowast+(αδ)(m) Sup αisin[u1]

δisin[v2π]

Aξlowast+(αδ)(m)⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Sup αisin[0u)δisin[0v)

αeiδ

ueiv

μAξ(m)

(19)

e decomposition of ξ CFS Aξ as a union of level setscan be established by the following result

Theorem 12 (third decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisinΩAξ

δisinΩAξ

Aξlowast(αδ)

Proof e proof is analogous to that of eorem 10e following example illustrates the algebraic fact stated

in first decomposition theorem

Example 1 Consider the ξ-CFS

Aξ

02e

i05π

m1+04e

iπ

m2+06e

i12π

m3+08e

i18π

m41113896 1113897 (20)

For α 02 and δ 05π ξ-CFS Aξlowast(αδ) with respect to

ξ-CFS Aξ is given by

Aξlowast(0205π)

02ei05π

m1+02e

i05π

m2+02e

i05π

m3+02e

i05π

m41113896 1113897

(21)

Corresponding to α 04 and δ π we have

Aξlowast(04π)

0ei0

m1+04e

iπ

m2+04e

iπ

m3+04e

iπ

m41113896 1113897 (22)

Corresponding to α 06 and δ 12π

Aξlowast(0612π)

0ei0

m1+0e

i0

m2+06e

i12π

m3+06e

i12π

m41113896 1113897 (23)

Also for α 08 and δ 18π

Aξlowast(0818π)

0ei0

m1+0e

i0

m2+0e

i0

m3+08e

i18π

m41113896 1113897 (24)

Consequently Aξ cupAξlowast(αδ)

In the following example we apply the concept of ξ-CFSto judge the performance of an artist in an art competition

Example 2 Let X a b c d e f be the list of 10 artistscompeting in an art competition After the initial screeningbased on sketch designing the performance of each artist isgiven in Table 1

Table 1 Performance of all artists after initial screening

Artists Performance of the artistsrA(a) 02rA(b) 091rA(c) 05rA(d) 07rA(e) 04rA(f) 061rA(g) 072rA(h) 03rA(i) 08rA(j) 09

6 Complexity

e graphical interpretation of the above performance ofthe artists is displayed in Figure 1

Let ξ αeiβ be a parameter to select a candidate based onthe performance to draw a sketch of a healthy environmente judgment procedure has two phases α and β where αdenotes the level of roughness of the drawing and β denotesthe use of inappropriate color in the drawing Table 2 in-dicates the performance of the artists after the first phase forα 07

e graphical interpretation of the qualified artists ofstage one is presented in Figure 2

Table 3 indicates the performances of the qualified artistsfor phase two

e graphical interpretation of the artists of stage two ispresented in Figure 3

e following outcomes indicate the performance of thequalified artists after phase two for β 05π areωAξ(a) 05πa dm ωAξ(h) 04π and final score is μAξ(a)

02ei05π and μAξ(h) 03ei04π At this stage we will use thescore function to compare the performance of the artists Forthis we may take |a| 02276 and |h| 02998 e aboveinformation shows that artist ldquohrdquo may be considered the bestof all the artists

4 Algebraic Attributes of ξ-ComplexFuzzy Subgroups

In this section we innovate the notion of ξ-CFSG defined onξ-CFS and establish fundamental algebraic characteristics ofthis phenomenon

Definition 15 A homogeneous ξ-CFS Aξ of a group G iscalled ξ-complex fuzzy subgroup (ξ-CFSG) if Aξ admits thefollowing conditions

(1) μAξ(mn)gemin μAξ(m) μAξ(n)1113864 1113865

(2) μAξ(mminus 1)ge μAξ(m) forallm n isin G

e family of all ξ-CFSG defined on the group G isdenoted by Fξ(G)

Proposition 2 Each ξ-CFSG (G) Aξ satisfies the followingproperties

(1) μAξ(m)le μAξ(e)

(2) μAξ(mnminus 1) μAξ(e)⟹ μAξ(m) μAξ(n)forallm n isin G

Proof

(1) Let m isin G then

μAξ(e) μAξ mmminus 1

1113872 1113873

gemin μAξ(m) μAξ mminus 1

1113872 11138731113872 1113873

μAξ(m)

(25)

(2) Let m n isin G then

μAξ(m) μAξ mnminus 1

n1113872 1113873

gemin μAξ mnminus 1

1113872 1113873 μAξ(n)1113966 1113967

min μAξ(e) μAξ(n)1113864 1113865

μAξ(n)

(26)

In the following result we investigate the conditionunder which a given CFS is ξ-CFSG

Proposition 3 Let A be a CFS (G) such that μA(mminus 1)

μA(m) forallm isin GMoreover ξ le q where q inf μA(m) m isin G1113864 1113865 lten Aξ

is ξ-CFSG (G)

Proof By using the given conditions for any m isin G weobtain μA(m)ge ξ e application of Definition 7 in theabove inequality yields that μAξ(m) ξ ereforeμAξ(mn) min μA(mn) ξ1113864 1113865 and μAξ(mn)gemin μAξ1113864

(m) μAξ(n) for all m n isin G Moreover by using the givencondition μA(mminus 1) μA(m) we get μAξ(mminus 1) μAξ(m)

Perfo

rman

ce o

f the

artis

ts

0010203040506070809

1

b c d e f g h i jaArtists

Figure 1 A graphical overview of Table 1

Table 2 Performance of the artists after the first phase

Artists Roughness of the sketchrAξ (a) 02rAξ (c) 05rAξ (d) 07rAξ (e) 04rAξ (f) 061rAξ (h) 03

Complexity 7

e following result shows that every CFSG is alwaysξ-CFSG

Proposition 4 Every CFSG A is ξ-CFSG of a group G

Proof By using Definition 7 for all m n isin G we haveμAξ(mn) min μA(mn) ξ1113864 1113865 e application of Definition13 in the above relation gives us μAξ(mn)geminμAξ(m) μAξ(n)1113864 1113865

Moreover

μAξ mminus 1

1113872 1113873 min μA mminus 1

1113872 1113873 ξ1113966 1113967

gemin μA(m) ξ1113864 1113865

μAξ(m)

(27)

Hence A is a ξ-CFSG (G)

Remark 2 e converse of Proposition 4 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 3 eCFS A defined on a G 1 minus1 i minus i is givenas

A(m) 02e

iπ

1+04e

iπ

minus1+04e

i12π

minusi+03e

i09π

i1113896 1113897 (28)

e ξ-CFSG (G) corresponding to the value ξ 01ei05π

is given by

Aξ(m)

01ei05π

1+01e

i05π

minus1+01e

i05π

minusi+01e

i05π

i1113896 1113897

(29)

Moreover A is not CFSG (G) as A does not satisfyDefinition 4

e following result indicates that intersection of anytwo ξ-CFSG is also ξ-CFSG

Proposition 5 For any two Aξ Bξ isin Fξ(G) (AcapB)ξ Aξ

capBξ

Proof By using Proposition 1 for any two elementsm n isin G

μ(AcapB)ξ(mn) μAξcapBξ(mn)

min μAξ(mn) μBξ(mn)1113864 1113865(30)

e application of Definition 13 in the above relationgives that

μ(A capΒ)ξ(mn) min μ

(AcapΒ)ξ(m) μ(AcapΒ)ξ(n)1113966 1113967 (31)

Moreover

c d e f haArtists

0

01

02

03

04

05

06

07

08

Roug

hnes

s of t

he sk

etch

Figure 2 A graphical interpretation of Table 2

Table 3 Performance of the artists after the second phase

Artists Performance of the artistsωA(a) 05πωA(c) 06πωA(d) 15πωA(e) 06πωA(f) 07πωA(h) 04π

8 Complexity

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

Definition 6 (see [5]) A CFSG A of a group Gis CFNSG (G)if mA Amforallm isin G

3 Decomposition Theorems of ξ-ComplexFuzzy Sets

In this section we initiate the idea of ξ-CFS as a powerfulextension of classical fuzzy sets We also define the conceptsof (α δ)-cut sets and strong (α δ)-cut sets of ξ-CFS andestablish fundamental properties of these phenomena Wealso prove three decomposition theorems of ξ-CFS

Definition 7 Let A be a CFS of a universe U and ξ αeiδ bean element of a unit circle with 0le αle 1 and 0le δ le 2πenthe CFS Aξ is called the ξ-complex fuzzy set (ξ-CFS) withrespect to CFS A and is expressed as μAξ(m) minμA(m) ξ1113864 1113865 forallm isin U

e family of all ξ-CFS defined on the universe U isdenoted by Fξ(U)

Definition 8 For any Aξ and Βξ isin Fξ(U)

(1) e union of ξ-CFS Aξ and Βξ is denoted by Aξ cupΒξand is defined as follows

μAξ cupΒξ(m) rAξ cupΒξ(m)eiω

Aξ cupΒξ (m)

max rAξ(m) rΒξ(m)( 1113857eimax ω

Aξ (m)ωBξ (m)( )

forallm isin U

(1)

(2) e intersection of ξ-CFS Aξ and Bξ is denoted byAξ capBξ and is defined as follows

μAξ capBξ(m) rAξ capBξ(m)eiω

Aξ capBξ (m)

min rAξ(m) rBξ(m)( 1113857eimin ω

Aξ (m)ωBξ (m)( )

forallm isin U

(2)

(3) e complement of ξ-CFS Aξ is denoted by Aξrsquo and isdescribed as follows

μAξrsquo (m) r

Aξrsquo (m)eiω

Aξrsquo (m)

1 minus rAξ(m)( 1113857ei 2πminusω

Aξ (m)( ) forallm isin U(3)

(4) e product of ξ-CFS Aξ and Bξ is represented byAξ ∘Bξ and is expressed as follows

μAξ ∘Bξ(m) rAξ ∘Bξ(m)eiω

Aξ ∘Bξ (m)

rAξ(m)rBξ(m)( 1113857ei2π ω

Aξ (m)2π( ) ωBξ (m)2π( )( )

forallm isin U

(4)

(5) Let Aξn n isin N be ξ-CFS of a universe U en

the Cartesian product of ξ-CFS Aξn is represented by

Aξ1 times A

ξ2 times middot middot middot times Aξ

n and is defined in the followingway

μAξ1timesA

ξ2timestimesA

ξn(m) r

Aξ1timesA

ξ2timestimesA

ξn(m)e

iωAξ1timesA

ξ2timestimesA

ξn

(m)

min rAξ1

m1( 1113857 rAξ2

m2( 1113857 rAξn

mn( 11138571113874 1113875eimin ω

Aξ1

m1( )ωAξ2

m2( )ωAξn

mn( )1113874 1113875

(5)

where m (m1 m2 middot middot middot mn) mi isin Ui i isin N

Definition 9 Let Aξ and Bξ be any two ξ-CFS of a universeU en

(1) Aξ is homogeneous ξ-CFS if rAξ(m)le rAξ(n) impliesωAξ(m)leωAξ(n) and vice versa forallm n isin U

(2) Aξ is homogeneous ξ-CFS with Bξ if rAξ(m)le rBξ(n)

implies ωAξ(m)leωBξ(n) and vice versa forallm n isin U

e next result illustrates that the intersection of any twoξ-CFS is also ξ-CFS

Proposition 1 For any two ξ-CFS Aξand Bξ (AcapB)ξ

Aξ capBξ

Proof Consider μ(AcapB)ξ

(m) min μAcapB(m) ξ1113864 1113865 m isin U

By applying Definition 8 (2) we have μ(AcapB)ξ

(m)

min(μAξ(m) μBξ(m)) μAξ capBξ(m)

Remark 1 e union of any two ξ-CFS is also ξ-CFS

Definition 10 e (α δ)-cut set of ξ-CFS Aξ is representedby A

ξ(αδ) and is defined as follows

Aξ(αδ) m isin U rAξ(m)ge αωAξ(m)ge δ 0le αle 1 0le δ le 2π1113864 1113865

(6)

Definition 11 For any Aξ isin Fξ(U) strong (α δ)-cut set ofAξ is defined as A

ξ+(αδ) m isin U rAξ(m)gt αωAξ1113864

(m)gt δ 0le αle 1 0le δ le 2π

Complexity 3

Definition 12 e level set ΩAξ of Aξ can be described asΩAξ m isin U rAξ(m) αωAξ(m) δ1113864 1113865 where α isin [0 1]

and δ isin [0 2π]

Theorem 1 Let Aξ and Bξ be any two ξ-CFS lten thefollowing attributes hold for any α αprime isin [0 1] andβ βprime isin [0 2π]

(1) Aξ+(αδ) subeA

ξ(αδ)

(2) αle αprime δ le δprime implies Aξ(αrsquoδrsquo) subeA

ξ(αδ)

(3) (Aξ capBξ)(αδ) Aξ(αδ) capB

ξ(αδ)

