Abstract
We apply the classical theory of hyperrings to vague soft sets to derive the concepts of vague soft hyperrings, vague soft hyperideals, and vague soft hyperring homomorphism. The properties and structural characteristics of these concepts are also studied and discussed. Furthermore, the relationship between the concepts introduced here and the corresponding concepts in classical hyperring theory and soft hyperring theory is studied and investigated.
1. Introduction
Hyperstructure theory which is a generalization of the notion of classical algebraic structures was introduced by Marty in 1934 (see [1]). Since its inception, hyperstructure theory has seen tremendous development. Corsini (see [2]) introduced the concept of a hyperring and gave the definition of the most general form of hyperring, besides being responsible for introducing the notion of hyperring homomorphism. In 1994, Vougiouklis (see [3]) introduced another type of hyperrings called the -ring which is a generalization of the notion of hyperring introduced by [2]. Vougiouklis (see [3]) also introduced the notion of -subring and left and right -ideal of a -ring.
The theory of hyperstructures has also been actively applied to various mathematical theories such as the theory of fuzzy sets introduced by Zadeh in 1965 (see [4]), intuitionistic fuzzy theory introduced by Atanassov in 1986 (see [5]), and soft set theory introduced by Molodtsov in 1999 (see [6]), to produce several important concepts and theories. The study of fuzzy algebra began with the introduction of the notion of a fuzzy subgroup of a group by Rosenfeld (see [7]). Consequently, this led to the study of fuzzy hyperalgebra beginning with the introduction of the notion of a fuzzy subhyperring of a hyperring (resp., fuzzy -subrings) of a hyperring (resp., -ring) and fuzzy hyperideals (resp., fuzzy -ideals) of hyperrings (resp., -rings). The notion of fuzzy hyperideals introduced by Davvaz (see [8]) is a generalization of the concept of Liu’s (see [9]) definition of a fuzzy ideal of a ring.
Soft set theory is a general mathematical tool which was designed to deal with uncertainties and is widely acknowledged as an effective alternative to the classical mathematical tools that were used to deal with uncertainties and imprecision. Prior to the introduction of soft set theory, mathematical theories such as the theory of fuzzy sets, rough sets, and probability theory were used as tools to deal with uncertainties, imprecision, and vagueness. However, soft set theory has been proven to be a superior alternative to these classical theories as it is free from the inherent difficulties that affect these classical mathematical tools. Soft set theory has been successfully applied in various areas of mathematics such as in classical algebra and hyperalgebra, fuzzy algebra, and fuzzy hyperalgebra to develop various algebraic structures. However, an inherent weakness of soft set theory is that it is difficult to be used to represent the vagueness of problem parameters, particularly in problem-solving and decision making contexts. To overcome this disadvantage, various generalizations of soft sets such as fuzzy soft sets, vague soft sets, and intuitionistic fuzzy soft sets were introduced as better alternatives to the concept of soft sets. In the context of this paper, the notion of vague soft sets is used as a base to develop the initial theory of vague soft hyperrings.
In this paper, we develop the initial theory of vague soft hyperrings in Rosenfeld’s sense as a continuation to the theory of vague soft hypergroups introduced in [10]. Subsequently some of the fundamental properties and structural characteristics of these concepts are studied and discussed. Lastly, we prove that there exists a one-to-one correspondence between some of these concepts and their corresponding concepts in soft hyperring theory as well as classical hyperring theory.
2. Preliminaries
In this section, some well-known and useful definitions pertaining to the development of the theory of vague soft hyperrings introduced here will be presented. These definitions and concepts will be used throughout this chapter.
Definition 1 (see [6]). A pair is called a soft set over , where is a mapping given by . In other words, a soft set over is a parameterized family of subsets of the universe . For , may be considered as the set of -elements of the soft set or as the -approximate elements of the soft set.
Definition 2 (see [6]). For a soft set , the set is called the support of the soft set . Thus a null soft set is a soft set with an empty support and a soft set is said to be nonnull if .
Definition 3 (see [11]). Let be a space of points (objects) with a generic element of denoted by . A vague set in is characterized by a truth membership function and a false membership function . The value is a lower bound on the grade of membership of derived from the evidence for and is a lower bound on the negation of derived from the evidence against . The values and both associate a real number in the interval with each point in , where . This approach bounds the grade of membership of to a subinterval of . Hence a vague set is a form of fuzzy set, albeit a more accurate form of fuzzy set.
