July 2020
Intermediate to advanced
384 pages
11h 25m
English
Content preview from A Modern Introduction to Fuzzy Mathematics
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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 ...
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