(4) (Aξ cupBξ)(αδ) Aξ(αδ) cupB

ξ(αδ)

(5) Aξrsquo(αδ) (Aξrsquo )+(1minusα2πminusδ)

Proof

(1) In view of Definition 10 for any element m isin UrAξ(m)gt α and ωAξ(m)gt δ It means that rAξ(m)ge αand ωAξ(m)ge δ us A

ξ+(αδ)sube A

ξ(αδ)

(2) Let m isin U and by applying Definition 10 we haverAξ(m)ge αωAξ(m)ge δ erefore A

ξ(αprime δprime)subeA

ξ(αδ)

(3) For any m isin (Aξ capBξ)(αδ) we have rAξ capBξ(m)ge αand ωAξ capBξ(m)ge δ is implies that min rAξ(m)1113864

rBξ(m)ge α and min ωAξ(m)ωBξ1113864 (m)ge δ enclearly rAξ(m)ge α rBξ(m)ge α and ωAξ(m)ge δωBξ

(m)ge δ erefore m isin Aξ(αδ) capB

ξ(αδ) and ulti-

mately we obtain

Aξ capB

ξ1113872 1113873

(αδ)subeAξ

(αδ) capBξ(αδ) (7)

Let m isin Aξ(αδ) capB

ξ(αδ)

By applying Definition 10 in the above relations weobtain

rAξ(m)ge αωAξ(m)ge δ rBξ(m)ge αωBξ(m)ge δ (8)

is shows that min rAξ(m) rBξ(m)1113864 1113865ge α and min ωAξ1113864

(m)ωBξ(m)ge δConsequently

Aξ(αδ) capB

ξ(αδ) sube A

ξ capBξ

1113872 1113873(αδ)

(9)

From (7) and (9) the required equality holds(4) For any m isin (Aξ cupBξ)(αδ) we obtain rAξ cupBξ(m)ge α

and ωAξ cupBξ(m)ge δ erefore max rAξ(m) rBξ1113864

(m)ge α and max ωAξ(m)ωBξ(m)1113864 1113865ge δ is meansthat rAξ(m)ge α rBξ(m)ge α and ωAξ(m)ge δ

ωBξ(m)ge δConsequently

Aξ cupB

ξ1113872 1113873

(αδ)subeAξ

(αδ) cupBξ(αδ) (10)

Now suppose that m isin Aξ(αδ) cupB

ξ(αδ) then m isin A

ξ(αδ)

or m isin Bξ(αδ) By applying Definition 10 in the above

relations we get rAξ(m)ge αωAξ(m)ge δ orrBξ(m)ge αωBξ(m)ge δ It further shows that

max rAξ(m) rBξ(m)1113864 1113865ge α and max ωAξ(m)ωBξ1113864 (m)

ge δConsequently

Aξ(αδ) cupB

ξ(αδ)sube A

ξ cupBξ

1113872 1113873(αδ)

(11)

From (10) and (11) the required equality is satisfied(5) Let m isin A

ξprime(αδ) then μ

Aξprime(m) 1 minus rAξ(m)

ei(2πminusωAξ (m)) implying that 1 minus rAξ(m)ge α 2π minus ωAξ

(m)ge δ

It follows that rAξ(m)le 1 minus α ωAξ(m)le 2π minus δ whichshows that m notin (Aξ)+(1minusα2πminusδ)

erefore m isin (Aξprime)+(1minusα2πminusδ) and hence

Aξprime(αδ)sube A

ξprime1113874 1113875

+(1minusα2πminusδ) (12)

Now suppose m isin (Aξprime)+(1minusα2πminusδ) then m notin (Aξ)+

(1 minus α 2π minus δ)is implies that 1 minus αge rAξ(m) ωAξ(m)le 2π minus δ

αle 1 minus rAξ(m) and δ le 2π minus ωAξ(m)It means that m isin A

ξprime(αδ) therefore

Aξprime

1113874 1113875+(1minusα2πminusδ)

subeAξprime(αδ) (13)

From (12) and (13) the required result is satisfied

Theorem 2 Let Aξ and Bξ be any two ξ-CFS lten thefollowing attributes hold for all α αprime isin [0 1] andδ δprime isin [0 2π]

(1) αle αprime δ le δprime implies Aξ+(αprime δprime)subeA

ξ+(αδ)

(2) (Aξ capBξ)+(αδ) Aξ+(αδ) capB

ξ+(αδ)

(3) (Aξ cupBξ)(αδ) Aξ(αδ) cupB

ξ(αδ)

Theorem 3 lte following properties hold for any family ofξ-CFS A

ξi i isin I

(1) cup iisinI(Aξi )(αδ)subecup iisinI(A

ξi )(αδ)

(2) cap iisinI(Aξi )(αδ) cap iisinI(A

ξi )(αδ)

Proof

(1) Let m isin cup iisinI(Aξi )(αδ) in view of Definition 10 we

get rAξio

(m)ge α and ωAξio

(m)ge δe above relation holds only if Supr

Aξi

(m)ge α andSupω

Aξi

(m)ge δ that is cup iisinI(rξi )(m)ge α and

cup iisinI(ωξi )(m)ge δ

It follows that m isin cup iisinI(Aξi )(αδ)

(2) Let m isin cap iisinI(Aξi )(αδ) by using Definition 10 we

obtain rAξio

(m)ge α ωAξio

(m)ge δ

e above inequality holds only if infrAξi

(m)ge αinfω

Aξi

(m)ge δ that is cap iisinI(rξi )(m)ge α and cap iisinI(ω

ξi )(m)

ge δ is implies that m isin cap iisinI(Aξi )(αδ) Hence cap iisinI

(Aξi )(αδ)subecap iisinI(A

ξi )(αδ) Now suppose that m isin cap iisinI

(Aξi )(αδ) Again by applying Definition 10 we get

4 Complexity

rAξi

(m)ge α ωAξi

(m)ge δ en obviously m isin cap iisinI(Aξi )(αδ)

Consequently cap iisinI(Aξi )(αδ)subecap iisinI(A

ξi )(αδ)

Theorem 4 Any family of ξ-CFS Aξi i isin I admits the fol-

lowing properties

(1) cup iisinI(Aξi )+(αδ) cup iisinI(A

ξi )+(αδ)

(2) cup iisinI(Aξi )+(αδ)subecup iisinI(A

ξi )+(αδ)

Theorem 5 Any two ξ-CFS Aξ and Bξ satisfy the followingrelations

(1) AξsubeBξ if and only if Aξ(αδ)subeB

ξ(αδ)

(2) Aξ Bξ if and only if Aξ(αδ) B

ξ(αδ)

Proof

(1) Suppose AξsubeBξ Assume that there exist αo isin [0 1]

and δo isin [0 2π] such that Aξ(αoδo)subeB

ξ(αoδo)

It means that mo isin U such that mo isin Aξ(αoδo) but

mo notin Bξ(αoδo) en rAξ(mo)ge αo ωAξ(mo)ge δo rBξ

(mo)lt αo and ωBξ(mo)lt δo Hence rBξ(mo)ltrAξ(mo) and ωBξ(mo)ltωAξ(mo) which is contra-diction to our suppositionConversely let A

ξ(αδ)subeB

ξ(αδ) Consider AξsubeBξ im-

plying that mo isin U such thatrAξ(mo)gt rBξ(mo)ωAξ(mo)gtωBξ(mo) is further shows thatmo isin A

ξ(αδ) and mo notin B

ξ(αδ) which contradicts our

assumption(2) In a similarwaywe can obtain the required equality

Theorem 6 Any two ξ-CFS Aξ and Bξ satisfy the followingcharacteristics

(1) AξsubeBξ if and only if Aξ+(αδ)subeB

ξ+(αδ)

(2) Aξ Bξ if and only if Aξ+(αδ) B

ξ+(αδ)

Theorem 7 Every ξ-CFS Aξ satisfies the following relations

(1) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ(αprime δprime)

(2) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ(αprime δprime)

(3) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ+(αprimeδprime)

(4) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ+(αprime δprime)

Proof

(1) In view of eorem 1 (2)

Aξ(αδ) sube cap

αprimeltαδprimeltδ

Aξαprime δprime( )

(14)

for all αprime lt α and δprime lt δNext suppose m isin cap αprime lt α

δprime lt δA

ξ(αprime δprime) Again by applying

eorem 1 (2) we have rAξ(m)ge αrsquo and ωAξ(m)ge δrsquoe application of the given condition in the above

relation yields that rAξ(m)ge α and ωAξ(m)ge δ is impliesthat m isin A

ξ(αδ)

Hence

cap αprime lt αδprime lt δ

Aξαprime δprime( )subeAξ

(αδ) (15)

From (14) and (15) the required equality holds eremaining parts can be proved in a similar manner In thefollowing definitions we present a new approach to defineξ-CFS which is quite necessary to establish the proofs ofdecomposition theorems

Definition 13 Let Aξ(αδ) be a (α δ)-cut set of Aξ isin Fξ(U)

en the ξ-CFS Aξlowast(αδ) with respect to A

ξ(αδ) is defined as

follows

Aξlowast(αδ)(m)

αeiδ

0

if m isin Aξ(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Definition 14 Let Aξ+(αδ) be a strong (α δ) cut set of

Aξ isin Fξ(U) en the ξ-CFS Aξlowast+(αδ) with respect to A

ξ+(αδ)

can be described as follows

Aξlowast+(αδ)(m)

αeiδ

0

if m isin Aξ+(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

e subsequent result illustrates the decomposition of anξ-CFS as a union of ξ-CFS A

ξlowast(αδ)

Theorem 10 (first decomposition theorem) For everyAξ isin Fξ(U) Aξ cup

αisin[01]δisin[02π]

Aξlowast(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

Complexity 5

cupαisin[01]

δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup

αisin[01]

δisin[02π]

Aξlowast(αδ)(m)

max Supαisin[0u]

δisin[0v]

Aξlowast(αδ)(m) Sup

αisin(u1]

δisin(v2π]

Aξlowast(αδ)(m)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

Choose any α isin (u 1] and δ isin (v 2π] thenrAξ(m) ult α andωAξ(m) vlt δ erefore A

ξlowast(αδ)

(m)

0ei0 On the contrary for any choice of α isin [0 u] andβ isin [0 v] we have rAξ(m) uge α andωAξ(m) vge δ

erefore Aξlowast(αδ)(m) αeiδ which implies that

cupαisin[01]δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (m) Sup αisin[0 u]

δisin[0 v]

αeiδ ueiv μAξ (m)

e following result describes the decomposition of Aξ as aunion of A

ξlowast+(αδ)

Theorem 11 (second decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisin[0 1]

δisin[0 2π]

Aξlowast+(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

cupαisin[01]

δisin[02π]

Aξlowast+(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup αisin[01]δisin[02π]

Aξlowast+(αδ)(m)

max Sup αisin[0u]δisin[0v]

Aξlowast+(αδ)(m) Sup αisin[u1]

δisin[v2π]

Aξlowast+(αδ)(m)⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Sup αisin[0u)δisin[0v)

αeiδ

ueiv

μAξ(m)

(19)

e decomposition of ξ CFS Aξ as a union of level setscan be established by the following result

Theorem 12 (third decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisinΩAξ

δisinΩAξ

Aξlowast(αδ)

Proof e proof is analogous to that of eorem 10e following example illustrates the algebraic fact stated

in first decomposition theorem

Example 1 Consider the ξ-CFS

Aξ

02e

i05π

m1+04e

iπ

m2+06e

i12π

m3+08e

i18π

m41113896 1113897 (20)

For α 02 and δ 05π ξ-CFS Aξlowast(αδ) with respect to

ξ-CFS Aξ is given by

Aξlowast(0205π)

02ei05π

m1+02e

i05π

m2+02e

i05π

m3+02e

i05π

m41113896 1113897

(21)

Corresponding to α 04 and δ π we have

Aξlowast(04π)

0ei0

m1+04e

iπ

m2+04e

iπ

m3+04e

iπ

m41113896 1113897 (22)

Corresponding to α 06 and δ 12π

Aξlowast(0612π)

0ei0

m1+0e

i0

m2+06e

i12π

m3+06e

i12π

m41113896 1113897 (23)

Also for α 08 and δ 18π

Aξlowast(0818π)

0ei0

m1+0e

i0

m2+0e

i0

m3+08e

i18π

m41113896 1113897 (24)

Consequently Aξ cupAξlowast(αδ)

In the following example we apply the concept of ξ-CFSto judge the performance of an artist in an art competition

Example 2 Let X a b c d e f be the list of 10 artistscompeting in an art competition After the initial screeningbased on sketch designing the performance of each artist isgiven in Table 1