Definition 4 (see [12]). A pair is called a vague soft set over where is a mapping given by and is the power set of vague sets over In other words, a vague soft set over is a parameterized family of vague sets of the universe . Every set , for all , from this family may be considered as the set of -approximate elements of the vague soft set . Hence the vague soft set can be viewed as consisting of a collection of approximations of the following form:where is a subset of for all and for all .
Example 5 (see [12]). Consider a vague soft set , where is a set of six houses under consideration of a decision maker to purchase which is denoted by and is a parameter set, where = , , , , . The vague soft set describes the “attractiveness of the houses” to this decision maker.
Suppose that The vague soft set is a parameterized family of vague sets on , and
Definition 6 (see [10]). Let be a vague soft set over . The support of denoted by is defined asfor all .
It is to be noted that a null vague soft set is a vague soft set where both the truth and false membership functions are equal to zero. Therefore, a vague soft set is said to be nonnull if .
Definition 7 (see [12]). Let and be vague soft sets over The set is called a vague soft subset of if and, for every , and are the same approximation. In this case, is called the vague soft superset of and this relationship is denoted by .
Definition 8 (see [12]). If and are two vague soft sets over , the intersection of and denoted as “” is defined by , wherefor all and for all .
Definition 9 (see [12]). Let and be vague soft sets over Then “” denoted by is defined as , wherefor every and , where and are the truth membership function and false membership function of , respectively. This relationship can be written as , where represents the intersection operation between two vague soft sets as defined in the case of in Definition 8.
Definition 10 (see [10]). Let be a vague soft set over Then, for all , where , the -cut or the vague soft -cut of is a subset of which is as defined below:for all .
If , then it is called the vague soft -cut of or the -level set of , denoted by , is a subset of which is as defined below:for all
Definition 11 (see [10]). Let be a vague soft set over and let be a nonnull subset of Then is called a vague soft characteristic set over in and the lower bound and upper bound of are as defined below:where is a subset of , , and .
This paper pertains to the application of classical hyperring theory to the concept of vague soft sets. Furthermore, the relationships that exist between the concepts introduced in this paper and the corresponding concepts in classical hyperring theory are also studied and discussed. The definitions of several important concepts pertaining to the theory of hyperstructures are presented below.
Definition 12 (see [1]). A hypergroup is a set equipped with an associative hyperoperation which satisfies the reproduction axiom given by for all in .
Definition 13 (see [13]). A hyperstructure is called an -group if the following axioms hold:(i) for all ,(ii) for all in .If only satisfies the first axiom, then it is called a -semigroup.
Definition 14 (see [1]). A subset of is called a subhypergroup if is a hypergroup.
Definition 15 (see [3]). A -ring is a multivalued system which satisfies the following axioms:(i) is a -group,(ii) is a -semigroup,(iii)the hyperoperation “” is weak distributive over the hyperoperation “+”; that is, for each the following holds true:
Definition 16 (see [3]). A nonempty subset of is called a subhyperring of if is a subhypergroup of and for all , , where is the set of all nonempty subsets of .
Definition 17 (see [3]). Let be a -ring. A nonempty subset of is called a left (resp., right) -ideal if the following axioms hold:(i) is a -subgroup of ,(ii) ().If is both left and right -ideals of , then is said to be a -ideal of .
3. Vague Soft Hyperrings
In this section, the concept of vague soft hyperrings in Rosenfeld’s sense is introduced as a continuation to the theory of vague soft hypergroups introduced by [10]. We derive the definition of vague soft hyperrings by applying the theory of hyperrings to vague soft sets. The properties of this concept are studied and discussed.
From now on, let be a hyperring (or -ring), let be a set of parameters, and let For the sake of simplicity, we will denote by
Definition 18 (see [14]). Let be a nonnull soft set over Then is called a soft hyperring over if is a subhyperring of for all .
Definition 19 (see [10]). Let be a nonnull vague soft set over a hypergroup Then is called a vague soft semihypergroup over if the following conditions are satisfied for all :
For all , and ; that is, .
