## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

2.1 Fuzzy numbers and characterizing functions

In order to model one-dimensional fuzzy data the best up-to-date mathematical model is so-called fuzzy numbers.

Definition 2.1: A fuzzy number x* is determined by its so-called characterizing function ξ(·) which is a real function of one real variable x obeying the following:

1. ξ : → [0; 1].

2. ∀δ ∈ (0; 1] the so-called δ-cut Cδ(x*) :={x ∈ : ξ(x) ≥ δ} is a finite union of compact intervals, .

3. The support of ξ(·), defined by supp[ξ(·)] :={x ∈ : ξ(x) > 0} is bounded.

The set of all fuzzy numbers is denoted by .

For the following and for applications it is important that characterizing functions can be reconstructed from the family (Cδ(x*); δ ∈ (0; 1]), in the way described in Lemma 2.1.

Lemma 2.1:

For the characterizing function ξ(·) of a fuzzy number x* the following holds true: Proof:

For fixed x0 ∈ we have

Therefore we have for every ...

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required