*2.1. Fuzzy Sets*

Zadeh [1] introduced the concept of fuzzy set *A* over the universe *X* as a collection of pairs *x*, *<sup>μ</sup>A*(*x*) such that the first component of each pair *x* ∈ *X* is an element of the universe, while the second element *<sup>μ</sup>A*(*x*) ∈ [0, 1] is its corresponding membership degree. Function *μA* : *X* → [0, 1] is called the membership function of the fuzzy set *A*.

Atanassov [12] introduced the intuitionistic fuzzy sets as a generalization to the fuzzy sets. An intuitionistic fuzzy set *A<sup>I</sup>* of a universe *X* is a set of triples

$$\left(\mathbf{x}, \mu\_{\overline{A}^l}(\mathbf{x}), \nu\_{\overline{A}^l}(\mathbf{x})\right) \tag{1}$$

such that *x* ∈ *X*, *<sup>μ</sup>AI*(*x*), *<sup>ν</sup>AI*(*x*) ∈ [0, 1], and 0 ≤ *<sup>μ</sup>AI*(*x*) + *<sup>ν</sup>AI*(*x*) ≤ 1. The membership function of *A<sup>I</sup>* is *μA<sup>I</sup>* : *X* → [0, 1], and the nonmembership function of *A<sup>I</sup>* is *<sup>ν</sup>A<sup>I</sup>* : *X* → [0, 1] of *A<sup>I</sup>* in *X*. For each *x* ∈ *X* the value *<sup>μ</sup>AI*(*x*), called the membership degree, the value *<sup>ν</sup>AI*(*x*) is called the nonmembership degree, and the value

$$h(\mathbf{x}) = 1 - \mu\_{\tilde{A}^I}(\mathbf{x}) - \nu\_{\tilde{A}^I}(\mathbf{x}) \tag{2}$$

is called the degree of hesitancy of *x* in *AI*. Deng [13] proposed a new way to measure the information volume of fuzzy and intuitionistic fuzzy membership functions.