Definition 20 (see [10]). Let be a nonnull vague soft set over Then is called a vague soft hypergroup over if the following conditions are satisfied for all :(i)For all and ; that is, .(ii)For all , there exists such that and and ; that is, .(iii)For all , there exists such that and and ; that is, .Conditions (ii) and (iii) represent the left and right reproduction axioms, respectively. Then is a nonnull vague subhypergroup of (in Rosenfeld’s sense) for all .
Definition 21. Let be a nonnull vague soft set over Then is called a vague soft hyperring over if, for all , the following conditions are satisfied: (i)For all , and , which means .(ii)For all , there exists such that and and ; that is, .(iii)For all , there exists such that and and ; that is, .(iv)For all , and ; that is, .That is, is a nonnull vague subhyperring (in Rosenfeld’s sense) of for each .
Conditions (i), (ii), and (iii) indicate that is a vague subhypergroup of while condition (iv) indicates that is a vague subsemihypergroup of . Therefore it can be concluded that is a vague soft hyperring over if is a vague soft hypergroup over and a vague soft semihypergroup over .
Theorem 22. Let be a nonnull vague soft set over Then the necessary and sufficient condition for to be a vague soft hyperring over is for to be a vague soft semihypergroup over and also a vague soft hypergroup over .
Proof. The proof is straightforward by Definition 21.
Example 23. A vague soft hyperring for which is a singleton is a vague subhyperring. Hence vague subhyperrings and classical hyperrings are a particular type of vague soft hyperrings.
Example 24. Let be a nonnull vague soft set over a hyperring which is as defined below:Then is a vague soft hyperring over .
Example 25. Let be a vague subhyperring of a hyperring This means that satisfies the axioms stated in Definition 21. Now consider the family of -level sets for given byfor all and . Then, for all , is a subhyperring of . Hence is a soft hyperring over .
Example 26. Any soft hyperring is a vague subhyperring since any characteristic function of a subhyperring is a vague subhyperring.
Proposition 27. Let and be vague soft hyperrings over Then is a vague soft hyperring over if it is nonnull.
Proof. Suppose that and are vague soft hyperrings over Now let . Then for all and for all we havethat is,Since and are both nonnull and represents the basic intersection between and it follows that must be nonnull. For the sake of similarity, we only show the proof for the truth membership function of given by . The proof for can be derived in the same manner.
Hence for all and we obtainFurthermore for all we havewhere such that Similarly, it can also be proved that , where such that Therefore it has been proved that is a vague subhypergroup of , which means that is a vague soft hypergroup over .
Lastly for all we haveThis proves that is a vague subhyperring of for all . Hence is a vague soft hyperring over .
Theorem 28. Let be a vague soft set over . Then is a vague soft hyperring over if and only if, for every , is a soft hyperring over .
Proof. Since is a hyperring, is a commutative hypergroup and is a semihypergroup. Let be a vague soft hyperring over and Then, for all , the corresponding is a vague subhyperring of Now let Then and , , which means that and . Furthermore since is a vague subhypergroup of , we have . This implies that and therefore, for every , we obtain . As such, for every , we obtain . Now let . Then and , which means that and . Due to the fact that is a vague subhypergroup of , there exists such that and . Since , we obtain and this implies that and as a result, Therefore, we obtain . As such, we obtain . As a result, is a subhypergroup of . Hence is a soft hypergroup over . Furthermore, since is a semihypergroup, is a nonnull vague subsemihypergroup of for all and . Now let . Then and , which means that and . Therefore it follows that and this implies that . This in turn implies that and consequently . Therefore, for every , we obtain . As such, is a subhyperring of . Hence is a soft hyperring over .
Assume that is a soft hyperring over . Thus for every , is a nonnull subhyperring of Now for every , we have or . So if we let , then and therefore . Therefore, for every , we have which implies that . As such, condition (i) of Definition 21 is verified. Next, let and . This implies that which means that there exists such that . Since , we have and which means that and thus which means that . Therefore condition (ii) of Definition 21 has been verified. Condition (iii) of Definition 21 can be verified in a similar manner. Thus it has been proven that is a vague subhypergroup of . Now is a subsemihypergroup of the semihypergroup . Let for every . Thus and therefore we have . Thus, for every , we obtain . As a result, we obtain which proves that condition (iv) of Definition 21 has been verified. As such, is a vague subhyperring of and therefore is a vague soft hyperring over .
