Set-valued set theory. I.
E. William Chapin · Notre Dame Journal of Formal Logic · 1974
The Informal Systems In this section, we will consider the original exposition of Zadeh [3] and one of the more recent versions of the theory by Brown [l], both for the purpose of introducing the reader to the intuitive ideas to be formalized and so that the reader may see first hand some of the difficulties involved in the earlier formulations of the theory.The Original System of Zadeh In [3], Zadeh defines fuzzy sets to be functions from some ordinary set X to the unit interval [0, l].(Strictly speaking, Zadeh says that fuzzy sets are characterized by these functions, but for our purposes, we may identify the set and the function that characterizes it.)Thus these functions are generalizations of the ordinary characteristic functions of the subsets of X. Zadeh tells us that such fuzzy sets are to represent classes of elements "with a continuum of grades of membership" ([3], p. 339), "classes of objects encountered in the real physical world [which] do not have precisely defined criteria of membership" ([3], p. 338).Among his examples is the "class of all real numbers which are much greater than one"; he indicates that such classes "play an important role in human thinking, particularly in the domains of pattern recognition, communication of information, and abstraction" ([3], p. 338).He further notes that "the notion of a fuzzy set is completely nonstatistical in nature" (£3], p. 340) and that the concept of fuzzy set "provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables" ([3], p. 339).Zadeh continues ([3], pp.340-341) with the following definitions:A fuzzy set is empty if it is the constant function zero.Two fuzzy sets are equal if they are equal as functions.The complement of a fuzzy set / is the function defined by: fThe fuzzy set/ is a subset of the fuzzy set g if and only if, for all x in X, Ax)ίg{x).The union of the fuzzy sets / and g is the function / U g defined by (fug)(x) = maχ[f(x), g(x)] (equivalently, the smallest fuzzy set having both /and g as subsets).