Table 1 Performance of all artists after initial screening

Artists Performance of the artistsrA(a) 02rA(b) 091rA(c) 05rA(d) 07rA(e) 04rA(f) 061rA(g) 072rA(h) 03rA(i) 08rA(j) 09

6 Complexity

e graphical interpretation of the above performance ofthe artists is displayed in Figure 1

Let ξ αeiβ be a parameter to select a candidate based onthe performance to draw a sketch of a healthy environmente judgment procedure has two phases α and β where αdenotes the level of roughness of the drawing and β denotesthe use of inappropriate color in the drawing Table 2 in-dicates the performance of the artists after the first phase forα 07

e graphical interpretation of the qualified artists ofstage one is presented in Figure 2

Table 3 indicates the performances of the qualified artistsfor phase two

e graphical interpretation of the artists of stage two ispresented in Figure 3

e following outcomes indicate the performance of thequalified artists after phase two for β 05π areωAξ(a) 05πa dm ωAξ(h) 04π and final score is μAξ(a)

02ei05π and μAξ(h) 03ei04π At this stage we will use thescore function to compare the performance of the artists Forthis we may take |a| 02276 and |h| 02998 e aboveinformation shows that artist ldquohrdquo may be considered the bestof all the artists

4 Algebraic Attributes of ξ-ComplexFuzzy Subgroups

In this section we innovate the notion of ξ-CFSG defined onξ-CFS and establish fundamental algebraic characteristics ofthis phenomenon

Definition 15 A homogeneous ξ-CFS Aξ of a group G iscalled ξ-complex fuzzy subgroup (ξ-CFSG) if Aξ admits thefollowing conditions

(1) μAξ(mn)gemin μAξ(m) μAξ(n)1113864 1113865

(2) μAξ(mminus 1)ge μAξ(m) forallm n isin G

e family of all ξ-CFSG defined on the group G isdenoted by Fξ(G)

Proposition 2 Each ξ-CFSG (G) Aξ satisfies the followingproperties

(1) μAξ(m)le μAξ(e)

(2) μAξ(mnminus 1) μAξ(e)⟹ μAξ(m) μAξ(n)forallm n isin G

Proof

(1) Let m isin G then

μAξ(e) μAξ mmminus 1

1113872 1113873

gemin μAξ(m) μAξ mminus 1

1113872 11138731113872 1113873

μAξ(m)

(25)

(2) Let m n isin G then

μAξ(m) μAξ mnminus 1

n1113872 1113873

gemin μAξ mnminus 1

1113872 1113873 μAξ(n)1113966 1113967

min μAξ(e) μAξ(n)1113864 1113865

μAξ(n)

(26)

In the following result we investigate the conditionunder which a given CFS is ξ-CFSG

Proposition 3 Let A be a CFS (G) such that μA(mminus 1)

μA(m) forallm isin GMoreover ξ le q where q inf μA(m) m isin G1113864 1113865 lten Aξ

is ξ-CFSG (G)

Proof By using the given conditions for any m isin G weobtain μA(m)ge ξ e application of Definition 7 in theabove inequality yields that μAξ(m) ξ ereforeμAξ(mn) min μA(mn) ξ1113864 1113865 and μAξ(mn)gemin μAξ1113864

(m) μAξ(n) for all m n isin G Moreover by using the givencondition μA(mminus 1) μA(m) we get μAξ(mminus 1) μAξ(m)

Perfo

rman

ce o

f the

artis

ts

0010203040506070809

1

b c d e f g h i jaArtists

Figure 1 A graphical overview of Table 1

Table 2 Performance of the artists after the first phase

Artists Roughness of the sketchrAξ (a) 02rAξ (c) 05rAξ (d) 07rAξ (e) 04rAξ (f) 061rAξ (h) 03

Complexity 7

e following result shows that every CFSG is alwaysξ-CFSG

Proposition 4 Every CFSG A is ξ-CFSG of a group G

Proof By using Definition 7 for all m n isin G we haveμAξ(mn) min μA(mn) ξ1113864 1113865 e application of Definition13 in the above relation gives us μAξ(mn)geminμAξ(m) μAξ(n)1113864 1113865

Moreover

μAξ mminus 1

1113872 1113873 min μA mminus 1

1113872 1113873 ξ1113966 1113967

gemin μA(m) ξ1113864 1113865

μAξ(m)

(27)

Hence A is a ξ-CFSG (G)

Remark 2 e converse of Proposition 4 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 3 eCFS A defined on a G 1 minus1 i minus i is givenas

A(m) 02e

iπ

1+04e

iπ

minus1+04e

i12π

minusi+03e

i09π

i1113896 1113897 (28)

e ξ-CFSG (G) corresponding to the value ξ 01ei05π

is given by

Aξ(m)

01ei05π

1+01e

i05π

minus1+01e

i05π

minusi+01e

i05π

i1113896 1113897

(29)

Moreover A is not CFSG (G) as A does not satisfyDefinition 4

e following result indicates that intersection of anytwo ξ-CFSG is also ξ-CFSG

Proposition 5 For any two Aξ Bξ isin Fξ(G) (AcapB)ξ Aξ

capBξ

Proof By using Proposition 1 for any two elementsm n isin G

μ(AcapB)ξ(mn) μAξcapBξ(mn)

min μAξ(mn) μBξ(mn)1113864 1113865(30)

e application of Definition 13 in the above relationgives that

μ(A capΒ)ξ(mn) min μ

(AcapΒ)ξ(m) μ(AcapΒ)ξ(n)1113966 1113967 (31)

Moreover

c d e f haArtists

0

01

02

03

04

05

06

07

08

Roug

hnes

s of t

he sk

etch

Figure 2 A graphical interpretation of Table 2

Table 3 Performance of the artists after the second phase

Artists Performance of the artistsωA(a) 05πωA(c) 06πωA(d) 15πωA(e) 06πωA(f) 07πωA(h) 04π

8 Complexity

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

Definition 12 e level set ΩAξ of Aξ can be described asΩAξ m isin U rAξ(m) αωAξ(m) δ1113864 1113865 where α isin [0 1]

and δ isin [0 2π]

Theorem 1 Let Aξ and Bξ be any two ξ-CFS lten thefollowing attributes hold for any α αprime isin [0 1] andβ βprime isin [0 2π]

(1) Aξ+(αδ) subeA

ξ(αδ)

(2) αle αprime δ le δprime implies Aξ(αrsquoδrsquo) subeA

ξ(αδ)

(3) (Aξ capBξ)(αδ) Aξ(αδ) capB

ξ(αδ)

(4) (Aξ cupBξ)(αδ) Aξ(αδ) cupB

ξ(αδ)

(5) Aξrsquo(αδ) (Aξrsquo )+(1minusα2πminusδ)

Proof

(1) In view of Definition 10 for any element m isin UrAξ(m)gt α and ωAξ(m)gt δ It means that rAξ(m)ge αand ωAξ(m)ge δ us A

ξ+(αδ)sube A

ξ(αδ)

(2) Let m isin U and by applying Definition 10 we haverAξ(m)ge αωAξ(m)ge δ erefore A

ξ(αprime δprime)subeA

ξ(αδ)

(3) For any m isin (Aξ capBξ)(αδ) we have rAξ capBξ(m)ge αand ωAξ capBξ(m)ge δ is implies that min rAξ(m)1113864

rBξ(m)ge α and min ωAξ(m)ωBξ1113864 (m)ge δ enclearly rAξ(m)ge α rBξ(m)ge α and ωAξ(m)ge δωBξ

(m)ge δ erefore m isin Aξ(αδ) capB

ξ(αδ) and ulti-

mately we obtain

Aξ capB

ξ1113872 1113873

(αδ)subeAξ

(αδ) capBξ(αδ) (7)

Let m isin Aξ(αδ) capB

ξ(αδ)

By applying Definition 10 in the above relations weobtain

rAξ(m)ge αωAξ(m)ge δ rBξ(m)ge αωBξ(m)ge δ (8)

is shows that min rAξ(m) rBξ(m)1113864 1113865ge α and min ωAξ1113864

(m)ωBξ(m)ge δConsequently

Aξ(αδ) capB

ξ(αδ) sube A

ξ capBξ

1113872 1113873(αδ)

(9)

From (7) and (9) the required equality holds(4) For any m isin (Aξ cupBξ)(αδ) we obtain rAξ cupBξ(m)ge α

and ωAξ cupBξ(m)ge δ erefore max rAξ(m) rBξ1113864

(m)ge α and max ωAξ(m)ωBξ(m)1113864 1113865ge δ is meansthat rAξ(m)ge α rBξ(m)ge α and ωAξ(m)ge δ

ωBξ(m)ge δConsequently

Aξ cupB

ξ1113872 1113873

(αδ)subeAξ

(αδ) cupBξ(αδ) (10)

Now suppose that m isin Aξ(αδ) cupB

ξ(αδ) then m isin A

ξ(αδ)

or m isin Bξ(αδ) By applying Definition 10 in the above

relations we get rAξ(m)ge αωAξ(m)ge δ orrBξ(m)ge αωBξ(m)ge δ It further shows that

max rAξ(m) rBξ(m)1113864 1113865ge α and max ωAξ(m)ωBξ1113864 (m)

ge δConsequently

Aξ(αδ) cupB

ξ(αδ)sube A

ξ cupBξ

1113872 1113873(αδ)

(11)

From (10) and (11) the required equality is satisfied(5) Let m isin A

ξprime(αδ) then μ

Aξprime(m) 1 minus rAξ(m)

ei(2πminusωAξ (m)) implying that 1 minus rAξ(m)ge α 2π minus ωAξ

(m)ge δ

It follows that rAξ(m)le 1 minus α ωAξ(m)le 2π minus δ whichshows that m notin (Aξ)+(1minusα2πminusδ)

erefore m isin (Aξprime)+(1minusα2πminusδ) and hence

Aξprime(αδ)sube A

ξprime1113874 1113875

+(1minusα2πminusδ) (12)

Now suppose m isin (Aξprime)+(1minusα2πminusδ) then m notin (Aξ)+

(1 minus α 2π minus δ)is implies that 1 minus αge rAξ(m) ωAξ(m)le 2π minus δ

αle 1 minus rAξ(m) and δ le 2π minus ωAξ(m)It means that m isin A

ξprime(αδ) therefore

Aξprime

1113874 1113875+(1minusα2πminusδ)

subeAξprime(αδ) (13)

From (12) and (13) the required result is satisfied

Theorem 2 Let Aξ and Bξ be any two ξ-CFS lten thefollowing attributes hold for all α αprime isin [0 1] andδ δprime isin [0 2π]

(1) αle αprime δ le δprime implies Aξ+(αprime δprime)subeA

ξ+(αδ)

(2) (Aξ capBξ)+(αδ) Aξ+(αδ) capB

ξ+(αδ)

(3) (Aξ cupBξ)(αδ) Aξ(αδ) cupB

ξ(αδ)

Theorem 3 lte following properties hold for any family ofξ-CFS A

ξi i isin I

(1) cup iisinI(Aξi )(αδ)subecup iisinI(A

ξi )(αδ)

(2) cap iisinI(Aξi )(αδ) cap iisinI(A

ξi )(αδ)

Proof

(1) Let m isin cup iisinI(Aξi )(αδ) in view of Definition 10 we

get rAξio

(m)ge α and ωAξio

(m)ge δe above relation holds only if Supr

Aξi

(m)ge α andSupω

Aξi

(m)ge δ that is cup iisinI(rξi )(m)ge α and

cup iisinI(ωξi )(m)ge δ

It follows that m isin cup iisinI(Aξi )(αδ)

(2) Let m isin cap iisinI(Aξi )(αδ) by using Definition 10 we

obtain rAξio

(m)ge α ωAξio

(m)ge δ

e above inequality holds only if infrAξi

(m)ge αinfω

Aξi

(m)ge δ that is cap iisinI(rξi )(m)ge α and cap iisinI(ω

ξi )(m)

ge δ is implies that m isin cap iisinI(Aξi )(αδ) Hence cap iisinI

(Aξi )(αδ)subecap iisinI(A

ξi )(αδ) Now suppose that m isin cap iisinI

(Aξi )(αδ) Again by applying Definition 10 we get

4 Complexity

rAξi

(m)ge α ωAξi

(m)ge δ en obviously m isin cap iisinI(Aξi )(αδ)

Consequently cap iisinI(Aξi )(αδ)subecap iisinI(A

ξi )(αδ)

Theorem 4 Any family of ξ-CFS Aξi i isin I admits the fol-

lowing properties

(1) cup iisinI(Aξi )+(αδ) cup iisinI(A

ξi )+(αδ)

(2) cup iisinI(Aξi )+(αδ)subecup iisinI(A

ξi )+(αδ)

Theorem 5 Any two ξ-CFS Aξ and Bξ satisfy the followingrelations

(1) AξsubeBξ if and only if Aξ(αδ)subeB

ξ(αδ)

