Applying Fuzzy Logic and Fuzzy Methods to Marketing

Crisp Sets Vs. Fuzzy Sets

The concept of a set or a collection of objects is common in marketing and relationship management. For instance, all customers with their properties, such as name, age, address, and customer value are stored in the customer database or data warehouse. The objects in the set are called the elements of the set. Traditional data sets are also called ordinary or crisp sets in order to distinguish them from fuzzy sets.

An important notion in set theory is that of membership. If an element x belongs to a set A then x ∈ A, otherwise x ∉ A. For each element x of a set A there are only two possibilities: either x belongs to A or it does not. The membership rule that characterizes the elements of a set A can be described by the characteristic function. The characteristic function χ takes only two values 1 and 0 or ‘true’ and ‘false’. More precisely: if X is the universal set (universe of discourse) and A is a subset A ⊂ X then the characteristic function χ of a set A indicates whether or not x belongs to A, i.e., χ_A: X −> {0,1}. In other words every crisp set is uniquely defined by its characteristic function.