A Modern Introduction to Fuzzy Mathematics

3Fuzzy Numbers and Their Arithmetic

Fuzzy numbers are a mathematical formulation of vague statements about real numbers. For example, a statement expressing that a number is approximately 2 is modeled by a fuzzy number. Fuzzy numbers are of great importance in fuzzy systems and toward the end of this chapter, we are going to discuss how we can construct fuzzy numbers from real‐world data, and how these numbers can be used.

3.1 Fuzzy Numbers

Fuzzy (real) numbers are a special kind of fuzzy sets that have been introduced by Zadeh [311].

Definition 3.1.1  A fuzzy subset is called a fuzzy (real) number when the universe on which is defined is the set of all real numbers and it satisfies the following conditions:

1. all the α‐cuts of are not empty for ;
2. all the α‐cuts of are closed intervals of ;
3. the support of , that is, the set
   is bounded.

Clearly, every fuzzy number is a convex ...