Fuzzy Set-Theoretic Operators and Quantifiers [chapter]

János Fodor, Ronald R. Yager
2000 Fundamentals of Fuzzy Sets  
This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets. Since these extensions are mainly pointwisely defined, we review basic results on the underlying unary or binary operations on the unit interval such as negations, t-norms, t-conorms, implications, coimplications and equivalences. Some strongly related connectives (means, OWA, weighted, and prioritized
more » ... operations) are also considered, emphasizing the essential differences between these and the formerly investigated operator classes. We also show other operations which have no counterpart in the classical theory but play some important role in fuzzy sets (like symmetric sums, weak t-norms and conorms, compensatory AND). review the main results related to such extensions. In addition, an overview of similar problems concerning extensions of quantifiers is also given. Let X be a given set. In classical set theory intersection, union and complement of subsets of X are defined in a unique way. If A, B ~ X are crisp sets then for any a E X we have a E A n B if and only if a E A AND a E B, D. Dubois et al. (eds.), Fundamentals of Fuzzy Sets © Kluwer Academic Publishers 2000 126 FUNDAMENTALS OF FUZZY SETS a E AuB if and only if a E A OR a E B, a E AC if and only if NOT a E A. The uniqueness is due to the fact that AND, OR and NOT are two-valued logic operations. For example, if P1 and P2 are propositions being either TRUE or FALSE then P 1 AND ['2 is TRUE if and only if both P1 and P2 are TRUE. We can use a valuation of any proposition P in the following natural way: v(P) := 1 if and only if P is TRUE, v(P) := 0 if and only if P is FALSE. In fact, this valuation corresponds to the characte1'istic function XA : X ---+ {O, I} of a subset A ~ X when the proposition 'a E A' is considered for any By using characteristic functions and the usual connectives used in Boolean logic (A, V and -'), set-theoretic operations can be represented for all a E X as follows: XAnB ( a) XAU13 (a) Xk(a) XA(a) A XB(a), XA(a) V XB(a), -'XA(a). (2.1) Expressions in (2.1) imply that classical set-theoretic operations are defined pointwise. Note that we use the symbol := indicating that the item on its lefthand side is equal, by definition, to the already known item on its right-hand side. Also, m, n := {m, m + 1, ... , n -1, n} is used for any m,n E lN, m < n. As it is well-known, classical subsets of a given non-empty set X endowed with set-theoretic operations form a Boolean algebra structure (Sikorski, 1964) .
doi:10.1007/978-1-4615-4429-6_3 fatcat:obcyvr6pzfbttdlpmtlfjenyaa