# Fuzzy and Rough Set Theory in Treatment of Elderly Gastric Cancer Patients

Sammanfattning: Fuzzy set theory was presented for the first time by Professor Lotfi A. Zadeh from Berkeley University in 1965 as his vision of research involved in the analysis of complex systems [Zadeh, 1965]. Fuzzy mathematics constitutes a new tool of dealing with imprecise or vague data. The definition of a fuzzy set is an extension of the classical Cantor set. We are still accustomed to our traditional bases of reasoning being strict and precise. In conventional binary logic a statement can be true or false, and there is no place for even a little uncertainty in this judgment. By looking at sets, we can state that an element either belongs to a set or does not. We call these kinds of sets crisp sets. In practice we often experience those real situations that are represented by crisp sets as impossible to describe accurately. If we assign a truth-value of one to the element that is included in the set, and a truth-value comparable to zero to such an element that lies outside the set, then we will create the range of two-valued logic. This sort of logic assumes that precise symbols must be employed, and it is therefore not applicable to the real existence but only to an imagined existence [Zadeh, 1965; Zimmermann, 2001; Rakus-Andersson, 2007]. Lotfi Zadeh referred to the last hypothesis when he wrote: “As the complexity of a system increases, our ability to make precise and yet significant statements about its behaviour diminishes until the threshold is reached beyond which precision and significance become almost mutually exclusive characteristics” [Zadeh, 1965]. If we consider the characteristic features of real world systems, we will conclude that real situations are very often uncertain or vague in a number of ways. If the information demanded by a system is lacking, the future state of such a system may not be known completely. This type of uncertainty has been handled by probability theories and statistics, and it is called stochastic uncertainty. The vagueness, concerning the description of the semantic meaning of the events, phenomena, or statements themselves, is called fuzziness [Zadeh, 1965]. Fuzziness can be found in many areas of daily life, especially in medicine. We look for the methods that help us to express the borders of such sets as “young”, “middle-aged”, “old”, “seldom”, “rarely”, “often” and the like. Thus we introduce the fuzzy apparatus to extend a notion of the set under the circumstances of vagueness [Rakus-Andersson, 2007]. Since the introduction in 1965, fuzzy set theory has been frequently applied in a wide range of areas like, e.g., dynamic systems, militaries, medicine and other domains. One of the successful trials of technical adaptations of the theory is the construction of the smoothest subway developed in Japan. Another theory, which copes with the problem of imprecision, is known as rough set theory [Pawlak, 1984, 1997, 2004; Małuszyński and Vitόria, 2002, Skowron, 2001]. Rough set theory was proposed by Professor Zdzisław Pawlak in Warsaw in the 1980ties. Whereas imprecision is expressed in the category of a membership degree in fuzzy set theory, this is a matter of the set approximation in rough set theory. Due to the definition of a rough set formulated by means of the decision attribute value, two approximate sets of the rough set are determined. These contain sure and possible members of the universe considered, in which the rough set has been defined. The objective of this study is to apply some classical methods of fuzzy set theory to medicine in order to estimate the survival length of gastric cancer patients and, by means of rough set classification, to verify the types of operations. These items will be discussed in conformity with the physicians’ wishes to support results of statistical investigations. The current research is funded by the scientific grant obtained from Blekinge Research Board.

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