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 ==Insieme sfocato <math>\tilde{A}</math> e funzione di appartenenza <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>==
-We choose - as a formalism - to represent a fuzzy set with the 'tilde':<math>\tilde{A}</math>. A fuzzy set is a set where the elements have a 'degree' of belonging (consistent with fuzzy logic): some can be included in the set at 100%, others in lower percentages.
+Scegliamo - come formalismo - di rappresentare un insieme sfocato con la 'tilde':<math>\tilde{A}</math>. Un insieme fuzzy è un insieme in cui gli elementi hanno un 'grado' di appartenenza (coerente con la logica fuzzy): alcuni possono essere inclusi nell'insieme al 100%, altri in percentuali inferiori.
-To mathematically represent this degree of belonging is the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> called ''''Membership Function''''. The function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> is a continuous function defined in the interval <math>[0;1]</math>where it is:
-*<math>\mu_ {\tilde {A}}(x) = 1\rightarrow </math> if <math>x</math> is totally contained in <math>A</math> (these points are called 'nucleus', they indicate <u>plausible</u> predicate values).
+A rappresentare matematicamente questo grado di appartenenza è la funzione <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> chiamata ''''Funzione di appartenenza'''<nowiki/>'. La funzione <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> è una funzione continua definita nell'intervallo <math>[0;1]</math> dove:
-*<math>\mu_ {\tilde {A}}(x) = 0\rightarrow </math> if <math>x</math> is not contained in <math>A</math>
-*<math>0<\mu_ {\tilde {A}}(x) < 1 \;\rightarrow </math> if <math>x</math> is partially contained in <math>A</math> (these points are called 'support', they indicate the <u>possible</u> predicate values).
-The graphical representation of the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> can be varied; from those with linear lines (triangular, trapezoidal) to those in the shape of bells or 'S' (sigmoidal) as depicted in Figure 1, which contains the whole graphic concept of the function of belonging.<ref>{{Cite book
+*<math>\mu_ {\tilde {A}}(x) = 1\rightarrow </math> se <math>x</math> è totalmente contenuta in <math>A</math> (questi punti sono chiamati 'nucleus', essi indicano i valori ''plausibili'' del predicato ).
+*<math>\mu_ {\tilde {A}}(x) = 0\rightarrow </math> se <math>x</math> non è contenuto in <math>A</math>
+*<math>0<\mu_ {\tilde {A}}(x) < 1 \;\rightarrow </math> se <math>x</math> è parzialmente contenuto in <math>A</math> (questi punti sono chiamati 'Support' ed indicano i valori possibili del predicato <u>possible</u> predicate values).
+La rappresentazione grafica della funzione <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> può essere variato; da quelli con linee lineari (triangolari, trapezoidali) a quelli a forma di campana o 'S' (sigmoidale) come rappresentato in Figura 1, che racchiude l'intero concetto grafico della funzione di appartenenza.<ref>{{Cite book
   | autore = Zhang W
   | autore2 = Yang J
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 [[File:Fuzzy_crisp.svg|alt=|left|thumb|400px|'''Figure 1:''' Types of graphs for the membership function.]]
-The '''support set''' of a fuzzy set is defined as the zone in which the degree of membership results <math>0<\mu_ {\tilde {A}}(x) < 1</math>; <!--131-->on the other hand, the '''core''' is defined as the area in which the degree of belonging assumes value <math>\mu_ {\tilde {A}}(x) = 1</math>
+The '''support set''' of a fuzzy set is defined as the zone in which the degree of membership results <math>0<\mu_ {\tilde {A}}(x) < 1</math>; on the other hand, the '''core''' is defined as the area in which the degree of belonging assumes value <math>\mu_ {\tilde {A}}(x) = 1</math>
 The 'Support set' represents the values of the predicate deemed '''possible''', while the 'core' represents those deemed more '''plausible'''.
-If <math>{A}</math> <!--134-->represented a set in the ordinary sense of the term or classical language logic previously described, its membership function could assume only the values <math>1</math> <!--135-->or <math>0</math>, <math>\mu_{\displaystyle {{A}}}(x)= 1 \; \lor \;\mu_{\displaystyle {{A}}}(x)= 0</math> <!--136-->depending on whether the element <math>x</math> <!--137-->belongs to the whole or not, as considered. <!--138-->Figure 2 shows a graphic representation of the crisp (rigidly defined) or fuzzy concept of membership, which clearly recalls Smuts's considerations.<ref name=":0">•SMUTS J.C. 1926, [[wikipedia:Holism_and_Evolution|<!--139-->Holism and Evolution]], London: Macmillan.</ref>
+If <math>{A}</math> represented a set in the ordinary sense of the term or classical language logic previously described, its membership function could assume only the values <math>1</math> or <math>0</math>, <math>\mu_{\displaystyle {{A}}}(x)= 1 \; \lor \;\mu_{\displaystyle {{A}}}(x)= 0</math> depending on whether the element <math>x</math> belongs to the whole or not, as considered. Figure 2 shows a graphic representation of the crisp (rigidly defined) or fuzzy concept of membership, which clearly recalls Smuts's considerations.<ref name=":0">•SMUTS J.C. 1926, [[wikipedia:Holism_and_Evolution|<!--139-->Holism and Evolution]], London: Macmillan.</ref>
 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:
 [[File:Fuzzy1.jpg|thumb|400x400px|'''<!--141-->Figure 2:''' <!--142-->Representation of the comparison between a classical and fuzzy ensemble.]]
-'''Figure 2:''' Let us imagine the Science Universe <math>U</math> <!--145-->in which there are two parallel worlds or contexts, <math>{A}</math> <!--146-->and <math>\tilde{A}</math>.
+'''Figure 2:''' Let us imagine the Science Universe <math>U</math> in which there are two parallel worlds or contexts, <math>{A}</math> and <math>\tilde{A}</math>.
-<math>{A}=</math>  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 <math>KB</math>  <!--148-->with a clear dividing line that we have named <math>KB_c</math>.
+<math>{A}=</math>  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 <math>KB</math>  with a clear dividing line that we have named <math>KB_c</math>.
-<math>\tilde{A}=</math> In another scientific context called  ‘fuzzy logic’, and in which there is a union between the subset <math>{A}</math> <!--150-->in <math>\tilde{A}</math> <!--151-->that we can go so far as to say: union between <math>KB_c</math>.
+<math>\tilde{A}=</math> In another scientific context called  ‘fuzzy logic’, and in which there is a union between the subset <math>{A}</math> in <math>\tilde{A}</math> that we can go so far as to say: union between <math>KB_c</math>.
 We will remarkably notice the following deductions: