# 2. What are fuzzy sets and who uses them?

Before talking about how to use fuzzy sets for pattern classification, we must first define what we mean by fuzzy sets. A great source of information on fuzzy sets and fuzzy logic can be found in a collection of frequently asked questions and corresponding answers.

## 2.1 Fuzzy sets vs. crisp sets

Crisp sets are the sets that we have used most of our life. In a crisp set, an element is either a member of the set or not. For example, a jelly bean belongs in the class of food known as candy. Mashed potatoes do not.

Fuzzy sets, on the other hand, allow elements to be partially in a set. Each element is given a degree of membership in a set. This membership value can range from 0 (not an element of the set) to 1 (a member of the set). It is clear that if one only allowed the extreme membership values of 0 and 1, that this would actually be equivalant to crisp sets. A membership function is the relationship between the values of an element and its degree of membership in a set. An example of membership functions are shown in Figure 2-1. In this example, the sets (or classes) are numbers that are negative large, negative medium, negative small, near zero, positive small, positive medium, and positive large. The value, µ, is the amount of membership in the set.

Figure 2-1: Membership Functions for the Set of All Numbers (N = Negative, P = Positive, L = Large, M = Medium, S = Small)

## 2.2 Applications

Fuzzy sets are appropriate for pattern classification because a given gesture or pattern may in fact have partial membership in many different classes. Several companies already have products based on fuzzy pattern recognition:

1. Hand Writing Recognition: CSK, Hitachi

2. Hand Printed Character Recognition: Sony

3. Voice Recognition: Ricoh, Hitachi