Appl. and Comput. Math., V.8, N.2, 2009, pp.203-215 CLASSIFICATION BASED ON SIMILARITY AND DISSIMILARITY THROUGH EQUIVALENCE CLASSES M.KESHAVARZI †, M.A.DEHGHAN ‡, M.MASHINCHI ††, § Abstract. In this paper we first briefly review the literature on similarity relations, and then introduce a dissimilarity relation between a pair of elements in a specific domain. Since in some cases recognizing dissimilarity is easier than similarity, we try to find a connection between these relations based on specific functions. Keywords: fuzzy sets, similarity relations, dissimilarity relations. AMS Subject Classification: 62H30, 03E20, 03B52. 1. Introduction The most primary observation we can make when studying a group of objects or phenomena is that some are similar and others dissimilar. Similar objects can be grouped together to form a class, and we define a class as a set of similar objects. We always compare two objects with respect to a frame of reference, such as basic characteristics, context, or point of view. In other words, background information, or the existence of other classes, will affect the way objects are compared. The concept of similarity has been studied by many researchers. Similarity-based clustering is described in [5, 12, 26]. Chakraborty and Das [6], Valverede, Trillas and Jakas [11, 25, 27], Ovchinikov [16, 17] have widely studied the similarities in various contexts. The notion of similarity originated in psychology and was established to determine why and how entities are grouped to categories, and why some categories are comparable to each other while others are not [9, 10]. The main challenge in semantic similarity measurement is the comparison of meanings. Human judgments of similarity have been subject to research in psy- chology for more than fifty years [9]. Different approaches to modelling similarity including feature-based, network-based, and geometric approaches have been developed. More recently, the Artificial Intelligence (AI) community started investigations on computational similarity models as a new method for information retrieval [23]. The Matching Distance Similarity Mea- sure (MDSM) [24] was the first similarity-based model that has been developed specifically for the geospatial domain. V. Loia et al. in [14] used similarity relations in an internet e-mail application which is concerned with finding people who are interested in receiving a particular message via e-mail. Also similarity measures have been used extensively in text summarization as in [1] where the performance of different similarity measures have been evaluated in the con- text of document summarization. In this paper we basically study similarity and dissimilarity relations based on equivalence classes, but it is time worthy to pay attention to the work done on the application of similarity measures to fuzzy sets[28], which is an important tool in fuzzy †Faculty of Mathematics, Vali-e-Asr University, Rafsanjan, Iran, e-mail: mkeshavarzi@mail.vru.ac.ir ‡Faculty of Mathematics, Vali-e-Asr University, Rafsanjan, Iran, e-mail: dehghan@mail.vru.ac.ir ††Faculty of Mathematics and Computer Science, Shahid Bahonar University, Kerman, Iran, e-mail: mashinchi@mail.uk.ac.ir §Manuscript received 3 July, 2008. 203