By R. Lowen
The function of this e-book is to supply the reader who's drawn to purposes of fuzzy set thought, within the first position with a textual content to which she or he can refer for the elemental theoretical principles, ideas and methods during this box and within the moment position with an unlimited and recent account of the literature. even though there at the moment are many books approximately fuzzy set thought, and regularly approximately its functions, e. g. up to the mark concept, there isn't particularly a e-book on hand which introduces the simple thought of fuzzy units, in what i need to name "a reliable measure of generality". to jot down a booklet which might deal with the whole variety of effects in regards to the simple theoretical innovations in nice element and which might additionally take care of all attainable versions and choices of the idea, corresponding to e. g. tough units and L-fuzzy units for arbitrary lattices L, with the possibility-probability theories and interpretations, with the basis of fuzzy set concept through multi-valued good judgment or through express tools etc, may were an altogether diversified venture. This booklet is much extra modest in its mathematical content material and in its scope.
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Extra info for Fuzzy Set Theory: Basic Concepts, Techniques and Bibliography
All properties follow easily from the definitions. 4 Proposition lf X and Y are sets, I:X-Y is afunction, and arefamilies offuzzy sets on respectively X and Y, (/lj)jEJ' (Vj)jEJ then: 1. I (su p p,j) = sup I(p,j). JEJ 2. I 3. JEJ (j~} p,j) ~ j~} I(p,j). 1-1(SUP Vj) JEJ 4. ) JEJ J 1 =supl- (vj). JEJ = inf 1- 1(v·). JEJ J Proof. All properties follow at onee from the definitions. 0 At this point we would like to introduee what by many people working in fuzzy set theory is ealled the Extension Principle.
These are , intersection, union and complementation. These operations are important for set theory itself, but also because if one looks at the logical aspect of these operations, then they represent "and", "or" and "not". As is well known, from these basic logical connectives all other logical connectives are derived. We shall return to these aspects in detail in the chapter on fuzzy logic. For the moment it is sufficient to realize the importance of extending these three set theoretical operations to the setting of fuzzy sets.
Chapter 1. 6 Notes Set Theory There are many excellent textbooks on elementary set theory. One book in particular is worth mentioning since it is both interesting and pleasant to read, namely, "Naive Set Theory" by Paul R. Halmos [Van Nostrand 1960]. The reader should however be warned that the elementary set theory which we presented in this section is more elementary and naive than what is contained in the aforementioned book. We have not touched on matters such as axiomatics of set theory, or ordinals and cardinals since for our purposes this is not relevant.
Fuzzy Set Theory: Basic Concepts, Techniques and Bibliography by R. Lowen