Intuitionistic fuzzy sets

Intuitionistic fuzzy sets are sets whose elements have degrees of membership and non-membership. Intuitionistic fuzzy sets have been introduced by Krassimir Atanassov (1983) as an extension of Lotfi Zadeh's notion of fuzzy set, which itself extends the classical notion of a set.
 * In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set.
 * As an extension, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1].
 * The theory of intuitionistic fuzzy sets further extends both concepts by allowing the assessment of the elements by two functions: $$\mu$$ for membership and $$\nu$$ for non-membership, which belong to the real unit interval [0, 1] and whose sum belongs to the same interval, as well.

Intuitionistic fuzzy sets generalize fuzzy sets, since the indicator functions of fuzzy sets are special cases of the membership and non-membership functions $$\mu$$ and $$\nu$$ of intuitionistic fuzzy sets, in the case when the strict equality exists: $$\nu = 1 - \mu$$, i.e. the non-membership function fully complements the membership function to 1, not leaving room for any uncertainty.

Formal definition
Let us have a fixed universe $$E$$ and its subset $$A$$. The set

$$A = \lbrace \langle x, \mu_A(x), \nu_A(x) \rangle \ | \ x \in E \rbrace$$

where $$0 \leq \mu_A(x) + \nu_A(x) \leq 1$$ is called intuitionistic fuzzy set.

Functions $$\mu_A: E \to [0,1]$$ and $$\nu_A: E \to [0,1]$$ represent degree of membership (validity, etc.) and non-membership (non-validity, etc.).

We can define also function $$\pi_A: E \to [0,1]$$ through $$\pi(x) = 1 - \mu (x) - \nu (x)$$ and it corresponds to degree of indeterminacy (uncertainty, etc.).

Obviously, for every ordinary fuzzy set $$A$$: $$\pi_A(x) = 0$$ for each $$x \in E$$ and these sets have the form $$\lbrace \langle x, \mu_{A}(x), 1-\mu_{A}(x)\rangle  |x \in E \rbrace.$$