# Multiple Relations and its Application in Medical Diagnosis

V. Shijina (Department of Mathematics, National Institute of Technology Calicut, Kozhikode, India) and Sunil Jacob John (Department of Mathematics, National Institute of Technology Calicut, Kozhikode, India)
DOI: 10.4018/IJFSA.2017100104
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## Abstract

This paper develops the notion of multiple relations as multiple sets over the Cartesian product of the universal sets. This contribution focuses mainly on binary multiple relation - multiple relation over the Cartesian product of two universal sets. Some special binary multiple relations on a single universal set are discussed and their properties are investigated. This paper also has a discussion on multiple equivalence relation and some of its properties. A special attention is paid to the concept of distance multiple relation. Finally, this paper proposes an algorithm to diagnose diseases using distance multiple relation and the method is illustrated with an example.
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## Introduction

Uncertain or imprecise data are inherent and pervasive in many important applications in the areas such as economics, engineering, medical science and business management. Uncertain data in these applications could be caused by data randomness, incomplete information, limitations of measuring instruments, delayed data updates, etc. This problem of uncertainty, vagueness, imperfect knowledge, etc. has been tackled for a long time by philosophers, logicians and mathematicians. A great amount of research and applications have been developed to represent these concepts mathematically. The notion of fuzzy set proposed by Zadeh (1965) is a very successful approach to vagueness. A fuzzy set is defined as a generalization of an ordinary crisp set and is characterized by a membership function which assigns to each object a grade of membership ranging between zero and one. Fuzzy sets have advanced in a variety of ways and have applications in many disciplines which include medicine (Polat, Gunes & Tosun, 2006), decision theory (Liao, 2015), game theory (Fei, 2015b; Li, 2016) and many others (Klir & Yuan, 1995; Fei, W. 2015a; Li, 2017). Several other theories have been introduced to represent imprecision, uncertainty, etc. Some of these theories are extensions of fuzzy set theory and others try to handle these concepts in a different way. Such theories include rough set theory (Pawlak, 1982; Pawlak, 1998), vague set theory (Gau & Buehrer, 1993), intuitionistic fuzzy set theory (Atanassov, 1986), L-fuzzy sets (Goguen, 1967), multi sets (Cerf, Fernandez, Gostelow & Volausky, 1971), fuzzy multi sets (Yager, 1986), multi fuzzy sets (Sebastian & Ramakrishnan, 2010; Sebastian & Ramakrishnan, 2011), etc. Each of these theories has its advantages in dealing with uncertainties and has found applications in diverse domains such as decision support, engineering, environment, banking, medicine and others.

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