(2) Aξ Bξ if and only if Aξ(αδ) B

ξ(αδ)

Proof

(1) Suppose AξsubeBξ Assume that there exist αo isin [0 1]

and δo isin [0 2π] such that Aξ(αoδo)subeB

ξ(αoδo)

It means that mo isin U such that mo isin Aξ(αoδo) but

mo notin Bξ(αoδo) en rAξ(mo)ge αo ωAξ(mo)ge δo rBξ

(mo)lt αo and ωBξ(mo)lt δo Hence rBξ(mo)ltrAξ(mo) and ωBξ(mo)ltωAξ(mo) which is contra-diction to our suppositionConversely let A

ξ(αδ)subeB

ξ(αδ) Consider AξsubeBξ im-

plying that mo isin U such thatrAξ(mo)gt rBξ(mo)ωAξ(mo)gtωBξ(mo) is further shows thatmo isin A

ξ(αδ) and mo notin B

ξ(αδ) which contradicts our

assumption(2) In a similarwaywe can obtain the required equality

Theorem 6 Any two ξ-CFS Aξ and Bξ satisfy the followingcharacteristics

(1) AξsubeBξ if and only if Aξ+(αδ)subeB

ξ+(αδ)

(2) Aξ Bξ if and only if Aξ+(αδ) B

ξ+(αδ)

Theorem 7 Every ξ-CFS Aξ satisfies the following relations

(1) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ(αprime δprime)

(2) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ(αprime δprime)

(3) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ+(αprimeδprime)

(4) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ+(αprime δprime)

Proof

(1) In view of eorem 1 (2)

Aξ(αδ) sube cap

αprimeltαδprimeltδ

Aξαprime δprime( )

(14)

for all αprime lt α and δprime lt δNext suppose m isin cap αprime lt α

δprime lt δA

ξ(αprime δprime) Again by applying

eorem 1 (2) we have rAξ(m)ge αrsquo and ωAξ(m)ge δrsquoe application of the given condition in the above

relation yields that rAξ(m)ge α and ωAξ(m)ge δ is impliesthat m isin A

ξ(αδ)

Hence

cap αprime lt αδprime lt δ

Aξαprime δprime( )subeAξ

(αδ) (15)

From (14) and (15) the required equality holds eremaining parts can be proved in a similar manner In thefollowing definitions we present a new approach to defineξ-CFS which is quite necessary to establish the proofs ofdecomposition theorems

Definition 13 Let Aξ(αδ) be a (α δ)-cut set of Aξ isin Fξ(U)

en the ξ-CFS Aξlowast(αδ) with respect to A

ξ(αδ) is defined as

follows

Aξlowast(αδ)(m)

αeiδ

0

if m isin Aξ(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Definition 14 Let Aξ+(αδ) be a strong (α δ) cut set of

Aξ isin Fξ(U) en the ξ-CFS Aξlowast+(αδ) with respect to A

ξ+(αδ)

can be described as follows

Aξlowast+(αδ)(m)

αeiδ

0

if m isin Aξ+(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

e subsequent result illustrates the decomposition of anξ-CFS as a union of ξ-CFS A

ξlowast(αδ)

Theorem 10 (first decomposition theorem) For everyAξ isin Fξ(U) Aξ cup

αisin[01]δisin[02π]

Aξlowast(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

Complexity 5

cupαisin[01]

δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup

αisin[01]

δisin[02π]

Aξlowast(αδ)(m)

max Supαisin[0u]

δisin[0v]

Aξlowast(αδ)(m) Sup

αisin(u1]

δisin(v2π]

Aξlowast(αδ)(m)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

Choose any α isin (u 1] and δ isin (v 2π] thenrAξ(m) ult α andωAξ(m) vlt δ erefore A

ξlowast(αδ)

(m)

0ei0 On the contrary for any choice of α isin [0 u] andβ isin [0 v] we have rAξ(m) uge α andωAξ(m) vge δ

erefore Aξlowast(αδ)(m) αeiδ which implies that

cupαisin[01]δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (m) Sup αisin[0 u]

δisin[0 v]

αeiδ ueiv μAξ (m)

e following result describes the decomposition of Aξ as aunion of A

ξlowast+(αδ)

Theorem 11 (second decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisin[0 1]

δisin[0 2π]

Aξlowast+(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

cupαisin[01]

δisin[02π]

Aξlowast+(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup αisin[01]δisin[02π]

Aξlowast+(αδ)(m)

max Sup αisin[0u]δisin[0v]

Aξlowast+(αδ)(m) Sup αisin[u1]

δisin[v2π]

Aξlowast+(αδ)(m)⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Sup αisin[0u)δisin[0v)

αeiδ

ueiv

μAξ(m)

(19)

e decomposition of ξ CFS Aξ as a union of level setscan be established by the following result

Theorem 12 (third decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisinΩAξ

δisinΩAξ

Aξlowast(αδ)

Proof e proof is analogous to that of eorem 10e following example illustrates the algebraic fact stated

in first decomposition theorem

Example 1 Consider the ξ-CFS

Aξ

02e

i05π

m1+04e

iπ

m2+06e

i12π

m3+08e

i18π

m41113896 1113897 (20)

For α 02 and δ 05π ξ-CFS Aξlowast(αδ) with respect to

ξ-CFS Aξ is given by

Aξlowast(0205π)

02ei05π

m1+02e

i05π

m2+02e

i05π

m3+02e

i05π

m41113896 1113897

(21)

Corresponding to α 04 and δ π we have

Aξlowast(04π)

0ei0

m1+04e

iπ

m2+04e

iπ

m3+04e

iπ

m41113896 1113897 (22)

Corresponding to α 06 and δ 12π

Aξlowast(0612π)

0ei0

m1+0e

i0

m2+06e

i12π

m3+06e

i12π

m41113896 1113897 (23)

Also for α 08 and δ 18π

Aξlowast(0818π)

0ei0

m1+0e

i0

m2+0e

i0

m3+08e

i18π

m41113896 1113897 (24)

Consequently Aξ cupAξlowast(αδ)

In the following example we apply the concept of ξ-CFSto judge the performance of an artist in an art competition

Example 2 Let X a b c d e f be the list of 10 artistscompeting in an art competition After the initial screeningbased on sketch designing the performance of each artist isgiven in Table 1

Table 1 Performance of all artists after initial screening

Artists Performance of the artistsrA(a) 02rA(b) 091rA(c) 05rA(d) 07rA(e) 04rA(f) 061rA(g) 072rA(h) 03rA(i) 08rA(j) 09

6 Complexity

e graphical interpretation of the above performance ofthe artists is displayed in Figure 1

Let ξ αeiβ be a parameter to select a candidate based onthe performance to draw a sketch of a healthy environmente judgment procedure has two phases α and β where αdenotes the level of roughness of the drawing and β denotesthe use of inappropriate color in the drawing Table 2 in-dicates the performance of the artists after the first phase forα 07

e graphical interpretation of the qualified artists ofstage one is presented in Figure 2

Table 3 indicates the performances of the qualified artistsfor phase two

e graphical interpretation of the artists of stage two ispresented in Figure 3

e following outcomes indicate the performance of thequalified artists after phase two for β 05π areωAξ(a) 05πa dm ωAξ(h) 04π and final score is μAξ(a)

02ei05π and μAξ(h) 03ei04π At this stage we will use thescore function to compare the performance of the artists Forthis we may take |a| 02276 and |h| 02998 e aboveinformation shows that artist ldquohrdquo may be considered the bestof all the artists

4 Algebraic Attributes of ξ-ComplexFuzzy Subgroups

In this section we innovate the notion of ξ-CFSG defined onξ-CFS and establish fundamental algebraic characteristics ofthis phenomenon

Definition 15 A homogeneous ξ-CFS Aξ of a group G iscalled ξ-complex fuzzy subgroup (ξ-CFSG) if Aξ admits thefollowing conditions

(1) μAξ(mn)gemin μAξ(m) μAξ(n)1113864 1113865

(2) μAξ(mminus 1)ge μAξ(m) forallm n isin G

e family of all ξ-CFSG defined on the group G isdenoted by Fξ(G)

Proposition 2 Each ξ-CFSG (G) Aξ satisfies the followingproperties

(1) μAξ(m)le μAξ(e)

(2) μAξ(mnminus 1) μAξ(e)⟹ μAξ(m) μAξ(n)forallm n isin G

Proof

(1) Let m isin G then

μAξ(e) μAξ mmminus 1

1113872 1113873

gemin μAξ(m) μAξ mminus 1

1113872 11138731113872 1113873

μAξ(m)

(25)

(2) Let m n isin G then

μAξ(m) μAξ mnminus 1

n1113872 1113873

gemin μAξ mnminus 1

1113872 1113873 μAξ(n)1113966 1113967

min μAξ(e) μAξ(n)1113864 1113865

μAξ(n)

(26)

In the following result we investigate the conditionunder which a given CFS is ξ-CFSG

Proposition 3 Let A be a CFS (G) such that μA(mminus 1)

μA(m) forallm isin GMoreover ξ le q where q inf μA(m) m isin G1113864 1113865 lten Aξ

is ξ-CFSG (G)

Proof By using the given conditions for any m isin G weobtain μA(m)ge ξ e application of Definition 7 in theabove inequality yields that μAξ(m) ξ ereforeμAξ(mn) min μA(mn) ξ1113864 1113865 and μAξ(mn)gemin μAξ1113864

(m) μAξ(n) for all m n isin G Moreover by using the givencondition μA(mminus 1) μA(m) we get μAξ(mminus 1) μAξ(m)

Perfo

rman

ce o

f the

artis

ts

0010203040506070809

1

b c d e f g h i jaArtists

Figure 1 A graphical overview of Table 1

Table 2 Performance of the artists after the first phase

Artists Roughness of the sketchrAξ (a) 02rAξ (c) 05rAξ (d) 07rAξ (e) 04rAξ (f) 061rAξ (h) 03

Complexity 7

e following result shows that every CFSG is alwaysξ-CFSG

Proposition 4 Every CFSG A is ξ-CFSG of a group G

Proof By using Definition 7 for all m n isin G we haveμAξ(mn) min μA(mn) ξ1113864 1113865 e application of Definition13 in the above relation gives us μAξ(mn)geminμAξ(m) μAξ(n)1113864 1113865

Moreover

μAξ mminus 1

1113872 1113873 min μA mminus 1

1113872 1113873 ξ1113966 1113967

gemin μA(m) ξ1113864 1113865

μAξ(m)

(27)

Hence A is a ξ-CFSG (G)

Remark 2 e converse of Proposition 4 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 3 eCFS A defined on a G 1 minus1 i minus i is givenas

A(m) 02e

iπ

1+04e

iπ

minus1+04e

i12π

minusi+03e

i09π

i1113896 1113897 (28)

e ξ-CFSG (G) corresponding to the value ξ 01ei05π

is given by

Aξ(m)

01ei05π

1+01e

i05π

minus1+01e

i05π

minusi+01e

i05π

i1113896 1113897

(29)

Moreover A is not CFSG (G) as A does not satisfyDefinition 4

e following result indicates that intersection of anytwo ξ-CFSG is also ξ-CFSG

Proposition 5 For any two Aξ Bξ isin Fξ(G) (AcapB)ξ Aξ

capBξ

Proof By using Proposition 1 for any two elementsm n isin G

μ(AcapB)ξ(mn) μAξcapBξ(mn)

min μAξ(mn) μBξ(mn)1113864 1113865(30)

e application of Definition 13 in the above relationgives that

μ(A capΒ)ξ(mn) min μ

(AcapΒ)ξ(m) μ(AcapΒ)ξ(n)1113966 1113967 (31)

Moreover

c d e f haArtists

0

01

02

03

04

05

06

07

08

Roug

hnes

s of t

he sk

etch

Figure 2 A graphical interpretation of Table 2

Table 3 Performance of the artists after the second phase

Artists Performance of the artistsωA(a) 05πωA(c) 06πωA(d) 15πωA(e) 06πωA(f) 07πωA(h) 04π

8 Complexity

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

rAξi

(m)ge α ωAξi

(m)ge δ en obviously m isin cap iisinI(Aξi )(αδ)

Consequently cap iisinI(Aξi )(αδ)subecap iisinI(A

ξi )(αδ)

Theorem 4 Any family of ξ-CFS Aξi i isin I admits the fol-

lowing properties

(1) cup iisinI(Aξi )+(αδ) cup iisinI(A

ξi )+(αδ)

(2) cup iisinI(Aξi )+(αδ)subecup iisinI(A

ξi )+(αδ)

Theorem 5 Any two ξ-CFS Aξ and Bξ satisfy the followingrelations

(1) AξsubeBξ if and only if Aξ(αδ)subeB

ξ(αδ)

