[Prakash* et al., 5(8): August, 2016] ISSN: 2277-9655 IC™ Value: 3.00 Impact Factor: 4.116 http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology [913] IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY SOME PROPERTIES AND THEOREM ON FUZZY SUB-TRIDENT DISTANCE A. Praveen Prakash*, M. Geetha Lakshmi Professor, Department of Mathematics, Hindustan University, Padur, Chennai - 603103 Assistant Professor, Department of Mathematics, KCG College of Technology, Karapakkam, Chennai - 600097 DOI: 10.5281/zenodo.60831 ABSTRACT This paper introduces some simple properties and theorem based on Fuzzy Sub-Trident Distance along with the help of Trapezoidal Fuzzy Numbers. The results are discussed with suitable numerical example. KEYWORDS: Trapezoidal Fuzzy Number, Sub-Trident Distance, Positive, Negative. AMS Mathematics Subject Classification: 03E72, 94D05, 03B52. INTRODUCTION Fuzzy Set Theory is introduced by Lotfi.A.Zadeh in the year 1965 [1].Later Liem Tran and Lucien Duckstein gave the Comparison of fuzzy numbers using a fuzzy distance measure in the year 2002 [3]. Later Shanhuo Chen and Chienchung Wang introduced the Fuzzy Distance of Trapezoidal Fuzzy Numbers in the year 2008 [4]. In the year 2012, A.Nagoorgani [5] gave a new operation on Triangular Fuzzy number for solving Fuzzy Linear Programming Problem. A New Method for Rank, Mode, Divergence and spread on Generalized Exponential Trapezoidal Fuzzy Numbers is given by Salim Rezvani in the year 2012 [6]. Arithmetic Operations on Generalized Trapezoidal Fuzzy Number and its Applications is given by Sanhita Banerjee and Tapan Kumar Roy in the year 2012[7]. In the year 2014, Pardhasaradhi and Ravi Shankar gave an idea on Fuzzy Distance Measure [8]. In this Paper, Some simple properties and theorem based on Fuzzy Sub-Trident Distance along with the help of Trapezoidal Fuzzy Numbers are given. This Paper consists of five sections. The preliminaries in the first section, Defining Trapezoidal, Positive Trapezoidal, Negative Trapezoidal Fuzzy Numbers in the second section, Fuzzy Sub-Trident Distance in the third section, Properties and Theorem based on Fuzzy Sub-Trident Distance in the fourth section and finally, the results are discussed with suitable numerical examples. PRELIMINARIES Definition 1. The Characteristic function A ~  of a crisp set X A  ~ assigns a value either 0 or 1 to each member in X . This function can be generalized to a function A ~  such that the value assigned to the element of the universal set X fall within a specified range i.e., A ~  : X   . 1 , 0  The assigned value indicates the membership grade of the element in the set ~ A . The function A ~  is called the membership function and the set     X x x x A A   ; ) ( , ~ ~  defined by ) ( ~ x A  for each X x  is called a fuzzy set. [2]