Advances in Fuzzy Systems

Advances in Fuzzy Systems / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 968405 | 8 pages | https://doi.org/10.1155/2014/968405

Fuzzy Set Field and Fuzzy Metric

Academic Editor: Katsuhiro Honda
Received21 Apr 2014
Accepted12 Aug 2014
Published02 Sep 2014

Abstract

The notation of fuzzy set field is introduced. A fuzzy metric is redefined on fuzzy set field and on arbitrary fuzzy set in a field. The metric redefined is between fuzzy points and constitutes both fuzziness and crisp property of vector. In addition, a fuzzy magnitude of a fuzzy point in a field is defined.

1. Introduction

Different researchers introduced the concept of fuzzy field and notion of fuzzy metric on fuzzy sets. How to define a fuzzy metric on a fuzzy set is still active research topic in fuzzy set theory which is very applicable in fuzzy optimization and pattern recognition. The notion of fuzzy sets has been applied in recent years for studying sequence spaces by Tripathy and Baruah [1], Tripathy and Sarma [2], Tripathy and Borgohain [3], and others.

Wenxiang and Tu [4] introduced the concept of fuzzy field in field and fuzzy linear spaces over fuzzy field. Furthermore, different authors are attempting to define fuzzy normed linear spaces, fuzzy inner product space, fuzzy Hilbert space, fuzzy Banach spaces, and so forth (cf. [58]).

Many authors introduced different notion of fuzzy metric on a fuzzy set from different points of view. Kaleva and Seikkala [9] introduced the notion of a fuzzy metric space where metric was defined between fuzzy sets. The idea behind this notion was to fuzzify the classical metric by replacing real values of a metric by fuzzy values (fuzzy numbers). For the further research work and the properties of this type of fuzzy metric space see for instance Fang [10], Quan Xia and Guo [11], and others.

Wong [12] defined fuzzy point and discussed its topological properties and there after Deng [13] defined Pseudo-metric spaces where metric was defined between fuzzy points rather than between fuzzy sets. Hsu [14] introduced fuzzy metric space with metric defined between fuzzy points and examined the completion of fuzzy metric space. For different notions of fuzzy metric space and for further research work see for instance Shi [15], Shi and Zheng [16], Shi [17] and others.

This paper is an attempt to define a fuzzy set field in a field which is assumed to be the generalization of a fuzzy field introduced by [4]. We restate fuzzy set in more general form by allowing a particular fuzzy set to consist a family of membership functions. A fuzzy metric on fuzzy set and on fuzzy set field is reintroduced in such way that the classical metric is considered as a special type of fuzzy metric. In the sequel, a notion of magnitude of a fuzzy point in a field is introduced for the first time (up to our knowledge) and some of its properties are investigated.

2. Brief Summary of Fuzzy Set, Fuzzy Point, and Fuzzy Field

Definition 1 (see [18]). Let be a nonempty set and be a mapping. Then is said to be a fuzzy set in with membership function .

Definition 2. Let be a fuzzy in . Then (i) is called normal if there is at least one point with (see [19]).(ii) A fuzzy set is convex if for any and any , we have (see [19]).(iii) A fuzzy number is a fuzzy set on the real line that satisfies the conditions of normality and convexity (see [19]).(iv) is said to be fuzzy point if = singleton  set. It is usually denoted by , (see [12, 18]), where (v) Support of is the crisp set, (see [12, 18]).(vi)The fuzzy point is said to be contained in a fuzzy set, , or to belong to , denoted by , if and only if   (see [18]).

Definition 3 (see [20]). Let be a Cartesian product of universes , be fuzzy sets in , respectively, and be a mapping from to a universe , . Then the extension principle allows us to define a fuzzy set in given by , where is inverse of and

Lemma 4 (see [21]). Provided that is either a positive or a negative fuzzy number and that and are together either positive or negative fuzzy numbers, then .

Definition 5 (see [4]). Let be a field and let be a fuzzy set in with membership function . Suppose the following conditions hold: (1),(2),(3),(4)then is said to be a fuzzy field in and is denoted by .

Lemma 6 (see [4, 8]). If is a fuzzy field in with membership function , then (1), ,(2), ,(3), ,(4), .

3. Fuzzy Set Field and Fuzzy Metric on a Fuzzy Set in Fuzzy Set Field

3.1. Fuzzy Set Field

In this section the concept of fuzzy set field is introduced which is assumed to generalize a fuzzy field defined by [4]. Since we are dealing with a subset of original field, the axioms imposed on it are similar to that of the ordinary field. However, the operations (fuzzy sum and fuzzy product) are performed according to the extension principle.