(2) Aξ Bξ if and only if Aξ(αδ) B

ξ(αδ)

Proof

(1) Suppose AξsubeBξ Assume that there exist αo isin [0 1]

and δo isin [0 2π] such that Aξ(αoδo)subeB

ξ(αoδo)

It means that mo isin U such that mo isin Aξ(αoδo) but

mo notin Bξ(αoδo) en rAξ(mo)ge αo ωAξ(mo)ge δo rBξ

(mo)lt αo and ωBξ(mo)lt δo Hence rBξ(mo)ltrAξ(mo) and ωBξ(mo)ltωAξ(mo) which is contra-diction to our suppositionConversely let A

ξ(αδ)subeB

ξ(αδ) Consider AξsubeBξ im-

plying that mo isin U such thatrAξ(mo)gt rBξ(mo)ωAξ(mo)gtωBξ(mo) is further shows thatmo isin A

ξ(αδ) and mo notin B

ξ(αδ) which contradicts our

assumption(2) In a similarwaywe can obtain the required equality

Theorem 6 Any two ξ-CFS Aξ and Bξ satisfy the followingcharacteristics

(1) AξsubeBξ if and only if Aξ+(αδ)subeB

ξ+(αδ)

(2) Aξ Bξ if and only if Aξ+(αδ) B

ξ+(αδ)

Theorem 7 Every ξ-CFS Aξ satisfies the following relations

(1) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ(αprime δprime)

(2) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ(αprime δprime)

(3) Aξ(αδ) cap

αprime lt αδprime lt δ

Aξ+(αprimeδprime)

(4) Aξ+(αδ) cup

αltαprimeδ lt δprime

Aξ+(αprime δprime)

Proof

(1) In view of eorem 1 (2)

Aξ(αδ) sube cap

αprimeltαδprimeltδ

Aξαprime δprime( )

(14)

for all αprime lt α and δprime lt δNext suppose m isin cap αprime lt α

δprime lt δA

ξ(αprime δprime) Again by applying

eorem 1 (2) we have rAξ(m)ge αrsquo and ωAξ(m)ge δrsquoe application of the given condition in the above

relation yields that rAξ(m)ge α and ωAξ(m)ge δ is impliesthat m isin A

ξ(αδ)

Hence

cap αprime lt αδprime lt δ

Aξαprime δprime( )subeAξ

(αδ) (15)

From (14) and (15) the required equality holds eremaining parts can be proved in a similar manner In thefollowing definitions we present a new approach to defineξ-CFS which is quite necessary to establish the proofs ofdecomposition theorems

Definition 13 Let Aξ(αδ) be a (α δ)-cut set of Aξ isin Fξ(U)

en the ξ-CFS Aξlowast(αδ) with respect to A

ξ(αδ) is defined as

follows

Aξlowast(αδ)(m)

αeiδ

0

if m isin Aξ(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(16)

Definition 14 Let Aξ+(αδ) be a strong (α δ) cut set of

Aξ isin Fξ(U) en the ξ-CFS Aξlowast+(αδ) with respect to A

ξ+(αδ)

can be described as follows

Aξlowast+(αδ)(m)

αeiδ

0

if m isin Aξ+(αδ)

otherwise

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

e subsequent result illustrates the decomposition of anξ-CFS as a union of ξ-CFS A

ξlowast(αδ)

Theorem 10 (first decomposition theorem) For everyAξ isin Fξ(U) Aξ cup

αisin[01]δisin[02π]

Aξlowast(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

Complexity 5

cupαisin[01]

δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup

αisin[01]

δisin[02π]

Aξlowast(αδ)(m)

max Supαisin[0u]

δisin[0v]

Aξlowast(αδ)(m) Sup

αisin(u1]

δisin(v2π]

Aξlowast(αδ)(m)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

Choose any α isin (u 1] and δ isin (v 2π] thenrAξ(m) ult α andωAξ(m) vlt δ erefore A

ξlowast(αδ)

(m)

0ei0 On the contrary for any choice of α isin [0 u] andβ isin [0 v] we have rAξ(m) uge α andωAξ(m) vge δ

erefore Aξlowast(αδ)(m) αeiδ which implies that

cupαisin[01]δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (m) Sup αisin[0 u]

δisin[0 v]

αeiδ ueiv μAξ (m)

e following result describes the decomposition of Aξ as aunion of A

ξlowast+(αδ)

Theorem 11 (second decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisin[0 1]

δisin[0 2π]

Aξlowast+(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

cupαisin[01]

δisin[02π]

Aξlowast+(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup αisin[01]δisin[02π]

Aξlowast+(αδ)(m)

max Sup αisin[0u]δisin[0v]

Aξlowast+(αδ)(m) Sup αisin[u1]

δisin[v2π]

Aξlowast+(αδ)(m)⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Sup αisin[0u)δisin[0v)

αeiδ

ueiv

μAξ(m)

(19)

e decomposition of ξ CFS Aξ as a union of level setscan be established by the following result

Theorem 12 (third decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisinΩAξ

δisinΩAξ

Aξlowast(αδ)

Proof e proof is analogous to that of eorem 10e following example illustrates the algebraic fact stated

in first decomposition theorem

Example 1 Consider the ξ-CFS

Aξ

02e

i05π

m1+04e

iπ

m2+06e

i12π

m3+08e

i18π

m41113896 1113897 (20)

For α 02 and δ 05π ξ-CFS Aξlowast(αδ) with respect to

ξ-CFS Aξ is given by

Aξlowast(0205π)

02ei05π

m1+02e

i05π

m2+02e

i05π

m3+02e

i05π

m41113896 1113897

(21)

Corresponding to α 04 and δ π we have

Aξlowast(04π)

0ei0

m1+04e

iπ

m2+04e

iπ

m3+04e

iπ

m41113896 1113897 (22)

Corresponding to α 06 and δ 12π

Aξlowast(0612π)

0ei0

m1+0e

i0

m2+06e

i12π

m3+06e

i12π

m41113896 1113897 (23)

Also for α 08 and δ 18π

Aξlowast(0818π)

0ei0

m1+0e

i0

m2+0e

i0

m3+08e

i18π

m41113896 1113897 (24)

Consequently Aξ cupAξlowast(αδ)

In the following example we apply the concept of ξ-CFSto judge the performance of an artist in an art competition

Example 2 Let X a b c d e f be the list of 10 artistscompeting in an art competition After the initial screeningbased on sketch designing the performance of each artist isgiven in Table 1

Table 1 Performance of all artists after initial screening

Artists Performance of the artistsrA(a) 02rA(b) 091rA(c) 05rA(d) 07rA(e) 04rA(f) 061rA(g) 072rA(h) 03rA(i) 08rA(j) 09

6 Complexity

e graphical interpretation of the above performance ofthe artists is displayed in Figure 1

Let ξ αeiβ be a parameter to select a candidate based onthe performance to draw a sketch of a healthy environmente judgment procedure has two phases α and β where αdenotes the level of roughness of the drawing and β denotesthe use of inappropriate color in the drawing Table 2 in-dicates the performance of the artists after the first phase forα 07

e graphical interpretation of the qualified artists ofstage one is presented in Figure 2

Table 3 indicates the performances of the qualified artistsfor phase two

e graphical interpretation of the artists of stage two ispresented in Figure 3

e following outcomes indicate the performance of thequalified artists after phase two for β 05π areωAξ(a) 05πa dm ωAξ(h) 04π and final score is μAξ(a)

02ei05π and μAξ(h) 03ei04π At this stage we will use thescore function to compare the performance of the artists Forthis we may take |a| 02276 and |h| 02998 e aboveinformation shows that artist ldquohrdquo may be considered the bestof all the artists

4 Algebraic Attributes of ξ-ComplexFuzzy Subgroups

In this section we innovate the notion of ξ-CFSG defined onξ-CFS and establish fundamental algebraic characteristics ofthis phenomenon

Definition 15 A homogeneous ξ-CFS Aξ of a group G iscalled ξ-complex fuzzy subgroup (ξ-CFSG) if Aξ admits thefollowing conditions

(1) μAξ(mn)gemin μAξ(m) μAξ(n)1113864 1113865

(2) μAξ(mminus 1)ge μAξ(m) forallm n isin G

e family of all ξ-CFSG defined on the group G isdenoted by Fξ(G)

Proposition 2 Each ξ-CFSG (G) Aξ satisfies the followingproperties

(1) μAξ(m)le μAξ(e)

(2) μAξ(mnminus 1) μAξ(e)⟹ μAξ(m) μAξ(n)forallm n isin G

Proof

(1) Let m isin G then

μAξ(e) μAξ mmminus 1

1113872 1113873

gemin μAξ(m) μAξ mminus 1

1113872 11138731113872 1113873

μAξ(m)

(25)

(2) Let m n isin G then

μAξ(m) μAξ mnminus 1

n1113872 1113873

gemin μAξ mnminus 1

1113872 1113873 μAξ(n)1113966 1113967

min μAξ(e) μAξ(n)1113864 1113865

μAξ(n)

(26)

In the following result we investigate the conditionunder which a given CFS is ξ-CFSG

Proposition 3 Let A be a CFS (G) such that μA(mminus 1)

μA(m) forallm isin GMoreover ξ le q where q inf μA(m) m isin G1113864 1113865 lten Aξ

is ξ-CFSG (G)

Proof By using the given conditions for any m isin G weobtain μA(m)ge ξ e application of Definition 7 in theabove inequality yields that μAξ(m) ξ ereforeμAξ(mn) min μA(mn) ξ1113864 1113865 and μAξ(mn)gemin μAξ1113864

(m) μAξ(n) for all m n isin G Moreover by using the givencondition μA(mminus 1) μA(m) we get μAξ(mminus 1) μAξ(m)

Perfo

rman

ce o

f the

artis

ts

0010203040506070809

1

b c d e f g h i jaArtists

Figure 1 A graphical overview of Table 1

Table 2 Performance of the artists after the first phase

Artists Roughness of the sketchrAξ (a) 02rAξ (c) 05rAξ (d) 07rAξ (e) 04rAξ (f) 061rAξ (h) 03

Complexity 7

e following result shows that every CFSG is alwaysξ-CFSG

Proposition 4 Every CFSG A is ξ-CFSG of a group G

Proof By using Definition 7 for all m n isin G we haveμAξ(mn) min μA(mn) ξ1113864 1113865 e application of Definition13 in the above relation gives us μAξ(mn)geminμAξ(m) μAξ(n)1113864 1113865

Moreover

μAξ mminus 1

1113872 1113873 min μA mminus 1

1113872 1113873 ξ1113966 1113967

gemin μA(m) ξ1113864 1113865

μAξ(m)

(27)

Hence A is a ξ-CFSG (G)

Remark 2 e converse of Proposition 4 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 3 eCFS A defined on a G 1 minus1 i minus i is givenas

A(m) 02e

iπ

1+04e

iπ

minus1+04e

i12π

minusi+03e

i09π

i1113896 1113897 (28)

e ξ-CFSG (G) corresponding to the value ξ 01ei05π

is given by

Aξ(m)

01ei05π

1+01e

i05π

minus1+01e

i05π

minusi+01e

i05π

i1113896 1113897

(29)

Moreover A is not CFSG (G) as A does not satisfyDefinition 4

e following result indicates that intersection of anytwo ξ-CFSG is also ξ-CFSG

Proposition 5 For any two Aξ Bξ isin Fξ(G) (AcapB)ξ Aξ

capBξ

Proof By using Proposition 1 for any two elementsm n isin G

μ(AcapB)ξ(mn) μAξcapBξ(mn)

min μAξ(mn) μBξ(mn)1113864 1113865(30)

e application of Definition 13 in the above relationgives that

μ(A capΒ)ξ(mn) min μ

(AcapΒ)ξ(m) μ(AcapΒ)ξ(n)1113966 1113967 (31)

Moreover

c d e f haArtists

0

01

02

03

04

05

06

07

08

Roug

hnes

s of t

he sk

etch

Figure 2 A graphical interpretation of Table 2

Table 3 Performance of the artists after the second phase

Artists Performance of the artistsωA(a) 05πωA(c) 06πωA(d) 15πωA(e) 06πωA(f) 07πωA(h) 04π

8 Complexity

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

cupαisin[01]

δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup

αisin[01]

δisin[02π]

Aξlowast(αδ)(m)

max Supαisin[0u]

δisin[0v]

Aξlowast(αδ)(m) Sup

αisin(u1]

δisin(v2π]

Aξlowast(αδ)(m)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(18)

Choose any α isin (u 1] and δ isin (v 2π] thenrAξ(m) ult α andωAξ(m) vlt δ erefore A

ξlowast(αδ)

(m)

0ei0 On the contrary for any choice of α isin [0 u] andβ isin [0 v] we have rAξ(m) uge α andωAξ(m) vge δ

erefore Aξlowast(αδ)(m) αeiδ which implies that

cupαisin[01]δisin[02π]

