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Intuitionistic fuzzy sets: Difference between revisions
New page: Let us have a fixed universe <math>E</math> and its subset <math>A</math>. The set <div align="center"> <math>A^* = \lbrace \langle x, \mu_A(x), \nu_A(x) \rangle \ | \ x \in E \rbrace</m... |
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We can define also function <math>\pi_A: E \to [0,1]</math> through | We can define also function <math>\pi_A: E \to [0,1]</math> through | ||
<div align="center"><math>\pi(x) = 1 - \mu (x) - \nu (x)</math></div> | <div align="center"><math>\pi(x) = 1 - \mu (x) - \nu (x)</math></div> | ||
and it corresponds to ''degree of indeterminacy (uncertainty, etc.)''. | and it corresponds to ''degree of [[indeterminacy]] (uncertainty, etc.)''. | ||
For brevity, we shall write below <math>A</math> instead of <math>A^*</math>, whenever this is possible. | For brevity, we shall write below <math>A</math> instead of <math>A^*</math>, whenever this is possible. | ||
Revision as of 12:14, 13 October 2008
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
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]