Abstract
Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of -cut sets of -complex fuzzy sets and justify the representation of an -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an -complex fuzzy set is -complex fuzzy subgroup. Furthermore, we extend the idea of -complex fuzzy normal subgroup to define the quotient group of a group by this particular -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of by a specific normal subgroup .
1. Introduction
The competency of fuzzy logic to articulate steady adaptations from membership to nonmembership and the other way around has played an effective role to solve many physical problems. It provides us not only the powerful and meaningful representations of measuring uncertainty but also a useful approach to view the vague concepts expressed in the natural language. Despite all of the advantages of this logic, we still face immense complications to counter various physical situations based on a real-valued membership function. It is therefore quite necessary to propose an additional development of fuzzy set theory on account of set of complex numbers which is indeed an existing augmentation of real numbers. The complex fuzzy logic is linear augmentation of traditional fuzzy logic, which also allows a natural development of the difficulty based on the fuzzy logic that is impracticable to solve with superficial membership function. This particular set has a paramount part in numerous executions, in particular, advanced control systems and predicting of the periodic events, where multiple fuzzy variables are interrelated in a complex manner that cannot be effectively characterized by simple fuzzy operations. In addition, these sets are also used to solve several problems, especially the numerous periodic aspects and forecast problems. One of the far-reaching significances of studying the CFS is that they illustrate the data with uncertainty and periodicity in a much effective way.
Considering the inaccuracy in decision-making, Zadeh [1] popularized the concept of fuzzy sets for the first time, in 1965. Roenfeld [2] used Zadeh’s work to invent the approach of fuzzy subgroups in 1971. Mukherjee and Bhattacharya [3] initiated the study of the fuzzy cosets along with fuzzy normal subgroups in 1984. Mashour et al. [4] studied many important features of normal fuzzy subgroups. For more details about the recent development of fuzzy subgroups, we refer to [5–7]. In addition, another important aspect of fuzzy sets is the level sets of this notion. These level sets have an incredible significance as they set up a connection between concepts of crisp and fuzzy sets. This phenomenon is utilized to generalize various ideas and techniques for crisp set hypothesis as well as fuzzy sets. In the succession of fuzzification of subgroups, the idea of level subgroup was initiated by Das [8]. Yuan and Li [9] studied the impact of these level sets in the field of intuitionistic fuzzy sets. To see more on the level sets and level subgroups, we refer to [10–12]. Buckley [13] proposed the study of complex fuzzy numbers in 1989. The author [14] applied the theory of complex fuzzy numbers to set up new techniques of differentiation. Moreover, some fundamental properties of fuzzy counter integral in the complex plane were interpreted by the same author in [15]. Zhang [16] established many important properties of complex fuzzy numbers in 1992. Ascia et al. [17] designed a competent specific fuzzy processor which effectively deals with the complex fuzzy inference system. In [18], Ramote et al. launched the study of CFS in 2002. The two novel operations, namely, reflection and rotations, of these sets were introduced in [19]. Zhang et al. [20] formulated the -equalities of CFS. In 2011, Chen et al. [21] developed the adaptive neuro complex fuzzy inference system. In [22], Fu and Shen utilized CFS to design a new approach of linguistic evaluation of classifier performance. Tamir and Kandel [23] provided granted bases for first order estimation, complex class theory, and complex fuzzy logic in the same year. Ma et al. [24] defined product sum accruement operator for these particular sets in 2012. Li et al. [25] presented a new self-learning complex neuro fuzzy system using the same idea. In 2014, Alkouri and Salleh [26] illustrated many important multiple distance measures defined on CFS. In 2015, Tamir et al. [27] reintroduced the concept of CFS and complex fuzzy logic in a more logical way. Al-Husban and Salleh [28] interpreted the study of complex fuzzy hyper structure over complex fuzzy space in 2016. The phenomenon of CFSG over complex fuzzy space was discussed in [29]. The operator theory has an important role in modern mathematics because it is extensively used in fuzzy theory for approximate reasoning. The techniques of t-norms and t-co-norms with respect to complex fuzzy sets were presented by Nagarajan et al. [30]. For more on fuzzy set theory, we suggest reading of [31–35]. The competency of the complex fuzzy set has played an effective role to solve many physical problems. It provides us the meaningful representations of measuring uncertainty and periodicity. Despite all of these advantages, we still face vast complications to counter various physical situations based on a complex-valued membership function. This motivates us to define the notion of -complex fuzzy set (-CFS) through which one can have multiple options to investigate a specific real-world situation in much efficient way by choosing appropriate value of the parameter .
