A three-way, three-valued, or three-region approximation of a fuzzy set is constructed from a pair of thresholds ( ; ) with 0 < 1 on its membership function. An element whose membership grade equals to or is greater than is put into the positive region, an element whose membership grade equals to or is less than is put into the negative region, and an element whose membership grade is between and
is put into the boundary region. A fundamental issue in constructing such three-way approximations is the interpretation and determination of a pair of thresholds on the unit interval [0; 1]. Shadowed sets, proposed by Pedrycz, are an example of three- way approximations of fuzzy sets, in which one obtains an analytic solution of the thresholds by searching for a balance of uncertainty introduced by the three regions. In this thesis, we introduce a general framework for determining the thresholds by optimizing an objective function. Within the framework, we critically review existing studies and present new formulations according to three principles, i.e., a principle of uncertainty invariance, a principle of minimum distance, and a principle of least cost. When applying the principle of uncertainty invariance, we maintain the uncertainty of a fuzzy set in a three-valued set. When applying the principle of minimum distance, we compute thresholds by minimizing the distance between a fuzzy set and a three- valued set. When applying the principle of minimum cost, we compute the thresholds by minimizing the costs of three-way approximations of a fuzzy set. In our model, membership grades are mapped to three points, denoted as n,m and p. Membership grades of objects in the positive region are mapped to p, membership grades of the objects in the boundary region are mapped to m, and membership grades of the objects in negative region are mapped to n. In our new formulation, fn;m; pg is a di erent system. The set fn;m; pg denotes points that do not necessarily belong to the unit interval [0; 1]. We introduce a distance function between elements of [0; 1] and elements of fn;m; pg and study properties that should be satis ed. The problem of nding a pair of thresholds is transformed into a problem of minimizing the cost of a three-valued set with respect to a distance function. ii