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Conjunto difuso ${\tilde {A}}$ y función de pertenencia $\mu _{\displaystyle {\tilde {A}}}(x)$

Elegimos, como formalismo, representar un conjunto borroso con la 'tilde': ${\tilde {A}}$ Un conjunto borroso es un conjunto donde los elementos tienen un 'grado' de pertenencia (de acuerdo con la lógica borrosa): algunos pueden incluirse en el conjunto en 100%, otros en porcentajes menores.

Para representar matemáticamente este grado de pertenencia se encuentra la función $\mu _{\displaystyle {\tilde {A}}}(x)$ denominada 'Función de pertenencia'. La función $\mu _{\displaystyle {\tilde {A}}}(x)$ es una función continua definida en el intervalo $[0;1]$ donde es:

• $\mu _{\tilde {A}}(x)=1\rightarrow$ si $x$ está totalmente contenido en $A$ (estos puntos se llaman 'núcleo', indican valores predicados plausibles).
• $\mu _{\tilde {A}}(x)=0\rightarrow$si $x$ no está contenido en $A$
• $0<\mu _{\tilde {A}}(x)<1\;\rightarrow$si $x$ está parcialmente contenido en $A$ (estos puntos se llaman 'soporte', indican los posibles valores predicados).

La representación gráfica de la función $\mu _{\displaystyle {\tilde {A}}}(x)$ puede ser variado; desde los de líneas lineales (triangulares, trapezoidales) hasta los que tienen forma de campana o 'S' (sigmoidales) como se muestra en la Figura 1, que contiene todo el concepto gráfico de la función de pertenencia....

Figure 1: Types of graphs for the membership function.

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Let us go back to the specific case of our Mary Poppins, in which we see a discrepancy between the assertions of the dentist and the neurologist and we look for a comparison between classical logic and fuzzy logic:

Figure 2: Representation of the comparison between a classical and fuzzy ensemble.

Figure 2: Let us imagine the Science Universe $U$ in which there are two parallel worlds or contexts, ${A}$ and ${\tilde {A}}$.

${A}=$ In the scientific context, the so-called ‘crisp’, and we have converted into the logic of Classic Language, in which the physician has an absolute scientific background information $KB$ with a clear dividing line that we have named $KB_{c}$.

${\tilde {A}}=$ In another scientific context called ‘fuzzy logic’, and in which there is a union between the subset ${A}$ in ${\tilde {A}}$ that we can go so far as to say: union between $KB_{c}$.

We will remarkably notice the following deductions:

• Classical Logic in the Dental Context ${A}$ in which only a logical process that gives as results $\mu _{\displaystyle {A}}(x)=1$ will be possible, or $\mu _{\displaystyle {A}}(x)=0$ being the range of data $D=\{\delta _{1},\dots ,\delta _{4}\}$ reduced to basic knowledge $KB$ in the set ${A}$. This means that outside the dental world there is a void and that term of set theory is written precisely $\mu _{\displaystyle {A}}(x)=0$ and which is synonymous with a high range of:

«Differential diagnostic error»

• Fuzzy logic in a dental context ${\tilde {A}}$ in which they are represented beyond the basic knowledge $KB$ of the dental context also those partially acquired from the neurophysiological world $0<\mu _{\tilde {A}}(x)<1$ will have the prerogative to return a result $\mu _{\tilde {A}}(x)=1$ and a result $0<\mu _{\tilde {A}}(x)<1$ because of basic knowledge $KB$ which at this point is represented by the union of $KB_{c}$ dental and neurological contexts. The result of this scientific-clinical implementation of dentistry would allow a
  «Reduction of differential diagnostic error»