Theorem 28 proves that there exists a one-to-one correspondence between vague soft hyperrings and soft hyperrings over a hyperring.
Theorem 29. Let be a vague soft hyperring over . Then, for each , is a subhyperring of .
Proof. Let be a vague soft hyperring over . Then for each , the corresponding is a vague subhyperring of . Now let . Then , and , . Thus the following is obtained:However, since is a vague subhypergroup of , we have and . Therefore, for every , we obtain and thus . As such, it can be concluded that . Furthermore, since is a vague subhypergroup of , there exists such that and . Now let . Then the following is obtained:This means that and which implies that . Therefore for all , we have and . As such, it can be concluded that and this implies that, for any , . Hence it has been proven that is a subhypergroup of . Furthermore is a semihypergroup and is also a vague subsemihypergroup of for all . Now let . Then and and this results inThis implies that, for all , we obtain and therefore . Thus, for every , we obtain . As such, is a subsemihypergroup of the semihypergroup . Hence is a subhyperring of .
As a result of Theorem 29, we obtain Corollary 30.
Corollary 30. Let be a vague soft hyperring over . Then, for each , is a subhyperring of .
Theorem 31. Let be a nonempty subset of , let be a nonnull vague soft set over , and let be a vague soft characteristic set over . If is a vague soft hyperring over , then is a subhyperring of .
Proof. Let be a vague soft hyperring over . Then, by Definition 21, is a vague soft hypergroup over and a vague soft semihypergroup over and for each , the corresponding vague subset is a vague subhyperring of . Furthermore, since is a hyperring, is a commutative hypergroup and is a semihypergroup. Now let . Then and , which means that . Therefore, and so ( is a vague subhyperring of and a vague subhypergroup of . Thus, for every , we have and so it follows that . As such, for every , we have .
Now let , . Then and and so . Since is a vague subhypergroup of for all , there exists such that and . Thus we obtain and therefore which implies that . Since , this implies that and so we have for all . Therefore it has been proven that is a subhypergroup of . Let . Then and and thus . Now since is a semihypergroup, is a vague subhypergroup of for all . Thus it follows that and therefore too. As a result, for every , we have and therefore . As such, for every , we have . Hence is a subhyperring of .
Theorem 32. Let be a nonempty subset of , let be a nonnull vague soft set over , and let be a vague soft characteristic set over . Then is a vague soft hyperring over if and only if is a subhyperring of .
Proof. The proof is similar to that of Theorem 31.
Theorem 32 implies that there exists a one-to-one correspondence between vague soft hyperrings and classical subhyperrings of a hyperring.
4. Vague Soft Hyperideals
The ideal of a ring is an important concept in the study of rings. The concept of a fuzzy ideal of a ring was introduced by Liu (see [9]). Davvaz (see [8]) introduced the notion of a fuzzy hyperideal (resp., fuzzy -ideal) of a hyperring (resp., -ring). In this section the notion of vague soft hyperideals of a hyperring is defined using Rosenfeld’s approach.
Definition 33. Let be a nonnull soft set over . Then is called a soft left (resp., right) hyperideal of if is a left (resp., right) hyperideal of for all .
Definition 34. Let be a nonnull soft set over . Then is called a soft hyperideal of if is a hyperideal of ; that is, is both right and left hyperideals of for all .
Definition 35. Let be a nonnull vague soft set over . Then is called a vague soft left (resp., right) hyperideal of if the following axioms hold for all :(i)for all , and , which means that ,(ii)for all , there exists such that and and ; that is, ,(iii)for all , there exists such that and and ; that is, ,(iv)for all , (resp., ) and (resp., ); that is, (resp., ).That is, is a vague left (resp., right) hyperideal of (in Rosenfeld’s sense) for all .
Conditions (i), (ii), and (iii) prove that is a vague subhypergroup of . This means that is a vague soft left (resp., right) hyperideal of if, for all , is a vague subhypergroup of and also satisfies the condition (resp., ).
Definition 36. Let be a nonnull vague soft set over Then is called a vague soft hyperideal of if the following axioms hold for all :(i)for all , and , which means that ,(ii)for all , there exists such that and and ; that is, ,(iii)for all , there exists such that and and ; that is, ,(iv)for all , and ; that is, .That is, is a vague hyperideal of (in Rosenfeld’s sense) for all .
