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Fuzzy Systems - Fuzzy Set Theory

Fuzzy SystemsFuzzy Set TheoryProf. Dr. Rudolf Kruse Christian University of MagdeburgFaculty of Computer ScienceDepartment of Knowledge Processing and Language EngineeringR. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 2 Outline1. Representation of Fuzzy SetsCantor s TheoryAlpha-cutsProperties based on Alpha-cuts2. Fuzzy Set Operators based on Multi-valued LogicsDefinition of a set By a set we understand every collection made intoa whole of definite, distinct objects of our intuitionor of our thought. (Georg Cantor).For a set in Cantor s sense, the following propertieshold: x6={x}. Ifx XandX Y, thenx/ Y. The Set of all subsets ofXis denoted as 2X. is the empty set and thus very Cantor (1845-1918)R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 22 / 36 Extension to a Fuzzy setling. descriptionmodelall numbers smallerthan 10objective110)[characteristicfunction of asetall numbers almostequal to 10subjective110membershipfunction of a Fuzzy set DefinitionA Fuzzy set ofX6= is a function from the reference setXto theunit interval, :X [0,1].]

Fuzzy sets are usually stored as chain of linear lists. For each α-level, α 6= 0. A finite union of closed intervals is stored by their bounds. This data structure is appropriate for arithmetic operators. R. Kruse, C. Moewes FS – Fuzzy Set Theory Lecture 2 14 / 36

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