What is Fuzzy Set Theory?

Fuzzy set theory is the extension of conventional (crisp) set theory. It handles the concept of partial truth (truth values between 1 (completely true) and 0 (completely false)). It was introduced by Prof. Lotfi A. Zadeh of UC/Berkeley in 1965 as a mean to model the vagueness and ambiguity in complex systems.

The idea of fuzzy sets is simple and natural. For instance, we want to define a set of gray levels that share the property dark. In classical set theory, we have to determine a threshold, say the gray level 100. All gray levels between 0 and 100 are element of this set, the others do not belong to the set (left image in Fig.1). But the darkness is a matter of degree. So, a fuzzy set can model this property much better. To define this set, we also need two thresholds, say gray levels 50 and 150. All gray levels that are less than 50 are the full member of the set, all gray levels that are greater than 150 are not the member of the set. The gray levels between 50 and 150, however, have a partial memebrship in the set (right image in Fig.1).

Fig.1. Representation of "dark gray-levels" with a crisp and a fuzzy set (Adapted from: Tizhoosh, Fuzzy Image Processing, © CopyRight Springer,1997)

For a detailed tutorial see the following pages:

• A tutorial by Brule

• A tutorial by IIPL

• Fuzzy Logic & Probability Theory

See also

• FAQ (newsgroup comp.ai.fuzzy)