Notation 1. The following notations might be used (1)Every mapping from a a non-empty set into the unit interval, is denoted by small Greek letters;(2);  ;(3), we mean a fuzzy point with membership value ;(4);(5), we mean (in usual sense) and .

Definition 7. Let be a field and let be mapping. A point is said to be a fuzzy point of if and is said to be membership value of a fuzzy point .

Definition 8. Let is said to be a fuzzy set in .

In Definition 8, if , then is said to be ordinary fuzzy set on .

Notation 2. Let be a fuzzy set in a field . A fuzzy point belongs to a fuzzy (or ); we mean that there exists such that . Therefore, we will use and alternatively.

Definition 9. Two fuzzy points and are said to be equal if and only if (in usual sense) and and we denote it by .

Using Definition 3 (that is extension principle), we define operations fuzzy product, fuzzy sum, and fuzzy difference between fuzzy points according to the following definition.

Definition 10. Let  and be fuzzy points in a field . Then the fuzzy sum, fuzzy product, and fuzzy difference of fuzzy points are defined by(1), where .(2), where .(3), where .

Remark 11. Let be a fuzzy set in a field and let . Then (1);(2);(3) for all , .

Definition 12. Let be a fuzzy set in . Fuzzy points and in with respective membership values   and are said to be fuzzy additive and fuzzy multiplicative points, respectively, in if they satisfy the following conditions:(1)for each , ;(2)for each , .

Definition 13. Let be an ordinary fuzzy set in a field with membership function . Let . A fuzzy set, , is said to be a generated fuzzy set in with a generating membership function denoted  by . Alternatively, we call the collection of all fuzzy points generated by .

Theorem 14. Let be a generated fuzzy set in . Then for all (i), , and ;(ii), , and .

Proof. It follows from Definitions 10 and 13.

Theorem 15. Let be a fuzzy field where is as in Definition 13. If , , with respective membership values  , , and , then (1);(2);(3);(4);(5);(6);(7)for each , there exists such that ;(8)given , there exists an such that .

Proof. The results follow from Lemma 6 and Definition 10.

According to Theorem 15, if is a fuzzy field, , then represents almost a field over binary operations and . It is clear that if for all , then satisfies most axioms of a field under the binary operations and .

With motivation of Theorem 15 and discussion that follows it, we have the following definition.

Definition 16. Let be a field and let . A fuzzy set is said to be a fuzzy set field in if the binary operators and satisfy the following conditions: (1), , and for all , ;(2), , and for all ;(3), , and for all ;(4), , and for all , , ;(5)there exists unique with such that , , and for all ;(6)for each , there exists such that and for some , ;(7), ;(8)there exists with such that ;(9)given and , there exists an such that .

We call , the fuzzy additive identity and fuzzy unit points of , respectively.

Theorem 17. Let be a fuzzy set field in . Then (i) is the fuzzy additive identity point of if and only if , there exists such that , , ;(ii) is the fuzzy multiplicative identity point of if and only if and there exists such that , ;(iii), where is as in (i);(iv) or .

Proof. Let be a fuzzy set field in a field , let be arbitrary and let be as in Definitions 16 and 21 respectively. We need to show that (i) to (iv) of the theorem.(i)Let be the fuzzy additive identity point of with membership . Then we claim that . For if , then , hence the claim. Now let be arbitrary. Then = = . The converse is clear.(ii)Let be as stated in Definition 16 with membership value . Let with corresponding membership value . Then . Thus and . The converse holds trivially.(iii)Let be arbitrary with membership values and , respectively. Then by (5) and (7), we have by uniqueness of in .(iv)The proof of (iv) follows from (iii). Therefore, the theorem is proved.

Remark 18. If is a fuzzy field set in a field and is as stated in Theorem 17, then for each , either or .

Theorem 19. Let be a fuzzy set field in a field K. (1)If there exists a such that for all and for all , then with membership value is the additive element of .(2)If there exists a such that for all and for all , then with membership value is the multiplicative element of .

For ongoing discussion, we keep the notations and as they are stated in Theorems 17 and 19.

Remark 20. If is a fuzzy set field in , , , and be as stated in Theorem 19, then (1), .(2) for each .(3) for each .(4).(5) for each .(6), for some .(7) for all .

Definition 21. Let be a field. A function is said to have property (Du) if it satisfies the following conditions: (i), , and , is a subfield.(ii), .

