IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319–765X. Volume 10, Issue 3 Ver. V (May-Jun. 2014), PP 96-99 www.iosrjournals.org www.iosrjournals.org 96 | Page Practical Distributed Fuzzy Sets Mary Gracelet J 1 , Dr.G.Velammal 2 1,2 (Associate Professor, Department of Mathematics, Sri Meenakshi Governement Arts College for Women, Madurai,/India) Abstract : Membership function of a fuzzy set is the generalization of the characteristic function of a crisp set. Membership functions can be generalized using distributions or generalized functions. Distributed fuzzy sets are defined using distributions. In this paper we introduce the concept of Practical Distributed Fuzzy sets and extend the operations of usual fuzzy sets to practical distributed fuzzy sets Keywords: distributed fuzzy sets, fuzzy sets, generalized function , membership functions, practical distributed fuzzy sets I. INTRODUCTION The characteristic function of a crisp set is a function from a set to {0,1}. In 1965 Lotfi A Zadeh[5] introduced the concept of a fuzzy set. The membership function of a usual fuzzy set is a function from a set to the interval [0,1].This has been generalized in several ways. For example an L-fuzzy set has a membership function where the range is a partially ordered set [2]. Type 2 fuzzy sets, intuitionistic fuzzy sets, non stationary fuzzy sets, hesitant fuzzy sets are some of the other generalizations of fuzzy sets. In our previous paper[6] we had introduced the concept of distributed fuzzy sets which are defined over intervals in the real line For a distributed fuzzy set membership function can be a 'generalized function' or a distribution. In a fuzzy set the value of the membership function μ(x) is known at every point x. In contrast, in a distributed fuzzy set, only the average membership value over certain intervals may be known. As we have proved in [6], if the average membership value over every subinterval is known that will correspond to the usual fuzzy set. But in the real world this rarely happens. Precise data will often be unavailable. So μ(x) cannot be determined at every point. Only an estimate of the average membership value over a certain interval will be known. Distributed fuzzy sets model this situation very well. If it is too much to hope to determine μ(x) at every point it is too much to expect to know the distribution T which defines a particular distributed fuzzy set. In other words the distributed fuzzy set is a theoretical concept which models impreciseness of a certain sort. While constructing mathematical models to represent real life situations involving fuzziness of this type we need the concept of practical distributed fuzzy sets. II. PRELIMINARIES We briefly recall the definition of distributed fuzzy set. Definition 2.1: Consider a sequence of functions in with the following properties with respect to the interval [a,b]. P1 : out side P2 : in . P3 : 0 ≤ It is well known that such a sequence exists. Definition 2.2: Let be a sequence of functions in satisfying P1, P2, and P3 with respect to the interval [a, b] and let T be a distribution. T is said to define a distributed fuzzy set on [a, b] if i) T is a positive distribution ii) exists and is the same for any sequence of functions in satisfying the P1, P2, and P3 with respect to the interval [a,b] . iii) ≤ b-a Definition 2.3: Suppose T defines a distributed fuzzy set over an interval [a, b]. Then it is said to have average membership value m over the interval [a, b] where