# Fuzzy Sets

### 1. Introduction

Fuzzy set theory was introduced in 1965 by prof. Lotfi Zadeh of UC Berkeley, in his article "Fuzzy sets", published in Information and Control 8, 338-353 (1965).
However there is an older article in which this concept already is mentioned and described. It is RAND Memorandum RM-4307-PR, "Abstraction and Pattern Classification", by R. Bellman, R. Kalaba and L.A. Zadeh, October 1964. It was only published in 1966 in the Journal of mathematical analysis and applications 13, 1-7.
In his article "Fuzzy Sets" Zadeh describes some important features of these "fuzzy sets". He also proves some theses.

### 2. Fuzzy Sets

Citation of the summery of Zadeh's article "Fuzzy Sets":
"A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notations in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint."

### 3. A closer description

Fuzzy logic is an extension of conventional (Boolean) logic that uses to work with the values {0,1} to the interval [0,1]. In conventional (Boolean) logic something is true (1) or false (0). It knows the "Law of the excluded middle". Fuzzy logic uses truth-values that can vary between 0 en 1. So for example some object or statemant can have a truth value of 0.8 or 0.5 or 0.2. Those truth values are also refered to as grades of membership. The membershipfunction in fuzzy logic can be compared with the characteristic function in conventional (Boolean) logic. The set connectives as Intersection and Union can be modelled by a lot of operators. Triangular norms and their duals are important set connectives.

### 4. Fuzzy numbers

You can use fuzzy numbers for fuzzy arithmetic. To show you how this works I made a Fuzzy Calculator. It gives an approximation of fuzzy arithmetic.