Intuitionistic fuzzy sets

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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.