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Soft Set Theory First Results An Inlum~md computers & mathematics PERGAMON Computers and Mathematics with Applications 37 (1999) 19-31 Soft Set Theory First Results D. MOLODTSOV Computing Center of the R~msian Academy of Sciences 40 Vavilova Street, Moscow 117967, Russia Abstract--The soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the theory, and to discuss some problems of the future. (~) 1999 Elsevier Science Ltd. All rights reserved. Keywords--Soft set, Soft function, Soft game, Soft equilibrium, Soft integral, Internal regular- ization. 1. INTRODUCTION To solve complicated problems in economics, engineering, and environment, we cannot success- fully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Theory of probabilities can deal only with stochastically stable phenomena. Without going into mathematical details, we can say, e.g., that for a stochastically stable phenomenon there should exist a limit of the sample mean #n in a long series of trials. The sample mean #n is defined by 1 n IZn = -- ~ Xi, n i=l where x~ is equal to 1 if the phenomenon occurs in the trial, and x~ is equal to 0 if the phenomenon does not occur. To test the existence of the limit, we must perform a large number of trials. We can do it in engineering, but we cannot do it in many economic, environmental, or social problems. Interval mathematics have arisen as a method of taking into account the errors of calculations by constructing an interval estimate for the exact solution of a problem. This is useful in many cases, but the methods of interval mathematics are not sufficiently adaptable for problems with different uncertainties. They cannot appropriately describe a smooth changing of information, unreliable, not adequate, and defective information, partially contradicting aims, and so on. The most appropriate theory, for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [1]. We recall the definition of the notion of fuzzy set. For every set A C X, define its indicator function ~A 1, if x E A, I~A(X) = O, if x tg A. 0898-1221/99/$ - see front matter (~) 1999 Elsevier Science Ltd. All rights reserved. Typeset by AA4S-TEX PII: S0898-1221(99)00056-5 20 D. MOLODTSOV This correspondence between a set and its indicator function is obviously one-to-one corre- spondence. A fuzzy set F is described by its membership function #F. To every point x E X, this function associates a real number #F(X) in the interval [0, 1]. The number #F(X) is interpreted for the point as a degree of belonging x to the fuzzy set F. The theory of fuzzy sets offers natural, from the first glance, operations for fuzzy sets. Let F and G be fuzzy sets, and #F, #G be their membership functions. Then, the complement CF is defined by its membership function ,cF(x) = 1 - ,F(~). The intersection F N G can be defined by one of the following membership functions #FnV(X) = rain {pF(X), #G(X) } , ~FNG(X) = pF(X) " ]2G(Z), ~FnG(X) = m~x{0, ~F(X)+ ~,o(x) - 1}. There are three possibilities of membership functions for the union F U G PFuG(X) = max{#F(X), #G(X)}, ,FuG(X) = ,F(X) + ,G(X) -- ,F(X) " ,G(~), ~FuG(X) = min{1, #F(X), #G(X)}- At the present time, the theory of fuzzy sets is progressing rapidly. But there exists a difficulty: how to set the membership function in each particular case. We should not impose only one way to set the membership function. The nature of the membership function is extremely individual. Everyone may understand the notation #F(X) = 0.7 in his own manner. So, the fuzzy set operations based on the arithmetic operations with membership functions do not look natural. It may occur that these operations are similar to the addition of weights and lengths. The reason for these difficulties is, possibly, the inadequacy of the parametrization tool of the theory. In the next section, we propose a mathematical tool for dealing with uncertainties which is free of the difficulties mentioned above. 2. MAIN NOTIONS OF SOFT SET THEORY 2.1. Definition of the Soft Set To avoid difficulties, one must use an adequate parametrization. Let U be an initial universe set and let E be a set of parameters. DEFINITION 2.1. A pair (F,E) is called a soft set (over U) if and only if F is a mapping orE into the set of aft subsets of the set U. In other words, the soft set is a parametrized family of subsets of the set U. Every set F(e), e E E, from this family may be considered as the set of ~-elements of the soft set (F, E), or as the set of e-approximate elements of the soft set. As an illustration, let us consider the following examples. (1) A soft set (F, E) describes the attractiveness of the houses which Mr. X is going to buy. U - is the set of houses under consideration. E - is the set of parameters. Each parameter is a word or a sentence. E = {expensive; beautiful; wooden; cheap; in the green surroundings; modern; in good repair; in bad repair}. Soft Set Theory 21 In this case, to define a soft set means to point out expensive houses, beautiful houses, and so on. It is worth noting that the sets F(e) may be arbitrary. Some of them may be empty, some may have nonempty intersection. (2) Zadeh's fuzzy set may be considered as a special case of the soft set. Let A be a fuzzy set, and #A be the membership function of the fuzzy set A, that is ~A is a mapping of U into [0, 1]. Let us consider the family of c~-level sets for function ~tA F(c~) = {x E U[#A(x) >_ a}, c~ e [0,1]. If we know the family F, we can find the functions #A(X) by means of the following formulae: ~A(x)= sup a. ~e[0,1] xEF(ct) Thus, every Zadeh's fuzzy set A may be considered as the soft set (F, [0, 1]). (3) Let (X, r) - be a topological space, that is, X is a set and ~ is a topology, in other words, r is a family of subsets of X, called the open sets of X. Then, the family of open neighborhoods T(x) of point x, where T(x) = {V e r I x E V}, may be considered as the soft set (T(x), r). The way of setting (or describing) any object in the soft set theory principally differs from the way in which we use classical mathematics. In classical mathematics, we construct a mathematical model of an object and define the notion of the exact solution of this model. Usually the mathematical model is too complicated and we cannot find the exact solution. So, in the second step, we introduce the notion of approximate solution and calculate that solution. In the soft set theory, we have the opposite approach to this problem. The initial description of the object has an approximate nature, and we do not need to introduce the notion of exact solution. The absence of any restrictions on the approximate description in soft set theory makes this theory very convenient and easily applicable in practice. We can use any parametrization we prefer: with the help of words and sentences, real numbers, functions, mappings, and so on. It means that the problem of setting the membership function or any similar problem does not arise in the soft set theory. 2.2. Operations with Soft Sets Assume that we have a binary operation, denoted by *, for subsets of the set U. Let (F, A) and (G, B) be soft sets over U. Then, the operation * for soft sets is defined in the following way: (F, A) * (G, B) = (H, A × B), where H(~, ~) = F(a) • G(~), c~ 6 A,/3 6 B, and A x B is the Cartesian product of the sets A and B. This definition takes into account the individual nature of any soft set. If we produce a lot of operations with soft sets, the result will be a soft set with a very wide set of parameters. Sometimes such expansion of the set of parameters may be useful. So, the intersection of the soft set from Example 1 with itself gives the soft set with more detailed description. The resulting soft set points out the houses which are expensive and beautiful, modern and cheap, and so on. 22 D. MOLODTSOV In cases when such expansion of the set of parameters is not convenient, we may use a lot of cutting operations. Of course, the acceptance of these cutting operations depends on the special case and on the problem under consideration. If we want to construct a general mathematical tool, we do not introduce a universal cutting operation for the set of parameters. If we look at the operations with fuzzy sets from the point of view of the soft set theory, we will figure out that all binary operations with fuzzy sets include the universal cutting operation.
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