fuzzy relation

File:Cartesian Product qtl1.svg

A fuzzy relation is the cartesian product of mathematical fuzzy sets.{{cite book|author=Timothy J. Ross|title=Fuzzy Logic with Engineering Applications|url=https://books.google.com/books?id=3zcgIKPl8L0C|date=8 April 2005|publisher=John Wiley & Sons|isbn=978-0-470-86076-2|pages=59–}} Two fuzzy sets are taken as input, the fuzzy relation is then equal to the cross product of the sets which is created by vector multiplication. Usually, a rule base is stored in a matrix notation which allows the fuzzy controller to update its internal values.{{cite book|author=Galindo, Jos|title=Handbook of Research on Fuzzy Information Processing in Databases|url=https://books.google.com/books?id=vaY9PTkrn3gC&pg=PA17|date=31 May 2008|publisher=IGI Global|isbn=978-1-59904-854-3|pages=17–}}{{cite book|author=Witold Pedrycz|title=Fuzzy Modelling: Paradigms and Practice|url=https://books.google.com/books?id=31ngwZ8l-C0C&pg=PA39|date=31 March 1996|publisher=Springer Science & Business Media|isbn=978-0-7923-9703-8|pages=39–}}

From a historical perspective, the first fuzzy relation was mentioned in the year 1971 by Lotfi A. Zadeh.{{cite book|author1=Witold Pedrycz|author2=Fernando Gomide|title=Fuzzy Systems Engineering: Toward Human-Centric Computing|url=https://books.google.com/books?id=VIFUBIMDqLsC&pg=PA156|date=12 October 2007|publisher=John Wiley & Sons|isbn=978-0-470-16895-0|pages=156–}}

A practical approach to describe a fuzzy relation is based on a 2d table. At first, a table is created which consists of fuzzy values from 0..1. The next step is to apply the if-then-rules to the values. The resulting numbers are stored in the table as an array.

Fuzzy relations can be utilized in fuzzy databases.{{cite book|author=Zongmin Ma|title=Fuzzy Database Modeling with XML|url=https://books.google.com/books?id=cR4ByFywfAUC|date=30 March 2006|publisher=Springer Science & Business Media|isbn=978-0-387-24249-1|page=97–}}