Analysis of Fuzzy Logic Models

Abstract

Example 1.1. Let F be the set of all fuzzy subsets of a set Ω.I ff : Ω → [0, 1] then we define A =(f ,1− f), i.e. ν A = 1 − μ A. Example 1.2. Let (Ω, S) be a measurable space, S a σ-algebra, F the family of all pairs such that μ A : Ω → [0, 1], ν A : Ω → [0, 1] are measurable. Then F is closed under the operations ⊕, ⊙, ¬. Example 1.3. Let (Ω, T) be a topological space, F the family of all pairs such that μ A : Ω → [0, 1], ν A : Ω → [0, 1] are continuous. Then F is closed under the operations⊕, ⊙, ¬. Remark. Of course, in any case A ⊕ B, A ⊙ B, ¬A are IF-sets, if A, B are IF-sets. E.g.

Real functions and the extension of generalized probability measures II

“…In [31], R i eč a n defined states and observables for IF-events and succeeded in proving limit theorems for IF-probability (avoiding complement).…”

“…This subsection is devoted to relationships between IF-probability [31]- [33] and fuzzy probability [15]- [18], [28].…”

“…The random events in IF-probability [31] are pairs (u, v) of fuzzy random events, i.e., elements of probability domains of the form M(A) × M(A) such that u ≤ v, and some relevant constructions in the IF-probability can be carried out (cf. [31], [32]) as the corresponding constructions in fuzzy probability theory (via M(A) × M(A)).…”

“…[31], [32]) as the corresponding constructions in fuzzy probability theory (via M(A) × M(A)). This section is devoted to products of D-posets of fuzzy sets and some of the results will be used in the next section.…”

“…The last two sections are devoted to the ralationships between IF-probability, introduced by B. R i eč a n in [31], and fuzzy probability [8], [18], [34].…”