Intuitionistic fuzzy sets

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.).

For brevity, we shall write below $$A$$ instead of $$A^*$$, whenever this is possible.

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