Theorem 22. Let be an ordinary fuzzy set in a field , let be as in Definition 13. If has property (Du), then is a fuzzy set field.

Proof. Definition 16 ((1)–(4)) follows from Theorem 14 and Definition 16 (7) follows from Definition 21. Therefore, we need to verify (5), (6), (8), and (9) of Definition 16. From Definition 21 and definition of , for all , we can easily verify that(i), ,(ii),(iii), ,(iv).
Using (i) to (iv) and definition of and , we have for each , we have , , , and , . Consequently, , and  . Therefore, by Theorem 19, and with respective membership values   and are fuzzy zero and fuzzy unit of , respectively. Hence ((5) and (8)) of Definition 16 are proved. Since is subfield, given , , there exists such that and . Using ((iii) and (iv)) of this theorem, . Thus . Similarly, given , there is a unique such that , hence the theorem.

Theorem 23. Let and be fields and let be fuzzy set field of . If there exists as stated in Theorem 19 and is a homomorphism, then is a fuzzy set field in .

Proof. Let and be units (multiplicative) of and , respectively, and be neutral elements with respect to addition (+) of and , respectively and . Since is homomorphism, we have and . Since is fuzzy set in , is a corresponding fuzzy set in with membership function(s) given by [20].
Now, we shall verify Definition 16 ((1) to (9)). (1)Let with membership values and , respectively. Then = and there are such that = . But Thus .(2)Similar to proof of (1), we can easily verify that .(3)Suppose . Since , then there are fuzzy points such that = and . Since is a homomorphism, . So = . But = = . Thus, .(4)Similar to the proof of (3), we show that .(5)Let . Since and by Theorem 17, for every with corresponding membership value , there exists such that = . Since = , , we have = = = = . Therefore, by Theorem 17, with membership value is the fuzzy additive identity point of and for every , .(6)Let . Since , there exist such that and , by Definition 16. But , where , and . Since = and (be extension principle followed by Remak 20), it follows that = .Thus, . Since was arbitrary, so given there exists such that .(7)Similar to the proof of (3) above, we show that .(8)Since is a fuzzy field set, by Theorem 17, for every with corresponding membership value , there exists such that . Since , we have = = = = . Therefore, by Theorem 17, with membership value is the fuzzy multiplicative identity point of and for every .(9)Let . Since , there exist such that and and = , where . Thus . Since and (by extension prnciple followed by Remark 20), it follows that = . Thus, .

Corollary 24. Let , be fields, and let be a fuzzy set field in . If is injective homomorphism, then is a fuzzy set field in .

Example 25. Let be a field and let be given by . Then is a fuzzy set field in .

Proof. (1)–(9) of Definition 16 hold trivially, with and .

Example 26. Let and let be given by Let be as in Definition 13 and or , then is a fuzzy set field in .

Proof. First we show that has property (Du). Clearly, satisfies Definition 21(i). To verify Definition 21(ii), Lemma 4 will be applied. Since for all , we can consider only for . Moreover, , where and [20]. Therefore, and are both positive fuzzy numbers. Hence, by Lemma 4, = . Thus, =   . Therefore, applying Theorem 22 the result follows.

3.2. Fuzzy Metric of a Fuzzy Set and Fuzzy Magnitude of a Fuzzy Point in a Fuzzy Set Field

Now we introduce definition of a fuzzy magnitude of a fuzzy point and fuzzy metric on a fuzzy set in a fuzzy set field, , such that satisfies (1) in Theorem 19. For a fuzzy set field , such that satisfies (1) in Theorem 19, it follows that (Remark 20). That is for all and for each . Therefore, we assume that any fuzzy point, with a membership value (that is ) as a nonfuzzy point of a particular fuzzy set field under discussion. Hence we have the following definition.

Definition 27. Let be a fuzzy field set of a field as in Theorem 19, and let  . A fuzzy point with membership value is said to be nonfuzzy point of a fuzzy set field if and only if .

Definition 28. Let be a fuzzy set field of a field , , and let be a fuzzy point with membership value . Then the fuzzy magnitude of , denoted by , is given by . Where is as in Theorem 19.

From Definition 28, if and only is nonfuzzy in provided that is as in Theorem 19. Therefore one can redefine the magnitude of given vector in a field as a fuzzy magnitude. Moreover, if and only if and nonfuzzy and if and only if and is nonfuzzy. Furthermore, whenever .

