# 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 $\lbrace \langle x, \mu_{A}(x), 1-\mu_{A}(x)\rangle |x \in E \rbrace.$