A Modern Introduction to Fuzzy Mathematics

4Fuzzy Relations

Fuzzy relations are to (crisp) relations what fuzzy sets are to (crisp) sets. In particular, elements of some sets are related to a certain degree just like elements of some universe belong to a fuzzy set to some degree. Fuzzy relations have been introduced and further studied by Zadeh [305, 307]. Fuzzy relations are important tools that are used in fuzzy modeling, fuzzy diagnosis, and fuzzy control, which explains why it is useful to have a good understanding of fuzzy relations and their properties.

4.1 Crisp Relations

The concept of a relation is a basic concept in both mathematics and in everyday life. For example, when one has a set of keywords and a set of documents, she can create an association between the keywords and the documents, and this association is a relation. Fuzzy relations extend crisp relations just like fuzzy sets extend crisp sets. Thus, if the keywords and characterize document , then in a fuzzy theoretic setting one can say that these keywords characterize the document to some degree. In order to better understand the ideas about fuzzy relations, it is necessary to have a good understanding of crisp relations. In what follows, we will ...