Theorem 29. Let , , and be fuzzy points in a fuzzy set field . Then (1),(2),(3).

Proof. Let , , and be fuzzy points. Then (1),(2),(3).Hence the result is proved.

Definition 30. Let be a fuzzy set field of a field such that is as in Theorem 19, and be a nonempty fuzzy set in field . A fuzzy metric on is a mapping satisfying the following conditions: (1) for all ,(2) if and only if and both are nonfuzzy in ,(3),(4). A pair is said to be fuzzy metric spaces.

If is a metric on a field and is any subset of , then by considering characteristic function on as its membership function, defines a fuzzy metric on . Thus, one can consider an ordinary metric as fuzzy metric.

Theorem 31. Let be a fuzzy set field of a field such that is as in Theorem 19, and be a nonempty fuzzy set in field . A mapping given by defines a fuzzy metric on .

Proof. Let . We need to verify (1) to (4) of Definition 30.(1) (by Remark 11),(2), and = by Definition 27 both are nonfuzzy,(3),(4) + . Hence, the theorem is proved.

Corollary 32. Let be a fuzzy set field of a field such that is as in Theorem 19, and be a non empty fuzzy set in field . A mapping defined by is fuzzy metric on .

Theorem 33. Let be a fuzzy set of a field such that is as in Theorem 19, and be a nonempty fuzzy set in field . A mapping defined by is fuzzy metric on .

Proof. Let . We need to verify (1)–(4) of Definition 30. (1), (By Remark 11 ),(2) = and   = ,(3)By Theorem 29, ,(4) = .Hence, the theorem is proved.

4. Metric on Fuzzy Set of a Field

In Section 3, we have introduced the definition of fuzzy magnitude and a fuzzy metric of a fuzzy point and a fuzzy set, respectively, in a fuzzy set field, , in which for all and for all . In this section, we try to define a fuzzy magnitude and fuzzy metric of a fuzzy point and fuzzy set in a field, respectively. The definitions introduced differ slightly from that of preceding section due to targeted highest membership value which is assumed to be , where is as in Theorem 19. However, the two definitions coincide whenever .

Definition 34. Let be fuzzy set in a field. A fuzzy point is said to be nonfuzzy point if and only if .

Definition 35. Let be a fuzzy point in a field . Then the fuzzy magnitude of , denoted by , is given by

From Definition 34 a fuzzy magnitude of a nonfuzzy point coincides with that of usual magnitude. Therefore one can redefine the magnitude of a given vector in a field as a fuzzy magnitude.

Theorem 36. If , and are fuzzy points in a field . Then(1),(2),(3).

Proof. Let , , and be fuzzy points in a field . Then(1) (by Remark 11(3))   ,(2),(3).
Therefore, the theorem is proved.

Definition 37. Let be a nonempty fuzzy set in fuzzy field . A fuzzy metric on is a mapping satisfying the following conditions: (1),(2) if and only if is nonfuzzy in and ,(3) for all ,(4) for all .

A function is said to be fuzzy metric on and a pair is said to be fuzzy metric space.

Theorem 38. Let be a field, let be metric on and be a nonempty fuzzy set in . A mapping defined by is a fuzzy metric on .

Proof. Let with respective membership values  and . We will verify that (1) to (4) of Definition 37. (1), by Remark 11.(2), and by Definition 34 both are nonfuzzy.(3).(4) + .Hence the theorem is proved.

In the preceding theorem, provided that are nonfuzzy in . Therefore, if is characteristic function, then fuzzy metric coincides with usual metric of . So we call a fuzzy metric given as in Theorem 38 a standard fuzzy metric on a fuzzy set.

Corollary 39. Let be a fuzzy set in a field . A mapping given by defines a fuzzy metric on .

Proof. The result follows from Theorem 38.

Theorem 40. Let be a nonempty fuzzy set in a field . A mapping defined by is fuzzy metric on .

Proof. Proof is similar to Theorem 33, so it is omitted.

5. Conclusion

In this paper, fuzzy set field in a field is defined and some of its properties are discussed. We believe that our results will be helpful to develop similar notion for fuzzy linear space. The definition of the fuzzy metric introduced in this paper resembles distance between two fuzzy points and it involves both fuzziness and crisp property of fuzzy points that constitute classical metric properties. The definition is more general and it can be applied to define a fuzzy metric on linguistic variables also. We believe that our metric might be applied in fuzzy-decision theory, pattern recognition, and image processing.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2014 Gebru Gebray and B. Krishna Reddy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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