Intuitionistic fuzzy sets

Let us have a fixed universe [math]\displaystyle{ E }[/math] and its subset [math]\displaystyle{ A }[/math]. The set

[math]\displaystyle{ A^* = \lbrace \langle x, \mu_A(x), \nu_A(x) \rangle \ | \ x \in E \rbrace }[/math]

where [math]\displaystyle{ 0 \leq \mu_A(x) + \nu_A(x) \leq 1 }[/math] is called intuitionistic fuzzy set.

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

We can define also function [math]\displaystyle{ \pi_A: E \to [0,1] }[/math] through

[math]\displaystyle{ \pi(x) = 1 - \mu (x) - \nu (x) }[/math]

and it corresponds to degree of indeterminacy (uncertainty, etc.).

For brevity, we shall write below [math]\displaystyle{ A }[/math] instead of [math]\displaystyle{ A^* }[/math], whenever this is possible.

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