Aξlowast(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (m) Sup αisin[0 u]

δisin[0 v]

αeiδ ueiv μAξ (m)

e following result describes the decomposition of Aξ as aunion of A

ξlowast+(αδ)

Theorem 11 (second decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisin[0 1]

δisin[0 2π]

Aξlowast+(αδ)

Proof Suppose rAξ(m) u andωAξ(m) v for a particularm isin U then

cupαisin[01]

δisin[02π]

Aξlowast+(αδ)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(m) Sup αisin[01]δisin[02π]

Aξlowast+(αδ)(m)

max Sup αisin[0u]δisin[0v]

Aξlowast+(αδ)(m) Sup αisin[u1]

δisin[v2π]

Aξlowast+(αδ)(m)⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

Sup αisin[0u)δisin[0v)

αeiδ

ueiv

μAξ(m)

(19)

e decomposition of ξ CFS Aξ as a union of level setscan be established by the following result

Theorem 12 (third decomposition theorem) For everyAξ isin Fξ(U) Aξ cup αisinΩAξ

δisinΩAξ

Aξlowast(αδ)

Proof e proof is analogous to that of eorem 10e following example illustrates the algebraic fact stated

in first decomposition theorem

Example 1 Consider the ξ-CFS

Aξ

02e

i05π

m1+04e

iπ

m2+06e

i12π

m3+08e

i18π

m41113896 1113897 (20)

For α 02 and δ 05π ξ-CFS Aξlowast(αδ) with respect to

ξ-CFS Aξ is given by

Aξlowast(0205π)

02ei05π

m1+02e

i05π

m2+02e

i05π

m3+02e

i05π

m41113896 1113897

(21)

Corresponding to α 04 and δ π we have

Aξlowast(04π)

0ei0

m1+04e

iπ

m2+04e

iπ

m3+04e

iπ

m41113896 1113897 (22)

Corresponding to α 06 and δ 12π

Aξlowast(0612π)

0ei0

m1+0e

i0

m2+06e

i12π

m3+06e

i12π

m41113896 1113897 (23)

Also for α 08 and δ 18π

Aξlowast(0818π)

0ei0

m1+0e

i0

m2+0e

i0

m3+08e

i18π

m41113896 1113897 (24)

Consequently Aξ cupAξlowast(αδ)

In the following example we apply the concept of ξ-CFSto judge the performance of an artist in an art competition

Example 2 Let X a b c d e f be the list of 10 artistscompeting in an art competition After the initial screeningbased on sketch designing the performance of each artist isgiven in Table 1

Table 1 Performance of all artists after initial screening

Artists Performance of the artistsrA(a) 02rA(b) 091rA(c) 05rA(d) 07rA(e) 04rA(f) 061rA(g) 072rA(h) 03rA(i) 08rA(j) 09

6 Complexity

e graphical interpretation of the above performance ofthe artists is displayed in Figure 1

Let ξ αeiβ be a parameter to select a candidate based onthe performance to draw a sketch of a healthy environmente judgment procedure has two phases α and β where αdenotes the level of roughness of the drawing and β denotesthe use of inappropriate color in the drawing Table 2 in-dicates the performance of the artists after the first phase forα 07

e graphical interpretation of the qualified artists ofstage one is presented in Figure 2

Table 3 indicates the performances of the qualified artistsfor phase two

e graphical interpretation of the artists of stage two ispresented in Figure 3

e following outcomes indicate the performance of thequalified artists after phase two for β 05π areωAξ(a) 05πa dm ωAξ(h) 04π and final score is μAξ(a)

02ei05π and μAξ(h) 03ei04π At this stage we will use thescore function to compare the performance of the artists Forthis we may take |a| 02276 and |h| 02998 e aboveinformation shows that artist ldquohrdquo may be considered the bestof all the artists

4 Algebraic Attributes of ξ-ComplexFuzzy Subgroups

In this section we innovate the notion of ξ-CFSG defined onξ-CFS and establish fundamental algebraic characteristics ofthis phenomenon

Definition 15 A homogeneous ξ-CFS Aξ of a group G iscalled ξ-complex fuzzy subgroup (ξ-CFSG) if Aξ admits thefollowing conditions

(1) μAξ(mn)gemin μAξ(m) μAξ(n)1113864 1113865

(2) μAξ(mminus 1)ge μAξ(m) forallm n isin G

e family of all ξ-CFSG defined on the group G isdenoted by Fξ(G)

Proposition 2 Each ξ-CFSG (G) Aξ satisfies the followingproperties

(1) μAξ(m)le μAξ(e)

(2) μAξ(mnminus 1) μAξ(e)⟹ μAξ(m) μAξ(n)forallm n isin G

Proof

(1) Let m isin G then

μAξ(e) μAξ mmminus 1

1113872 1113873

gemin μAξ(m) μAξ mminus 1

1113872 11138731113872 1113873

μAξ(m)

(25)

(2) Let m n isin G then

μAξ(m) μAξ mnminus 1

n1113872 1113873

gemin μAξ mnminus 1

1113872 1113873 μAξ(n)1113966 1113967

min μAξ(e) μAξ(n)1113864 1113865

μAξ(n)

(26)

In the following result we investigate the conditionunder which a given CFS is ξ-CFSG

Proposition 3 Let A be a CFS (G) such that μA(mminus 1)

μA(m) forallm isin GMoreover ξ le q where q inf μA(m) m isin G1113864 1113865 lten Aξ

is ξ-CFSG (G)

Proof By using the given conditions for any m isin G weobtain μA(m)ge ξ e application of Definition 7 in theabove inequality yields that μAξ(m) ξ ereforeμAξ(mn) min μA(mn) ξ1113864 1113865 and μAξ(mn)gemin μAξ1113864

(m) μAξ(n) for all m n isin G Moreover by using the givencondition μA(mminus 1) μA(m) we get μAξ(mminus 1) μAξ(m)

Perfo

rman

ce o

f the

artis

ts

0010203040506070809

1

b c d e f g h i jaArtists

Figure 1 A graphical overview of Table 1

Table 2 Performance of the artists after the first phase

Artists Roughness of the sketchrAξ (a) 02rAξ (c) 05rAξ (d) 07rAξ (e) 04rAξ (f) 061rAξ (h) 03

Complexity 7

e following result shows that every CFSG is alwaysξ-CFSG

Proposition 4 Every CFSG A is ξ-CFSG of a group G

Proof By using Definition 7 for all m n isin G we haveμAξ(mn) min μA(mn) ξ1113864 1113865 e application of Definition13 in the above relation gives us μAξ(mn)geminμAξ(m) μAξ(n)1113864 1113865

Moreover

μAξ mminus 1

1113872 1113873 min μA mminus 1

1113872 1113873 ξ1113966 1113967

gemin μA(m) ξ1113864 1113865

μAξ(m)

(27)

Hence A is a ξ-CFSG (G)

Remark 2 e converse of Proposition 4 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 3 eCFS A defined on a G 1 minus1 i minus i is givenas

A(m) 02e

iπ

1+04e

iπ

minus1+04e

i12π

minusi+03e

i09π

i1113896 1113897 (28)

e ξ-CFSG (G) corresponding to the value ξ 01ei05π

is given by

Aξ(m)

01ei05π

1+01e

i05π

minus1+01e

i05π

minusi+01e

i05π

i1113896 1113897

(29)

Moreover A is not CFSG (G) as A does not satisfyDefinition 4

e following result indicates that intersection of anytwo ξ-CFSG is also ξ-CFSG

Proposition 5 For any two Aξ Bξ isin Fξ(G) (AcapB)ξ Aξ

capBξ

Proof By using Proposition 1 for any two elementsm n isin G

μ(AcapB)ξ(mn) μAξcapBξ(mn)

min μAξ(mn) μBξ(mn)1113864 1113865(30)

e application of Definition 13 in the above relationgives that

μ(A capΒ)ξ(mn) min μ

(AcapΒ)ξ(m) μ(AcapΒ)ξ(n)1113966 1113967 (31)

Moreover

c d e f haArtists

0

01

02

03

04

05

06

07

08

Roug

hnes

s of t

he sk

etch

Figure 2 A graphical interpretation of Table 2

Table 3 Performance of the artists after the second phase

Artists Performance of the artistsωA(a) 05πωA(c) 06πωA(d) 15πωA(e) 06πωA(f) 07πωA(h) 04π

8 Complexity

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

e graphical interpretation of the above performance ofthe artists is displayed in Figure 1

Let ξ αeiβ be a parameter to select a candidate based onthe performance to draw a sketch of a healthy environmente judgment procedure has two phases α and β where αdenotes the level of roughness of the drawing and β denotesthe use of inappropriate color in the drawing Table 2 in-dicates the performance of the artists after the first phase forα 07

e graphical interpretation of the qualified artists ofstage one is presented in Figure 2

Table 3 indicates the performances of the qualified artistsfor phase two

e graphical interpretation of the artists of stage two ispresented in Figure 3

e following outcomes indicate the performance of thequalified artists after phase two for β 05π areωAξ(a) 05πa dm ωAξ(h) 04π and final score is μAξ(a)

02ei05π and μAξ(h) 03ei04π At this stage we will use thescore function to compare the performance of the artists Forthis we may take |a| 02276 and |h| 02998 e aboveinformation shows that artist ldquohrdquo may be considered the bestof all the artists

4 Algebraic Attributes of ξ-ComplexFuzzy Subgroups

In this section we innovate the notion of ξ-CFSG defined onξ-CFS and establish fundamental algebraic characteristics ofthis phenomenon

Definition 15 A homogeneous ξ-CFS Aξ of a group G iscalled ξ-complex fuzzy subgroup (ξ-CFSG) if Aξ admits thefollowing conditions

(1) μAξ(mn)gemin μAξ(m) μAξ(n)1113864 1113865

(2) μAξ(mminus 1)ge μAξ(m) forallm n isin G

e family of all ξ-CFSG defined on the group G isdenoted by Fξ(G)

Proposition 2 Each ξ-CFSG (G) Aξ satisfies the followingproperties

(1) μAξ(m)le μAξ(e)

(2) μAξ(mnminus 1) μAξ(e)⟹ μAξ(m) μAξ(n)forallm n isin G

Proof

(1) Let m isin G then

μAξ(e) μAξ mmminus 1

1113872 1113873

gemin μAξ(m) μAξ mminus 1

1113872 11138731113872 1113873

μAξ(m)

(25)

(2) Let m n isin G then

μAξ(m) μAξ mnminus 1

n1113872 1113873

gemin μAξ mnminus 1

1113872 1113873 μAξ(n)1113966 1113967

min μAξ(e) μAξ(n)1113864 1113865

μAξ(n)

(26)

In the following result we investigate the conditionunder which a given CFS is ξ-CFSG

Proposition 3 Let A be a CFS (G) such that μA(mminus 1)

μA(m) forallm isin GMoreover ξ le q where q inf μA(m) m isin G1113864 1113865 lten Aξ

is ξ-CFSG (G)

Proof By using the given conditions for any m isin G weobtain μA(m)ge ξ e application of Definition 7 in theabove inequality yields that μAξ(m) ξ ereforeμAξ(mn) min μA(mn) ξ1113864 1113865 and μAξ(mn)gemin μAξ1113864

(m) μAξ(n) for all m n isin G Moreover by using the givencondition μA(mminus 1) μA(m) we get μAξ(mminus 1) μAξ(m)

Perfo

rman

ce o

f the

artis

ts

0010203040506070809

1

b c d e f g h i jaArtists

Figure 1 A graphical overview of Table 1

Table 2 Performance of the artists after the first phase

Artists Roughness of the sketchrAξ (a) 02rAξ (c) 05rAξ (d) 07rAξ (e) 04rAξ (f) 061rAξ (h) 03

Complexity 7

e following result shows that every CFSG is alwaysξ-CFSG

Proposition 4 Every CFSG A is ξ-CFSG of a group G

Proof By using Definition 7 for all m n isin G we haveμAξ(mn) min μA(mn) ξ1113864 1113865 e application of Definition13 in the above relation gives us μAξ(mn)geminμAξ(m) μAξ(n)1113864 1113865

Moreover

μAξ mminus 1

1113872 1113873 min μA mminus 1

1113872 1113873 ξ1113966 1113967

gemin μA(m) ξ1113864 1113865

μAξ(m)

(27)

Hence A is a ξ-CFSG (G)

Remark 2 e converse of Proposition 4 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 3 eCFS A defined on a G 1 minus1 i minus i is givenas

A(m) 02e

iπ

1+04e

iπ

minus1+04e

i12π

minusi+03e

i09π

i1113896 1113897 (28)

e ξ-CFSG (G) corresponding to the value ξ 01ei05π

is given by

Aξ(m)

