Fuzzy Logic with Engineering Applications, 4th Edition

2Classical Sets and Fuzzy Sets

Philosophical objections may be raised by the logical implications of building a mathematical structure on the premise of fuzziness, since it seems (at least superficially) necessary to require that an object be or not be an element of a given set. From an aesthetic viewpoint, this may be the most satisfactory state of affairs, but to the extent that mathematical structures are used to model physical actualities, it is often an unrealistic requirement….Fuzzy sets have an intuitively plausible philosophical basis. Once this is accepted, analytical and practical considerations concerning fuzzy sets are in most respects quite orthodox.

James Bezdek, 1981Professor, Computer Science

Quantum mechanics brought an unexpected fuzziness into physics because of quantum uncertainty, the Heisenberg uncertainty principle.

Edward Witten, twentieth‐century American Mathematician

As alluded to in Chapter 1, the universe of discourse is the universe of all available information on a given problem. Once this universe is defined we are able to define certain events on this information space. I will describe sets as mathematical abstractions of these events and of the universe itself. Figure 2.1a shows an abstraction of a universe of discourse, say X, and a crisp (classical) set A somewhere in this universe. A classical set is defined by crisp boundaries, that is, there is no uncertainty in the prescription or location of the boundaries of the set, as shown ...