In this article, we propose the idea of -CFS as a powerful extension of classical fuzzy set and use this idea to define the notion of the cut sets of these particular sets. We explore the importance of these cut sets by proving the decomposition theorems for a -CFS. Moreover, we introduce the phenomenon of -complex fuzzy subgroup (-CFSG) over an -complex fuzzy set and investigate some of their fundamental algebraic attributes.
After a brief discussion about the historical background and significance of CFS, the rest of the article is organized as follows. Section 2 contains some definitions of the basic introduction of the concept of -CFS and the study of -cut sets and strong -cut sets of this newly defined notion. In addition, we establish the importance of defining these ideas by viewing an -CFS as a union of nested sequence of both -cut sets and strong -cut sets, respectively. Moreover, we apply -CFS to a physical situation in which one may select the most suitable performance of a participant. In Section 4, we utilize the concept of -CFS to propose the study of the idea of -CFSG and prove that each CFSG is -CFSG. Moreover, we extend the importance of these fuzzy subgroups by introducing the notions of -complex fuzzy cosets and -complex fuzzy normal subgroup (-CFNSG) and investigate their many important algebraic aspects.
2. Preliminaries
This section contains a brief review of the notion of CFS and related ideas, which are quite essential to understand the novelty of this article.
Definition 1. (see [19]). A CFS defined on a universe of discourse is characterized by a membership function that allocates each element of to a unit circle in complex plane and is written as , where denotes the real-valued function from to the closed unit interval and is a periodic function whose periodic law and principal period are and , respectively.
Note that and is the principal argument.
Definition 2. (see [8]). Let and . Then, the -cut set of CFS is denoted by and is defined as .
Definition 3. (see [31]). Let A and B be any two CFS of a universe . Then,(1) is homogeneous CFS if implies and vice versa (2) is homogeneous CFS with if implies and vice versa
Definition 4. (see [31]) A homogeneous CFS of is said to be a CFSG if and .
Definition 5. (see [5]). Let be a CFSG (G) and . Then, the complex fuzzy left coset of in is represented by and is given by .
Similarly, one can define complex fuzzy right coset of in .
Definition 6. (see [5]). A CFSG of a group is CFNSG (G) if .
3. Decomposition Theorems of Complex Fuzzy Sets
In this section, we initiate the idea of -CFS as a powerful extension of classical fuzzy sets. We also define the concepts of -cut sets and strong -cut sets of -CFS and establish fundamental properties of these phenomena. We also prove three decomposition theorems of -CFS.
Definition 7. Let A be a CFS of a universe and be an element of a unit circle with and . Then, the CFS is called the -complex fuzzy set (-CFS) with respect to CFS and is expressed as , .
The family of all -CFS defined on the universe is denoted by .
Definition 8. For any and ,(1)The union of -CFS and is denoted by and is defined as follows:(2)The intersection of -CFS and is denoted by and is defined as follows:(3)The complement of -CFS is denoted by and is described as follows:(4)The product of -CFS and is represented by and is expressed as follows:(5)Let be -CFS of a universe . Then, the Cartesian product of -CFS is represented by and is defined in the following way:where
Definition 9. Let and be any two -CFS of a universe . Then,(1) is homogeneous -CFS if implies and vice versa (2) is homogeneous -CFS with if implies and vice versa The next result illustrates that the intersection of any two -CFS is also -CFS.