01ei05π

1+01e

i05π

minus1+01e

i05π

minusi+01e

i05π

i1113896 1113897

(29)

Moreover A is not CFSG (G) as A does not satisfyDefinition 4

e following result indicates that intersection of anytwo ξ-CFSG is also ξ-CFSG

Proposition 5 For any two Aξ Bξ isin Fξ(G) (AcapB)ξ Aξ

capBξ

Proof By using Proposition 1 for any two elementsm n isin G

μ(AcapB)ξ(mn) μAξcapBξ(mn)

min μAξ(mn) μBξ(mn)1113864 1113865(30)

e application of Definition 13 in the above relationgives that

μ(A capΒ)ξ(mn) min μ

(AcapΒ)ξ(m) μ(AcapΒ)ξ(n)1113966 1113967 (31)

Moreover

c d e f haArtists

0

01

02

03

04

05

06

07

08

Roug

hnes

s of t

he sk

etch

Figure 2 A graphical interpretation of Table 2

Table 3 Performance of the artists after the second phase

Artists Performance of the artistsωA(a) 05πωA(c) 06πωA(d) 15πωA(e) 06πωA(f) 07πωA(h) 04π

8 Complexity

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

e following result shows that every CFSG is alwaysξ-CFSG

Proposition 4 Every CFSG A is ξ-CFSG of a group G

Proof By using Definition 7 for all m n isin G we haveμAξ(mn) min μA(mn) ξ1113864 1113865 e application of Definition13 in the above relation gives us μAξ(mn)geminμAξ(m) μAξ(n)1113864 1113865

Moreover

μAξ mminus 1

1113872 1113873 min μA mminus 1

1113872 1113873 ξ1113966 1113967

gemin μA(m) ξ1113864 1113865

μAξ(m)

(27)

Hence A is a ξ-CFSG (G)

Remark 2 e converse of Proposition 4 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 3 eCFS A defined on a G 1 minus1 i minus i is givenas

A(m) 02e

iπ

1+04e

iπ

minus1+04e

i12π

minusi+03e

i09π

i1113896 1113897 (28)

e ξ-CFSG (G) corresponding to the value ξ 01ei05π

is given by

Aξ(m)

01ei05π

1+01e

i05π

minus1+01e

i05π

minusi+01e

i05π

i1113896 1113897

(29)

Moreover A is not CFSG (G) as A does not satisfyDefinition 4

e following result indicates that intersection of anytwo ξ-CFSG is also ξ-CFSG

Proposition 5 For any two Aξ Bξ isin Fξ(G) (AcapB)ξ Aξ

capBξ

Proof By using Proposition 1 for any two elementsm n isin G

μ(AcapB)ξ(mn) μAξcapBξ(mn)

min μAξ(mn) μBξ(mn)1113864 1113865(30)

e application of Definition 13 in the above relationgives that

μ(A capΒ)ξ(mn) min μ

(AcapΒ)ξ(m) μ(AcapΒ)ξ(n)1113966 1113967 (31)

Moreover

c d e f haArtists

0

01

02

03

04

05

06

07

08

Roug

hnes

s of t

he sk

etch

Figure 2 A graphical interpretation of Table 2

Table 3 Performance of the artists after the second phase

Artists Performance of the artistsωA(a) 05πωA(c) 06πωA(d) 15πωA(e) 06πωA(f) 07πωA(h) 04π

8 Complexity

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

μ(AcapΒ)ξ m

minus 11113872 1113873 μAξcapΒξ m

minus 11113872 1113873

gemin μAξ(m) μΒξ(m)1113864 1113865

μ(AcapΒ)ξ(m)

(32)

is concludes the proof

Theorem 13 Aξ is a ξ-CFSG (G) if and only if Aξprime is aξ-CFSG (G)

Proof Let Aξ be a ξ-CFSG en

μAξprime(mn) 1 minus rAξ(mn)e

i 2πminusωAξ (mn)( ) (33)

By using Definition 13 we obtain

μAξprime(mn)ge 1 minus min rAξ(m) rAξ(n)1113864 1113865e

i 2πminusmin ωAξ(m)ω

Aξ(n) ( )

max 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimax 2πminusω

Aξ(m)( ) 2πminusωAξ(n)( )

gemin 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865eimin 2πminusω

Aξ(m)( ) 2πminusωAξ (n)( )

(34)

By using Definition 8 (3) in the above relation we getμ

Aξprime(mn) min μAξprime(m) μ

Aξprime (n)1113966 1113967Moreover

μAξprime m

minus 11113872 1113873 1 minus rAξ m

minus 11113872 1113873e

i 2πminusωAξ mminus1( )( )

1 minus rAξ(m)ei 2πminusω

Aξ (m)( )

μAξprime (m)

(35)

Conversely let Aξprime be a ξ-CFSG Assume that

μAξ(mn) rAξ(mn)eiω

Aξ (mn)

1 minus 1 minus rAξ(mn)ei 2πminus 2πminusω

Aξ (mn)( )( )1113874 1113875

ge 1 minus min 1 minus rAξ(m)( 1113857 1 minus rAξ(n)( 11138571113864 1113865ei2πminusmin 2πminusω

Aξ (m)( ) 2πminusωAξ (n)( )

max rAξ(m) rAξ(n)( 11138571113864 1113865eimax ω

Aξ (m)ωAξ (n)

gemin rAξ(m) rAξ(n)( 11138571113864 1113865eimin ω

Aξ (m)ωAξ (n)

μAξ(mn) min μAξ(m) μAξ(n)1113864 1113865

(36)

Moreover

Perfo

rman

ce o

f the

artis

ts

c d e f haArtists

002040608

1121416

Figure 3 A graphical interpretation of Table 3

Complexity 9

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

μAξ mminus 1

1113872 1113873 1 minus 1 minus rAξ mminus 1

1113872 1113873ei2πminus 2πminusω

Aξ mminus1( )( )1113874 1113875 rAξ(m)eiω

Aξ (m) μAξ(m) (37)

Definition 16 Let Aξ isin Fξ(G) α isin [0 1] and δ isin [0 2π]en the subgroup A

ξ(αδ)

with rAξ(e)ge α and ωAξ(e)ge δ is

called the level subgroup of ξ-CFSG Aξ In the subsequent result we establish necessary and

sufficient condition for an ξ-CFS to be ξ-CFSG

Theorem 14 A ξ-CFS Aξ of G is a ξ-CFSG (G) if and only ifeach of its level set ΩAξ with rAξ(e)ge α and ω

Aξ(e)ge δ is asubgroup of G

Proof Suppose ΩAξ with rAξ(e)ge α and ωAξ(e)ge δ is a

subgroup of G Assume that rAξ(m) α rAξ(n) α1ωAξ(m) δ and ωAξ(n) δ1 for any elements m n isin G Byusing Definition 10 in the above relations we have m isin ΩAξ

and n isin ΩAξ1 By applying eorem 1 (2) for αlt α1 and

δ lt δ1 we have n isin ΩAξ since ΩAξ is a subgroup of Gerefore mn isin ΩAξ It shows that rAξ(mn)geminrAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)ωAξ1113864

(n) δIn view of Definition 10 we have

rAξ mminus 1

1113872 1113873ge α rAξ(m)

ωAξ mminus 1

1113872 1113873ge δ ωAξ(m)(38)

Consequently Aξ is a ξ CFSG (G) Conversely supposeAξ isin Fξ(G) Let ΩAξ be an arbitrary level subgroup of Aξ Obviously ΩAξ is nonempty as e isin ΩAξ where e is theidentity element of G For any elements m n isin ΩAξ andusing the fact that Aξ isin Fξ(G) we have rAξ(mn)gemin rAξ(m) rAξ(n)1113864 1113865 α and ωAξ(mn)gemin ωAξ(m)1113864

ωAξ(n) δ It follows that mn isin ΩAξ Moreover for anyelement m isin Aξ and using the fact that Aξ isin Fξ(G) we haverAξ(mminus1)ge rAξ(m) α andωAξ(mminus1)geωAξ(m) δ ere-fore mminus1 isin ΩAξ implying that ΩAξ is a subgroup of G

Definition 17 Let Aξ be a ξ-CFSG (G) and m isin G en theξ-complex fuzzy left coset of Aξ in G is represented bymAξand is given by mAξ(g) min μA(mminus 1g) ξ1113864 11138651113864

g isin G μAξ(mminus 1g)Similarly one can define the ξ-complex fuzzy right coset

of Aξ in G

Definition 18 A ξ-CFSG Aξ of a group G is ξ-complex fuzzynormal subgroup (ξ-CFNSG) of G if mAξ Aξm forallm isin G

e following result illustrates another characteristic ofξ-CFNSG

Proposition 6 Every ξ-CFNSG Aξ admits the followingproperty

μAξ(mn) μAξ(nm) for allm n isin G (39)

Proof By using Definition 16 we have mAξ Aξmforallm isin G

By using Definition 15 the above equation gives that

mAξ

1113872 1113873nminus 1

Aξm1113872 1113873n

minus 1foralln isin G

μAξ(nm)minus 1

μAξ(mn)minus 1

(40)

is shows that μAξ(nm) μAξ(mn) In the followingconsequence we explore the condition under which anξ-CFSG is ξ-CFNSG (G)

Proposition 7 For any Aξ isin Fξ(G) with ξ le q whereq Inf μA(m) m isin G1113864 1113865 then Aξ is a ξ-CFNSG (G)

Proof By using the given condition for any m isin G we haveμA(m)ge ξ e application of Definition 7 in the aboveinequality yields that μAξ(m) ξ

erefore

Aξm(g) mA

ξ(g) for allg isin G (41)

Hence

Aξm mA

ξ for allm isin G (42)

e following result shows that every CFNSG (G) isξ-CFNSG (G)

Proposition 8 Every CFNSG (G) A is ξ-CFNSG (G)

Proof By using Definition 6 for element m isin G we havemA(g) Am(g) By applying Definition 5 the above re-lation gives that μA(mminus 1g) μA(gmminus 1) So min μA1113864

(mminus 1g) ξ min μA(gmminus 1) ξ1113864 1113865 implying that mAξ(g)

Aξm(g) Consequently mAξ Aξm

Remark 3 e converse of Proposition 8 does not hold ingeneral is algebraic fact may be viewed in the followingexample

Example 4 e CFNSG Adefined on a group G langa b

a2 b2 (ab)2 1 ba a2brang is given by

A(m) 1e

i2π

1+09e

i18π

a+07e

i17π

a2 +

05ei16π

b+04e

iπ

ab+03e

i07π

a2b

1113896 1113897

(43)

e ξ-CFNSG (G) corresponding to the valueξ 03ei07π is given by

Aξ(m)

03ei07π

1+03e

i07π

a+03e

i07π

a2 +

03ei07π

b+03e

i07π

ab+03e

i07π

a2b

1113896 1113897 (44)

10 Complexity

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

Moreover A is not CFNSG (G) because

μA(ab) 04eiπ ne 03e

i07π μA(ba) (45)

Theorem 15 For any two ξ-CFNSG Aξ and Bξ (AcapB)ξ Aξ capBξ

Proof By using Proposition 1 for any element m isin G wehave

μ(AcapB)ξ

gmminus 1

1113872 1113873 μ Aξ capBξ( ) gmminus 1

1113872 1113873

min μAξ gmminus 1

1113872 1113873 μBξ gmminus 1

1113872 11138731113966 1113967(46)

e application of Definition 16 in the above relation isgiven as

μ(AcapB)ξ

gmminus 1

1113872 1113873 min μAξ mminus 1

g1113872 1113873 μBξ mminus 1

g1113872 11138731113966 1113967

μ Aξ capBξ( ) mminus 1

g1113872 1113873(47)

us μ(AcapB)ξ

(mminus 1g) μ(AcapB)ξ

(gmminus 1)

Theorem 16 Aξ is a ξ-CFNSG if and only if Aξprime is aξ-CFNSG

Proof Let Aξbe a ξ-CFNSG en

mAξprime

μAξprime m

minus 1g1113872 1113873

1 minus rAξ mminus 1

g1113872 1113873ei 2πminusω

Aξ mminus1g( )( )(48)

By using Definition 16 we obtain

mAξprime

1 minus rAξ gmminus 1

1113872 1113873ei 2πminusω

Aξ gmminus1( )( )

μAξprime gm

minus 11113872 1113873

(49)

Hence mAξprime AξprimemConversely let Aξprime be a ξ-CFNSG Assume that

mAξ

μAξ mminus 1

g1113872 1113873

rAξ mminus 1

g1113872 1113873eiω

Aξ mminus1g( )

