Mathematical Foundations of Fuzzy Sets

4Generalized Extension Principle

The main purpose of the extension principle is to fuzzify the crisp functions into fuzzy functions. This kind of fuzzification has been widely used to study economics and engineering problems under a fuzzy environment. Especially, the arithmetics of fuzzy sets are based on the extension principle.

4.1 Extension Principle Based on the Euclidean Space

The set of all fuzzy sets in is denoted by . We consider an onto function , which is also called a crisp function. We are going to fuzzify the function as a function , which is also called a fuzzy function. In other words, for any , we have . The principle for fuzzifying the crisp functions is called the extension principle.

For any set in , the set is defined by

Since is onto, we have that the set ...