Proposition 1. For any two -CFS and ,
Proof. Consider , .
By applying Definition 8 (2), we have .
Remark 1. The union of any two -CFS is also -CFS.
Definition 10. The -cut set of -CFS is represented by and is defined as follows:
Definition 11. For any strong -cut set of is defined as .
Definition 12. The level set of can be described as , where and .
Theorem 1. Let and be any two -CFS. Then, the following attributes hold for any and .(1)(2) implies (3)(4)(5)
Proof. (1)In view of Definition 10, for any element , and . It means that and . Thus, .(2)Let , and by applying Definition 10, we have . Therefore, (3)For any , we have and . This implies that and . Then, clearly, and . Therefore, , and ultimately, we obtain Let . By applying Definition 10, in the above relations, we obtain This shows that and . Consequently, From (7) and (9), the required equality holds.(4)For any we obtain and . Therefore, and . This means that and . Consequently, Now, suppose that ; then, or . By applying Definition 10 in the above relations, we get or . It further shows that and . Consequently, From (10) and (11), the required equality is satisfied.(5)Let , then implying that , .It follows that , which shows that .
Therefore, and henceNow, suppose ; then, .
This implies that , , , and .
It means that ; therefore,From (12) and (13), the required result is satisfied.
Theorem 2. Let and be any two -CFS. Then, the following attributes hold for all and (1) implies (2)(3)
Theorem 3. The following properties hold for any family of -CFS , .(1)(2)
Proof. (1)Let ; in view of Definition 10, we get and . The above relation holds only if and , that is, and . It follows that .(2)Let ; by using Definition 10, we obtain , .The above inequality holds only if , , that is, and . This implies that . Hence, . Now, suppose that . Again, by applying Definition 10, we get , . Then, obviously . Consequently, .
Theorem 4. Any family of CFS : admits the following properties:(1)(2)
Theorem 5. Any two -CFS and satisfy the following relations:(1) if and only if (2) if and only if
Proof. (1)Suppose . Assume that there exist and such that . It means that such that , but . Then, , , , and . Hence, and , which is contradiction to our supposition. Conversely, let . Consider , implying that , such that, . This further shows that and , which contradicts our assumption.(2)In a similar way, we can obtain the required equality.
Theorem 6. Any two -CFS and satisfy the following characteristics:(1) if and only if (2) if and only if
Theorem 7. Every -CFS satisfies the following relations:(1)(2)(3)(4)
Proof. (1)In view of Theorem 1 (2),for all and .
Next, suppose . Again, by applying Theorem 1 (2), we have and .
The application of the given condition in the above relation yields that and . This implies that .
Hence,From (14) and (15), the required equality holds. The remaining parts can be proved in a similar manner. In the following definitions, we present a new approach to define -CFS which is quite necessary to establish the proofs of decomposition theorems.
Definition 13. Let be a -cut set of . Then, the -CFS with respect to is defined as follows:
Definition 14. Let be a strong cut set of . Then, the -CFS with respect to can be described as follows:The subsequent result illustrates the decomposition of an -CFS as a union of -CFS .
Theorem 10. (first decomposition theorem). For every , .
Proof. Suppose for a particular ; then,Choose any and , then . Therefore, . On the contrary, for any choice of and , we have .
Therefore, , which implies that . The following result describes the decomposition of as a union of .
Theorem 11. (second decomposition theorem). For every , .
Proof. Suppose for a particular , thenThe decomposition of CFS as a union of level sets can be established by the following result.
Theorem 12. (third decomposition theorem). For every , .
Proof. The proof is analogous to that of Theorem 10.
The following example illustrates the algebraic fact stated in first decomposition theorem.
Example 1. Consider the -CFS:For and , -CFS with respect to -CFS is given byCorresponding to and , we haveCorresponding to and ,Also, for and ,Consequently, .