1 minus 1 minus rAξ mminus 1

g1113872 1113873ei 2πminus 2πminusω

Aξ mminus1g( )( )( )1113874 1113875

1 minus μAξprime m

minus 1g1113872 1113873

1 minus μAξprime gm

minus 11113872 1113873

μAξ gmminus 1

1113872 1113873

(50)

us mAξ Aξm

Proposition 9 Let Aξbe a ξ-CFSG (G) lten the set GAξ

m isin G μAξ(m) μAξ(e)1113864 1113865 is a normal subgroup of G

Proof Obviously GAξ neempty as e isin GAξ By applying Defini-tion 13 for any two elements mn isin G we have

μAξ mnminus 1

1113872 1113873gemin μAξ(m) μAξ nminus 1

1113872 11138731113966 1113967

μAξ(e)(51)

is shows that μAξ(mnminus 1)ge μAξ(e) but μAξ(mnminus 1)leμAξ(e) Consequently mnminus 1 isin GAξ Furthermore in view ofDefinition 16 for any element m isin GAξ and g isin G we obtain

μAξ gminus 1

mg1113872 1113873 μAξ(m)

μAξ(e)(52)

is implies that gminus 1mg isin GAξ Hence GAξ is a normalsubgroup of G

Proposition 10 Every ξ-CFNSG satisfies the followingrelation

If mAξ uAξ and nAξ vAξ then mnAξ uvAξ

Proof Since mAξ uAξ and nAξ vAξ thereforemminus 1u nminus 1v isin GAξ

Consider

(mn)minus 1

(uv) nminus 1

mminus 1

u1113872 1113873v nminus 1

mminus 1

u1113872 1113873 nnminus 1

1113872 1113873v

nminus 1

mminus 1

u1113872 1113873n1113960 1113961 nminus 1

v1113872 1113873 isin GAξ (53)

It follows that (mn)minus 1(uv) isin GAξ ConsequentlymnAξ uvAξ

Proposition 11 Every ξ CFNSG Aξ admits the followingcharacteristics

(1) mAξ nAξ if and only if mminus 1n isin GAξ

(2) Aξm Aξn if and only if mnminus 1 isin GAξ

Proof

(1) Let mAξ nAξand m n isin GAξ en by applyingDefinition 7 we have

μAξ mminus 1

n1113872 1113873 min μA mminus 1

n1113872 1113873 ξ1113966 1113967

mAξ(n)

(54)

By using the given condition in the above equations weobtain

μAξ mminus 1

n1113872 1113873 nAξ(n)

min μA nminus 1

n1113872 1113873 ξ1113966 1113967

min μA(e) ξ1113864 1113865

(55)

So μAξ(mminus 1n) μAξ(e) implying that mminus 1n isin GAξ Conversely suppose that mminus 1n isin GAξ is implies thatμAξ(mminus 1n) μAξ(e) By applying Definition 16 for any el-ement z isin G we have

Complexity 11

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

mAξ(z) min μA m

minus 1z1113872 1113873 ξ1113966 1113967

μAξ mminus 1

z1113872 1113873

μAξ mminus 1

n1113872 1113873 nminus 1

z1113872 11138731113872 1113873

(56)

By using Definition 13 in the above equation we obtain

mAξ(z) min μAξ(e) μAξ n

minus 1z1113872 11138731113966 1113967

μAξ nminus 1

z1113872 1113873(57)

Consequently mAξ(z) (nAξ)(z) e remaining partcan be proved as the first part

Definition 19 For any ξ-CFNSG Aξ of G we define the set ofall ξ-complex fuzzy left cosets of G by Aξ asGAξ mAξ m isin G1113966 1113967 is set forms a group under thefollowing binary operation (mAξ)(nAξ) mnAξ isparticular quotient group is called quotient group of G byξ-CFNSG Aξ

In the following result we establish a natural epi-morphism between group and its quotient group defined inDefinition 17

Theorem 17 For any ξ-CFNSG Aξ of G there exist a naturalepimorphism φ G⟶ GAξ defined by m⟶ mAξ m isin G

with kerφ GAξ

Proof e surjectivity of the function φ is quite obviousMoreover for any elements m n isin G we haveφ(mn) mAξnAξ φ(m)φ(n) erefore φ is an epi-morphism Moreover obvious φ is surjective Now

kerφ m isin G φ(m) eAξ

1113966 1113967

m isin G mAξ

eAξ

1113966 1113967

m isin G meminus 1 isin GAξ1113966 1113967

GAξ

(58)

In the following result we establish an isomorphiccorrespondence between quotient group of G by ξ-CFNSGAξ and quotient group G by GAξ

Theorem 18 Let Aξ be ξ-CFNSG and GAξ be normal sub-group of G lten there exist an isomorphism between GAξ

and GGAξ

Proof Define a mapping φ GAξ⟶ GGAξ asφ(mAξ) mGAξ For any xAξ yAξ isin GAξ we have

φ mAξnA

ξ1113872 1113873 φ mnA

ξ1113872 1113873

mnGAξ

mGAξnGAξ

φ mAξ

1113872 1113873φ nAξ

1113872 1113873

(59)

is shows that φ is homomorphism

Moreover for any mAξ nAξ isin GAξ we have

φ mAξ

1113872 1113873 φ nAξ

1113872 1113873

mGAξ nGAξ

nminus 1

mGAξ GAξ

(60)

which shows that nminus 1mGAξ isin GAξ By applyingeorem 16 in the above relation yields that

mAξ nAξ Moreover the surjective case is quite obviousConsequently φ is an isomorphism between GAξ andGGAξ

5 Conclusion

In this paper we first present the ξ-CFS which is completelya new notion We have utilized this phenomena to define the(α δ)-cut sets and strong (α δ)-cut sets and have proved therepresentation of an ξ-CFS in the framework of these setsMoreover the notions of ξ-CFSG and level subgroups ofthese groups have also been defined in this article In ad-dition a necessary and sufficient condition for an ξ-CFS tobe a ξ-CFSG has also been investigated Moreover an iso-morphism has been established between the quotient groupsof a group G by its ξ-CFNSG and a normal subgroup GAξ

Data Availability

Any type of data associated with this work can be obtainedfrom the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Research Center of theCenter for Female Scientific andMedical Colleges Deanshipof Scientific Research King Saud University

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of MathematicalAnalysis and Applications vol 35 no 3 pp 512ndash517 1971

[3] N Mukherjee and P Bhattacharya ldquoFuzzy normal subgroupsand fuzzy cosetsrdquo Information Sciences vol 34 no 3pp 225ndash239 1984

[4] A S Mashour H Ghanim and F I Sidky ldquoNormal fuzzysubgroupsrdquo Information Sciences vol 20 pp 53ndash59 1990

[5] N Ajmal and I Jahan ldquoA study of normal fuzzy subgroupsand characteristic fuzzy subgroups of a fuzzy grouprdquo FuzzyInformation and Engineering vol 4 no 2 pp 123ndash143 2012

[6] S Abdullah M Aslam T A Khan and M Naeem ldquoA newtype of fuzzy normal subgroups and fuzzy cosetsrdquo Journal ofIntelligent amp Fuzzy Systems vol 25 no 1 pp 37ndash47 2013

[7] M Tarnauceanu ldquoClassifying fuzzy normal subgroups of fi-nite groupsrdquo Iranian Journal of Fuzzy Systems vol 12pp 107ndash115 2015

12 Complexity

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13

[8] P S Das ldquoFuzzy groups and level subgroupsrdquo Journal ofMathematical Analysis and Applications vol 84 no 1pp 264ndash269 1981

[9] X H Yuan and H X Li ldquoCut sets on interval-valuedintuitionistic fuzzy setsrdquo Fuzzy Systems and Knowledge Dis-covery vol 6 pp 167ndash171 2009

[10] X H Yuan H Li and E S Lee ldquoree new cut sets of fuzzysets and new theories of fuzzy setsrdquo Computers amp Mathe-matics with Applications vol 57 no 2 pp 691ndash701 2009

[11] W Fengxia Z Cheng Y Xuehai and X Zunquan ldquoA theoryof fuzzy sets based on cut set with parametersrdquo InternationalJournal of Pure and Applied Mathematicsvol 106 no 2pp 355ndash364 2016

[12] H Aktas and N Cagman ldquoGeneralized product of fuzzysubgroups and t-level subgroupsrdquo Mathematical Communi-cations vol 11 no 2 pp 121ndash128 2006

[13] J J Buckley ldquoFuzzy complex numbersrdquo Fuzzy Sets andSystem vol 33 pp 333ndash345 1989

[14] J J Buckley and Y Qu ldquoFuzzy complex analysis I differ-entiationrdquo Fuzzy Sets and Systems vol 41 no 3 pp 269ndash2841991

[15] J J Buckley ldquoFuzzy complex analysis II integrationrdquo FuzzySets and Systems vol 49 no 2 pp 171ndash179 1992

[16] G Zhang ldquoFuzzy limit theory of fuzzy complex numbersrdquoFuzzy Sets amp Systems vol 46 pp 227ndash235 1992

[17] G Ascia V Catania and M Russo ldquoVLSI hardware archi-tecture for complex fuzzy systemsrdquo IEEE Transactions onFuzzy Systems vol 7 no 5 pp 553ndash570 1999

[18] D Ramot R Milo M Friedman and A Kandel ldquoComplexfuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol 10 no 2pp 171ndash186 2002

[19] D Ramot M Friedman G Langholz and A KandelldquoComplex fuzzy logicrdquo IEEE Transactions on Fuzzy Systemsvol 11 no 4 pp 450ndash461 2003

[20] Z Chen S Aghakhani J Man and S Dick ldquoANCFIS aneurofuzzy architecture employing complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 19 no 2 pp 305ndash3222011

[21] X Fu and Q Shen ldquoFuzzy complex numbers and their ap-plication for classifiers performance evaluationrdquo PatternRecognition vol 44 no 7 pp 1403ndash1417 2011

[22] D E Tamir and A Kandel ldquoAxiomatic theory of complexfuzzy logic and complex fuzzy classesrdquo International Journalof Computers Communications amp Control vol 6 no 3pp 562ndash576 2011

[23] J Ma G Zhang and J Lu ldquoA method for multiple periodicfactor prediction problems using complex fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 20 pp 32ndash45 2012

[24] C Li T Wu and F-T Chan ldquoSelf-learning complex neuro-fuzzy system with complex fuzzy sets and its application toadaptive image noise cancelingrdquo Neurocomputing vol 94pp 121ndash139 2012

[25] A U M Alkouri and A R Salleh ldquoLinguistic variable hedgesand several distances on complex fuzzy setsrdquo Journal of In-telligent amp Fuzzy Systems vol 26 no 5 pp 2527ndash2535 2014

[26] D E Tamir N D Rishe and A Kandel ldquoComplex fuzzy setsand complex fuzzy logic an overview of theory and appli-cationsrdquo Fifty Years of Fuzzy Logic and its Applicationsvol 26 pp 661ndash681 2015

[27] A Al-Husban and A R Salleh ldquoComplex fuzzy hypergroupsbased on complex fuzzy spacesrdquo International Journal of Pureand Applied Mathematics vol 107 pp 949ndash958 2016

[28] A Al-Husban and A R Salleh ldquoComplex fuzzy group basedon complex fuzzy spacerdquo Global Journal of Pure and AppliedMathematics vol 12 pp 1433ndash1450 2016

[29] R Nagarajan S SaleemAbdullah and K Balamurugan ldquoBriefdiscussions on T-level complex fuzzy subgrouprdquo IJAR vol 2pp 957ndash964 2016

[30] P irunavukarasu R Suresh and Pamilmani ldquoComplexneuro fuzzy system using complex fuzzy sets and update theparameters by PSO-GA and RLSE methodrdquo InternationalJournal of Pure and Applied Mathematics vol 3 no 1pp 117ndash122 2013

[31] R Al-Husban A R Salleh and A G B Ahmad ldquoComplexintuitionistic fuzzy normal subgrouprdquo International Journalof Pure and Applied Mathematics vol 115 no 3 pp 455ndash4662017

[32] T T Ngan L T H Lan M Ali et al ldquoLogic connectives ofcomplex fuzzy setsrdquo Romanian Journal of Information Scienceand Technology vol 21 pp 344ndash358 2018

[33] S Dai L Bi and B Hu ldquoDistance measures between theinterval-valued complex fuzzy setsrdquo Mathematics vol 7no 6 p 549 2019

[34] A Khan M Izhar and K Hila ldquoOn algebraic properties ofDFS sets and its application in decision making problemsrdquoJournal of Intelligent amp Fuzzy Systems vol 36 no 6pp 6265ndash6281 2019

[35] A Khan M Izhar and M M Khalaf ldquoGeneralised multi-fuzzy bipolar soft sets and its application in decision makingrdquoJournal of Intelligent amp Fuzzy Systems vol 37 no 2pp 2713ndash2725 2013

Complexity 13