Handling vagueness of verbal meanings and discreteness of word is very important to use verbal expressions in the field of engineering and so on. The former can be expressed by using fuzzy sets adequately, while suitable method to handle the latter has not been found yet. This paper aims to give the definition of the fuzzy set which expresses between one fuzzy set and the other in order to handle the discreteness. It is called "Between Set". After explaining mathematical conditions of support set and two original fuzzy sets and practical meanings of them, the membership function of between set is defined as the value subtracted absolute difference of membership values between two fuzzy sets from unity. It is described that between set has splendid characteristics as convexity, high performance to express elements and so on. Advantages of it are demonstrated in comparison it with the other sets calculated by another operators.

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Since fuzzy relation equations were firstly proposed by Sanchez in 1976,researches about the theory and applications of fuzzy relation equations were widely reported. In spite of this, it is difficult to find reports about methods to obtain only the set of independent solutions (solutions that are not contained in the others). In this paper, based on the method of solution defined by Tsukamoto and Tashiro, we propose the Interval Method. Using Tsukamoto's method, the Mid-martix , that is constructed as an intermediate step to find the solutions, may include empty set and one of the solutions which are included in another. In order to obtain easily and simply the solutions which are non-inclusive to one anothe, We introduce a greater/less Comparison Sign into a coefficient matrix, which is called Sign Matrix.An exampe shows that the proposed Interval Method gives effective solutions of the equations. In addition, by examining the Sign Matrix T, it is easy to determine the existance or non-existance of solutions.

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In the management and the social system which contain human factors, there exist many cases that data including the human vagueness are treated. In this paper, we propose the new methods of estimation of discriminant functions in cases of one variable and two variables for the approximate maximum likelihood estimation by the fuzzy data obtained from two groups. In usual observations, it is desired that we get accurate data as much as possible, but in the case treating the data with human vagueness, it is more realistic that we consider the event which is observed by the interval data such as classified as fuzzy intervals. In this paper, therefore, we defined the fuzzy data using the concept of Zadeh's probability of fuzzy events and proposed new methods, employing this fuzzy data, to estimate the discriminant functions by usual statistical calculation. Furthermore, we investigated the availability of our method by computer simulation under realistic situations. Consequently, it became clear that our methods are useful.

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This paper presents a study on hybrid linear-fuzzy model, based on the techniqueof fuzzy reasoning and mathematical quantification theory No.1. This model is for the subjective evaluation model of drivers of vehicles under different driving conditions. The linear equation generated by mathematical quantificationis for the linear factors of drivers' subjective evaluation. The other model, fuzzy reasoning model deals with its nonlinear factor. In order to easily realize this model into a simple system which does not require any specific exclusive fuzzy hardware like a fuzzy chip, the technique of simple forming fuzzy variables, namely fuzzified labels, and membership functions has been proposed. In generating fuzzy reasoning knowledge, average values and variant values of observed data are considered. The advantage of the hybrid model is best seen when the linear predictor cannot estimate nonlinear parameters in the simple modelling method. An example involving a vehicle-driversystem is given to show the merit of this modelling technique.

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The rapid and objective judgements of the freshness of raw oysters in Hiroshima district were carrid out using fuzzy reasoning based on the three kinds of biochemical indicator such as respiration activity of oyster gills (succinate dehydrogenase, SDH), pH and anmonia concentration of the body fluid. During the storage of oyster, SDH and pH were smoothly decreased with time elapsing, on the contraly, anmonia concentration was increased. It would suggest that these indicators were closely related to the freshness of oyster. By fuzzy reasoning based on these three indicators, a new system to estimate the freshness with the score of 0-100 point was developed. Estimated freshness points were in good agreement with the sensory judgements by seversl professionals of oyster farms. Moreover, the estimations of freshness by fuzzy reasoning indicated superior judgement compared to those by the conventional judgement using multiplicative average method which was developed previously.

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