A fuzzy set is a set that is defined by a membership function. By defining a set using a membership function, it is possible for an element to belong partially to a set. For example, if a door is slightly ajar, one might say that the door is open, with a membership grade of 0.2 to indicate that the door is slightly open. We might also say that the door is closed, with a membership grade of 0.8. By using a fuzzy set, we are able to indicate that the door is partially open or partially closed.
Fuzzy logic is an extension of classical logic and uses fuzzy sets rather than classical sets.
There are a few different explanations of what fuzzy logic is, so rather than add our own explanation, we will quote one explanation put forth by Lotfi A. Zadeh , the father of fuzzy logic. But in a broader and much more significant sense, fuzzy logic is coextensive with the theory of fuzzy sets, that is, classes of objects in which the transition from membership to nonmembership is gradual rather than abrupt. In its wider sense, fuzzy logic has many branches ranging from fuzzy arithmetic and fuzzy automata to fuzzy pattern recognition, fuzzy languages, and fuzzy expert systems. Still others are putting it to work in pattern recognition, economics, data analysis, and other areas that involve a high level of uncertainty, complexity, or nonlinearity. There are presently numerous applications that incorporate fuzzy logic control.