Axiomatic Fuzzy Set Theory and Its Applications
In the age of Machine Intelligence and computerized decision making, we have to deal with subjective imprecision inherently associated with human perception and described in natural language and uncertainty captured in the form of randomness. This treatis
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Fundamentals
The main objective of this chapter is to introduce some preliminaries regarding essential mathematical notions and mathematical structures that have been commonly encountered in the theory of topological molecular lattices, fuzzy matrices, AFS (Axiomatic Fuzzy Set) structures and AFS algebras. The proofs of some theorems or propositions which are not too difficult to be proved are left to the reader as exercises.
1.1
Sets, Relations and Maps
Set theory provides important foundations of contemporary mathematics. Even if one is not particularly concerned what sets actually are, sets and set theory still form a powerful language for reasoning about mathematical objects. The use of the theory has spilled over into a number of related disciplines. In this section, we recall and systematize various standard set-theoretical notations and underlying constructs [6], proceeding with the development of the subject as far as the study of maps and relations is concerned.
1.1.1
Sets
We view the idea of set as a collection of objects as being a fairly obvious and quite intuitive. For many purposes we want to single out such a collection for attention, and it is convenient to be able to regard it as a single set. Usually we name a set by associating with some meaningful label so that later on we can easily refer to it. The objects which have been collected into the set are then called its members , or elements, and this relationship of membership is designated by the “included in” symbol ∈. Thus, a ∈ X is reads as ‘a is a member of the set X’ or just ‘a is in the set X’. An important point worth stressing here is that everything we can know about a set is provided by being told what members it is composed of. Put it another way, ‘two sets are equal if and only if they have the same members’. This is referred to as X. Liu and W. Pedrycz: Axiomatic Fuzzy Set Theory and Its Applications, STUDFUZZ 244, pp. 3–60. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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1 Fundamentals
the axiom of extensionality since it tells us that how far a set ‘extends’ is determined by what its members are. The other main principle of set formation is called the axiom of comprehension, which states that ‘for any property we can form a set containing precisely those objects with the given property’. This is a powerful principle which allows us to form a great variety of sets for all sorts of purposes. Unfortunately, with too full an interpretation of the word ‘property’ this gives rise to the Russell’s paradox: Let the property p(x) be x ∈ / x, that is, ‘x is not a member of itself’ and R be a set containing precisely those objects with the property p(x). This means that for every y, y ∈ R if and only if y ∈ / y. If this is true for every y, then it must be true when R is substituted for y, so that R ∈ R if and only if R ∈ / R. Since one of R ∈ R and R ∈ / R must hold, we arrive at an obvious contradiction which is the crux of the Russel paradox. This means that the axiom of comprehension must be restricted in some way
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