Similar to Definition 35, conditions (i), (ii), and (iii) of Definition 36 indicate that is a vague subhypergroup of while condition (iv) is a combination of both the conditions given in Definition 35 (condition (iv)).
Theorem 37. Let be a nonnull vague soft set over . Then the necessary and sufficient condition for to be a vague soft hyperideal is for to be a vague soft hypergroup of and is both a vague soft left hyperideal and a vague soft right hyperideal of .
Proof. The proof is straightforward by Definitions 35 and 36.
Proposition 38. Let and be two vague soft hyperideals of . Then is a vague soft hyperideal of if it is nonnull.
Proof. Suppose that and are vague soft hyperideals of Now to prove that is a vague soft hyperideal of , we have to prove that all the conditions of Definition 36 have been satisfied. However, in Proposition 27, it has been proven that conditions (i), (ii), and (iii) have been satisfied. Therefore it is only necessary to prove that satisfies condition (iv) of Definition 36.
Thus for all and we haveSimilarly, it can be proven that the condition is satisfied. This means that . Thus it has been proven that is a vague hyperideal of for all and as such is a vague soft hyperideal of
Theorem 39. Let be a nonnull vague soft set over . Then is a vague soft hyperideal of if and only if, for each , is a soft hyperideal of .
Proof. Let be a vague soft hyperideal of and . Then is a vague subhypergroup of . Now let . Then and , which means that and . In Theorem 28, it was proved that if is a vague subhypergroup of , then is a subhypergroup of . This means that it is only necessary to prove that satisfies the following conditions:Now let and . Then and which means that . Since is a vague hyperideal of for all , must satisfy the following condition:which means . Therefore we obtain . This proves that for all and as a result we obtain . Thus for every and , we obtain and this proves that is a left hyperideal of . This implies that is a soft left hyperideal of . Similarly, it can been proven that is a soft right hyperideal of . Thus is a soft right hyperideal of Hence is a soft left hyperideal of and a soft right hyperideal of which proves that is a soft hyperideal of
Let be a soft hyperideal of . Then, for each , is a nonnull hyperideal of . Therefore by Definition 16, is a subhypergroup of . This implies that is a soft hypergroup over . In [10], it was proven that if is a soft hypergroup, then is a vague soft hypergroup. Thus it can be concluded that is a vague soft hypergroup over . Furthermore, since is both a left hyperideal and a right hyperideal of , the following conditions are satisfied:Now let and or and . Thus we obtain and which implies that , where . As such, we obtain . Now let . Then and . Therefore, we obtain as well as which means that and . Combining these conditions, we obtain . Thus condition (iv) of Definition 36 has also been verified. This means that is a vague hyperideal of and hence it proves that is a vague soft hyperideal of .
Theorem 39 proves that there exists a one-to-one correspondence between vague soft hyperideal and the soft hyperideal of a hyperring.
Theorem 40. Let be a nonnull vague soft set over left (resp., right) hyperideal of . Then, for all , is a left (resp., right) hyperideal of .
Proof. Let be a vague soft left (resp., right) hyperideal of . Then, by Definition 35, is a vague soft hypergroup over . Now in Theorem 29 it was proven that if is a vague soft hypergroup over , then is a subhypergroup of . Therefore, in order to prove that is a left (resp., right) hyperideal of , we only have to prove that (resp., ). Now let and . Then and . Since is a vague left ideal of for all , satisfies the conditions and . Therefore we obtain and which implies that, for all , . As such, we obtain . Thus for every and , we obtain . Hence is a left hyperideal of . Next let and . Then and . Similarly, we obtain and . This implies that for all and therefore we obtain . As such, for every and , we obtain which means is a right hyperideal of .
Theorem 41. Let be a nonnull vague soft set over . Then is a vague soft left (resp., right) hyperideal of if and only if, for all , is a left (resp., right) hyperideal of .
Proof. The proof is straightforward.
Theorem 42. Let be a nonempty subset of and be a vague soft characteristic set over . If is a vague soft hyperideal of , then is a hyperideal of .