In the following example, we apply the concept of -CFS to 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 artists competing in an art competition. After the initial screening based on sketch designing, the performance of each artist is given in Table 1.
The graphical interpretation of the above performance of the artists is displayed in Figure 1.
Let be a parameter to select a candidate based on the performance to draw a sketch of a healthy environment. The judgment procedure has two phases and , where denotes the level of roughness of the drawing and denotes the use of inappropriate color in the drawing. Table 2 indicates the performance of the artists after the first phase for .
The graphical interpretation of the qualified artists of stage one is presented in Figure 2.
Table 3 indicates the performances of the qualified artists for phase two.
The graphical interpretation of the artists of stage two is presented in Figure 3.
The following outcomes indicate the performance of the qualified artists after phase two for are a dm and final score is and . At this stage, we will use the score function to compare the performance of the artists. For this, we may take and . The above information shows that artist “” may be considered the best of all the artists.
4. Algebraic Attributes of -Complex Fuzzy Subgroups
In this section, we innovate the notion of -CFSG defined on -CFS and establish fundamental algebraic characteristics of this phenomenon.
Definition 15. A homogeneous -CFS of a group is called -complex fuzzy subgroup (-CFSG) if admits the following conditions:(1)(2), The family of all -CFSG defined on the group is denoted by .
Proposition 2. Each -CFSG (G) satisfies the following properties:(1)(2),
Proof. (1)Let ; then,(2)Let ; then,In the following result, we investigate the condition under which a given CFS is -CFSG.
Proposition 3. Let be a CFS (G), such that .
Moreover, , where . Then, is -CFSG (G).
Proof. By using the given conditions for any , we obtain . The application of Definition 7 in the above inequality yields that . Therefore, and , for all . Moreover, by using the given condition , we get . The following result shows that every CFSG is always -CFSG.
Proposition 4. Every CFSG is -CFSG of a group .
Proof. By using Definition 7, for all , we have . The application of Definition 13 in the above relation gives us .
Moreover,Hence, is a -CFSG (G).
Remark 2. The converse of Proposition 4 does not hold in general. This algebraic fact may be viewed in the following example.
Example 3. The CFS defined on a is given asThe -CFSG (G) corresponding to the value is given byMoreover, is not CFSG (G) as does not satisfy Definition 4.
The following result indicates that intersection of any two -CFSG is also -CFSG.
Proposition 5. For any two , , .
Proof. By using Proposition 1, for any two elements ,The application of Definition 13 in the above relation gives thatMoreover,This concludes the proof.
Theorem 13. is a -CFSG (G) if and only if is a -CFSG (G).
Proof. Let be a -CFSG. Then,By using Definition 13, we obtainBy using Definition 8 (3) in the above relation, we get .
Moreover,Conversely, let be a -CFSG. Assume thatMoreover,
Definition 16. Let , , and . Then, the subgroup with and is called the level subgroup of -CFSG .
In the subsequent result, we establish necessary and sufficient condition for an -CFS to be -CFSG.
Theorem 14. A -CFS of is a -CFSG (G) if and only if each of its level set with and is a subgroup of .
Proof. Suppose with and is a subgroup of . Assume that , , , and , for any elements . By using Definition 10 in the above relations, we have and . By applying Theorem 1 (2) for and , we have , since is a subgroup of . Therefore, . It shows that and .
In view of Definition 10, we haveConsequently, is a CFSG (G). Conversely, suppose . Let be an arbitrary level subgroup of . Obviously, is nonempty as , where is the identity element of . For any elements and using the fact that , we have and . It follows that . Moreover, for any element and using the fact that , we have and. Therefore, implying that is a subgroup of .
Definition 17. Let be a -CFSG (G) and . Then, the -complex fuzzy left coset of in is represented by and is given by .
Similarly, one can define the -complex fuzzy right coset of in .
