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The theory of the vague units is a theory Mathématique field of the abstract Algèbre . It was developed by Lotfi Zadeh in 1965 in order to mathematically represent the inaccuracy relative to certain classes of objects and is used as base with the fuzzy Logique.

Presentation

The vague units (or vague parts) were introduced in order to model the human representation of knowledge, and thus to improve the performances of the systems of decision which use this modeling.

The vague units are used either to model uncertainty and the inaccuracy, or to represent accurate informations in assimilable lexical form by a Expert system.

Definition

The vague parts (or vague units) are defined like units being able to contain elements in a partial way.

Properties

• a fuzzy part $A$ of $B$ is characterized by an application of $B$ in $$. This application, called function of membership and noted $\backslash \; mu\_A$ represents the degree of validity of the proposal “$x$ belongs to $A$” for each element $x$ of $B$. If $\backslash \; mu\_A\; (X)\; =\; 1$, the $x$ object belongs completely to $A$, and if $\backslash \; mu\_A\; (X)\; =\; 0$, it does not belong to him at all. For an element $x$ given, the value of the function of membership $\backslash \; mu\_A\; (X)$ is called degree of membership of the element $x$ to the subset $A$.

• the core of a fuzzy part $A$ is the whole of the elements which belong completely to $A$ i.e. of which the degree of membership of $A$ is worth 1.

  $n\; (A)\; =\; \backslash \; \{X\; \backslash \; in\; B\; \backslash \; mid\; \backslash \; mu\_A\; (X)\; =1\; \backslash \}$
• the support of a fuzzy part $A$ is the whole of the elements belonging, even very little, with $A$ i.e. from which the degree of membership of $A$ is different from 0.$\backslash \; operatorname\; \{supp\}\; (A)\; =\; \backslash \; \{X\; \backslash \; in\; B\; \backslash \; mid\; \backslash \; mu\_A\; (X)\; >0\; \backslash \}$
• the height froma fuzzy subset has E is defined by $h\; (A)=\; \backslash \; sup\; \backslash \; \{\backslash \; mu\_A\; (X)\; \backslash \; mid\; X\; \backslash \; in\; B\; \backslash \}$.
• a vague under-part $A$ of $B$ can also be characterized by the whole of its α-cuts. A α-cut of a vague unit $A$ is the subset Net (traditional) elements having a degree of membership equal to or higher than α. $\backslash \; operatorname\; \{\backslash \; alpha-cut\}\; (A)\; =\; \backslash \; \{X\; \backslash \; in\; B\; \backslash \; mid\; \backslash \; mu\_A\; (X)\; \backslash \; geqslant\; \backslash \; alpha\; \backslash \}$

Remarks

The vague set theory is very different from the set theory, though it is based on the Fondements of mathematics.

An important difference is that, thanks to the Axiome of foundation, a finished unit has a number finished of subsets whereas it has an infinite number of fuzzy subsets.

See too

• Logical fuzzy
• Theory of the possibilities

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