Proof. Let be a vague soft hyperideal of . Therefore, by Definition 36, is a vague soft hypergroup over and satisfies the condition . In order to prove that is a hyperideal of , we need to prove that is a subhypergroup of and and . In Theorem 31, it was proven that is a subhypergroup of if is a vague soft hypergroup over . Now let and . Then which means that . Since is a vague hyperideal of for all , it must satisfy the following conditions:Therefore since for all and , we obtain which means that . This implies that for all and this proves that for all and . Therefore we obtain and this implies that is a left hyperideal of . Now let and . Then which means that . Similarly, we obtain which means that . This implies that for all and consequently we obtain for all and . This proves that . As such, is a right hyperideal of . Hence it has been proven that is a hyperideal of .
Theorem 43. Let be a nonempty subset of , let be a nonnull vague soft set over , and let be a vague soft characteristic set over . Then is a vague soft hyperideal of if and only if is a hyperideal of .
Proof. The proof is straightforward.
5. Vague Soft Hyperring Homomorphism
In this section, we introduce the notion of vague soft hyperring homomorphism by applying the concept of vague soft functions as well as the notion of the image and preimage of a vague soft set under a vague soft function introduced by [10] in the context of vague soft hyperrings. Lastly, it is proven that the vague soft hyperring homomorphism preserves vague soft hyperrings.
Definition 44 (see [2]). Let and be two hyperrings. A map is called a hyperring homomorphism if, for all , one has and .
Definition 45 (see [10]). Let and be two functions, where and are parameter sets for the classical sets and , respectively. Let and be vague soft sets over and , respectively. Then the ordered pair is called a vague soft function from to and is denoted by .
Definition 46 (see [10]). Let and be vague soft sets over and , respectively. Let be a vague soft function. (i)The image of under the vague soft function , which is denoted by , is a vague soft set over , which is defined as where for all , , and .(ii)The preimage of under the vague soft function , which is denoted by , is a vague soft set over , which is defined as where for all , , and .If and are injective (surjective), then the vague soft function is said to be injective (surjective).
Definition 47. Let and be two hyperrings. Also let and be two vague soft hyperrings over and , respectively, and let be a vague soft function. Then is called a vague soft hyperring homomorphism if the following conditions are satisfied:(i) is a hyperring homomorphism from to ,(ii) is a function from to ,(iii) for all .
Theorem 48. Let and be two hyperrings. Let and be two vague soft hyperrings over and , respectively. Let be a vague soft hyperring homomorphism and let be an injective mapping. Then is a vague soft hyperring over .
Proof. Since is a vague soft hyperring homomorphism, it can be seen thatNow let . Then there exists such that and . Therefore, for all , we obtainwhere such that and .
Furthermore, for all , there exists such that and . Therefore, for all and for each , we obtainwhere such that and and alsowhere such that and .
Thus it has been proven that is a nonnull subhypergroup of and consequently is a vague soft hypergroup over .
Lastly, let . Then there exists such that and . Therefore for all and we obtainwhere such that and .
Therefore it has been proven that is a nonnull subsemihypergroup of and consequently is a vague soft semihypergroup over . As such, is a vague subhyperring of . Hence is a vague soft hyperring over .
Theorem 49. Let and be two hyperrings. Let and be two vague soft hypergroups over and , respectively. Let be a vague soft hyperring homomorphism. Then is a vague soft hyperring over .
Proof. The proof is similar to that of Theorem 48.
Theorems 48 and 49 prove that the homomorphic image and preimage of a vague soft hyperring are also a vague soft hyperring. This implies that the vague soft hyperring homomorphism preserves vague soft hyperrings.
Theorem 50. Let , , and be nonnull hyperrings and let , , and be vague soft hyperrings over , , and , respectively. Let and be vague soft hyperring homomorphisms. Then is a vague soft hyperring homomorphism.
Proof. The proof is straightforward.
6. Conclusion
In this paper we introduced and developed the initial theory of vague soft hyperrings as a continuation to the theory of vague soft hypergroups. It was proved that there exists a one-to-one correspondence between the concepts introduced here and the corresponding concepts in soft hyperring theory and classical hyperring theory. Lastly, it was proved that the notion of vague soft hyperring homomorphism introduced here preserves vague soft hyperrings. This is an important step in the formulation and advancement of knowledge in the area of vague soft hyperalgebra.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to gratefully acknowledge the financial assistance received from the Ministry of Education, Malaysia and Universiti Kebangsaan Malaysia under Grant no. FRGS/2/2013/SG04/UKM/02/3.