Definition 18. A -CFSG of a group is -complex fuzzy normal subgroup (-CFNSG) of if , .
The following result illustrates another characteristic of -CFNSG.
Proposition 6. Every -CFNSG admits the following property:
Proof. By using Definition 16, we have ,
By using Definition 15, the above equation gives thatThis shows that . In the following consequence, we explore the condition under which an -CFSG is -CFNSG (G).
Proposition 7. For any with , where , then is a -CFNSG (G).
Proof. By using the given condition for any , we have . The application of Definition 7 in the above inequality yields that .
Therefore,Hence,The following result shows that every CFNSG (G) is -CFNSG (G).
Proposition 8. Every CFNSG (G) is -CFNSG (G).
Proof. By using Definition 6 for element , we have . By applying Definition 5, the above relation gives that . So, implying that . Consequently, .
Remark 3. The converse of Proposition 8 does not hold in general. This algebraic fact may be viewed in the following example.
Example 4. The CFNSG defined on a group is given byThe -CFNSG (G) corresponding to the value is given byMoreover, is not CFNSG (G) because
Theorem 15. For any two -CFNSG and , .
Proof. By using Proposition 1 for any element , we haveThe application of Definition 16 in the above relation is given asThus, .
Theorem 16. is a -CFNSG if and only if is a -CFNSG.
Proof. Let be a -CFNSG. Then,By using Definition 16, we obtainHence, .
Conversely, let be a -CFNSG. Assume thatThus, .
Proposition 9. Let be a -CFSG (G). Then, the set is a normal subgroup of .
Proof. Obviously, as . By applying Definition 13 for any two elements , we haveThis shows that , but . Consequently, . Furthermore, in view of Definition 16 for any element and , we obtainThis implies that . Hence, is a normal subgroup of .
Proposition 10. Every -CFNSG satisfies the following relation;
If and , then .
Proof. Since and , therefore .
ConsiderIt follows that . Consequently, .
Proposition 11. Every CFNSG admits the following characteristics:(1) if and only if (2) if and only if
Proof. (1)Let and . Then, by applying Definition 7, we haveBy using the given condition in the above equations, we obtainSo, , implying that . Conversely, suppose that . This implies that . By applying Definition 16 for any element , we haveBy using Definition 13 in the above equation, we obtainConsequently, . The remaining part can be proved as the first part.
Definition 19. For any -CFNSG of , we define the set of all -complex fuzzy left cosets of by as . This set forms a group under the following binary operation . This particular quotient group is called quotient group of by -CFNSG .
In the following result, we establish a natural epimorphism between group and its quotient group defined in Definition 17.
Theorem 17. For any -CFNSG of , there exist a natural epimorphism , defined by with .
Proof. The surjectivity of the function is quite obvious. Moreover, for any elements , we have . Therefore, is an epimorphism. Moreover, obvious is surjective. Now,In the following result, we establish an isomorphic correspondence between quotient group of by -CFNSG and quotient group by .
Theorem 18. Let be -CFNSG and be normal subgroup of . Then, there exist an isomorphism between and .
Proof. Define a mapping as . For any , we haveThis shows that is homomorphism.
Moreover, for any , we havewhich shows that .
By applying Theorem 16 in the above relation yields that . Moreover, the surjective case is quite obvious. Consequently, is an isomorphism between and .
5. Conclusion
In this paper, we first present the -CFS which is completely a new notion. We have utilized this phenomena to define the -cut sets and strong -cut sets and have proved the representation of an -CFS in the framework of these sets. Moreover, the notions of -CFSG and level subgroups of these groups have also been defined in this article. In addition, a necessary and sufficient condition for an -CFS to be a -CFSG has also been investigated. Moreover, an isomorphism has been established between the quotient groups of a group by its -CFNSG and a normal subgroup .
Data Availability
Any type of data associated with this work can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Research Center of the Center for Female Scientific and Medical Colleges, Deanship of Scientific Research, King Saud University.