Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 40, 2021, pp. 67-75 67 Certain results on metric and norm in fuzzy multiset setting Johnson Mobolaji Agbetayo* Paul Augustine Ejegwa† Abstract Fuzzy multiset is an extension of fuzzy set in multiset framework. In this paper, we review the concept of fuzzy multisets and study the notions of metric and norm on fuzzy multiset. Some results on metric and norm are established in fuzzy multiset context. Keywords: fuzzy set; fuzzy multiset; metric; norm. 2010 AMS subject classification: 03E72, 08A72, 20N25. *Department of Mathematics/Statistics/Computer Science, Federal University of Agriculture, P.M.B. 2373, Makurdi, Nigeria; agbetayojohnson@gmail.com. † Department of Mathematics/Statistics/Computer Science, Federal University of Agriculture, P.M.B. 2373, Makurdi, Nigeria; ocholohi@gmail.com. § Received on January 12th, 2021. Accepted on May 12th, 2021. Published on June 30th, 2021. doi: 10.23755/rm.v40i1.582. ISSN: 1592-7415. eISSN: 2282-8214. ©J. M. Agbetayo and P. A. Ejegwa. This paper is published under the CC-BY licence agreement. J. M. Agbetayo and P. A. Ejegwa 68 1. Introduction The theory of set proposed by Cantor in 1915, is a collection of well-defined objects of thought and intuition. The limitation of set theory is its inability to deal with the vague properties of its member or element, and likewise its distinctness property which does not allows repetition in the collection. In other to handle the vague property of a set, Zadeh [19] proposed a mathematical model that deal with vagueness of a set known as fuzzy sets. The distinct property of crisp set has been violated by allowing repetition of an element in a collection. This gave birth to a set called multiset. The term multiset was first suggested by De Bruijn to Knuth in a private correspondence as noted in [13]. The theory of multisets has been studied [3, 4, 10, 11, 12, 16]. Lake [14] presented an abridge account on sets, fuzzy sets, multisets and functions. By synthesizing the concepts of fuzzy sets and multisets, Yager [18] introduced the concept of fuzzy multiset (FMS) that deal with vagueness property of a set and allowed the repetition of its membership function. In fact, fuzzy bag or fuzzy multiset generalizes fuzzy sets in such a way that the membership degree of a fuzzy set is allowed to repeat. Some fundamentals properties of fuzzy bags have been studied [6, 15]. The concept of fuzzy bags has been applied in multi-criteria decision-making [1, 2], sequences [5] and computational science [17]. Metric is a function that defines a concept of distance between any members of the set, which are usually called points. The notions of metric and norm have been extended to the environment of fuzzy sets [7, 8, 9]. In this work, we present the notions of norm and metric in fuzzy multiset context. 2. Preliminaries In this section, we review some definitions and result that are important for the main work. Certain results on metric and norm in fuzzy multiset setting 69 Definition 2.1 [18]. Assume is a set of elements. Then, a fuzzy bag/multiset, drawn from can be characterized by a count membership function such that where is the set of all crisp bags or multisets from the unit interval, According to Syropoulos [17], a fuzzy multiset can also be characterized by a high-order function. In particular, a fuzzy multiset can be characterized by a function or where and The count membership degrees, for is given as where ,… ∈ [0,1] such that ( ) ≥ ( ) ≥ ( ), ≥ … ≥ (x) ≥ …, whereas in a finite case, we write = { ( ), ( ),…, ( )} for ( ) ≥ ( )≥ … ≥ ( ). A fuzzy multiset can be represented in the form or In a simple term, a fuzzy multiset, of is characterized by the count membership function, for , that takes the value of a multiset of a unit interval . We denote the set of all fuzzy multisets by Example 2.2. Assume that is a set. Then for ={0.5,0.4,0.2}, A is a fuzzy multiset of written as Definition 2.3 [15]. Let Then, is called a fuzzy submultiset of written as if . Also, if and , then A is called a proper fuzzy submultiset of and denoted as . J. M. Agbetayo and P. A. Ejegwa 70 Definition 2.4 [15]. Let Then, and are comparable to each other if and only if Definition 2.5 [17]. Let .Then, the intersection and union of , denoted by and are defined by (i) (ii) Definition 2.6 [17]. Let . Then, the sum of and , denoted by , is defined by the addition operation in for crisp multiset. That is, The addition operation is carry out by merging the membership degree in a decreasing order. Definition 2.7 [6]. Let . Then, the difference of from is a fuzzy multiset such that . Definition 2.8 [6]. Let . Then, the complement of is a fuzzy multiset such that Metric and Norm defined over Fuzzy Multisets 3. Metric and norm defined over fuzzy multisets In this section, we present metrics and norm defined over fuzzy multiset. Definition 3.1. Let be an arbitrary non-empty set and let . A metric or distance function between A and B on is a function with the following properties: (i) . (ii) iff . (iii) . (iv) Certain results on metric and norm in fuzzy multiset setting 71 if Note: (i) The distance is a non-negative function and only zero at a single point. (ii) The distance is a symmetric function. (iii) The distance satisfy triangle. Proposition 3.2. Let Then is a metric defined on Proof. We use Definition 3.1: Axiom (i) . Axiom (ii) If . Conversely, if . Axiom (iii) . Axiom (iv) The following are distances between fuzzy multisets: Hamming distance; Euclidean distance; Normalized Hamming distance; . Normalized Euclidean distance; J. M. Agbetayo and P. A. Ejegwa 72 . Theorem 3.3. Let be non-empty set and then . Proof. We show that or . But and . Thus, = Hence Corollary 3.4. If is a distance of fuzzy multiset of and then Pr oof. Clearly, . Proposition 3.5. If is a metric of fuzzy multiset and then Proof. By Definition 3.1, if , so it follows that . Proposition 3.6. Let and is a metric defined on Then is also a metric. Certain results on metric and norm in fuzzy multiset setting 73 Proof. The proof is obvious, since Hence is a metric. Corollary 3.7. If λ then . Proof. The proof is straightforward. Corollary 3.8. If then . Proof. The proof is straightforward. Corollary 3.9. If then Proof. The proof is straightforward. Definition 3.10. Let be a non-empty set and be a fuzzy multiset of X. A non-negative real-valued function defined on is called a norm if the following properties are satisfied: (i) iff that is, iff (ii) which implies that (iii) which implies that . The Fuzzy multiset equipped with a norm is called Normed Fuzzy multiset. Proposition 3.11. Let then Proof. We show that . Now, . Proposition 3.12. Let and a norm define over . Proof. (i) . (ii) J. M. Agbetayo and P. A. Ejegwa 74 (iii) Hence is a norm defined over fuzzy multiset Corollary 3.13. If then and if , then . Proof. 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