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===<translate>Set operators</translate>===

===<translate>Set operators</translate>===

Given the whole universe <math>U</math> we indicate with <math>x</math> its generic element such that <math>x \in U</math>; then we consider two subsets <math>A</math> and <math>B</math>  internal to <math>U</math> such that <math>A \subset U</math> and <math>B \subset U</math>

<translate>Given the whole universe</translate> <math>U</math> <translate>we indicate with</translate> <math>x</math> <translate>its generic element so that</translate> <math>x \in U</math>; <translate>then, we consider two subsets</translate> <math>A</math> and <math>B</math> <translate>internal to</translate> <math>U</math> <translate>so that</translate> <math>A \subset U</math> <translate>and</translate> <math>B \subset U</math>

{|

{|

|[[File:Venn0111.svg|sinistra|80px]]

|[[File:Venn0111.svg|left|80px]]

|'''Union:''' represented by the symbol <math>\cup</math>, indicates the union of the two sets <math>A</math> and <math>B</math> <math>(A\cup B)</math>. It is defined by all the elements that belong to <math>A</math> and <math>B</math> or both:

|'''<translate>Union</translate>:''' <translate>represented by the symbol</translate> <math>\cup</math>, <translate>indicates the union of the two sets</translate> <math>A</math> <translate>and</translate> <math>B</math> <math>(A\cup B)</math>. <translate>It is defined by all the elements that belong to</translate> <math>A</math> <translate>and</translate> <math>B</math> <translate>or both</translate>:

<math>(A\cup B)=\{\forall x\in U \mid x\in A \land x\in B\}</math>

<math>(A\cup B)=\{\forall x\in U \mid x\in A \land x\in B\}</math>

|-

|-

|[[File:Venn0001.svg|sinistra|80px]]

|[[File:Venn0001.svg|sinistra|80px]]

|'''Intersection:''' represented by the symbol <math>\cap</math>, indicates the elements belonging to both sets:

|'''<translate>Intersection</translate>:''' <translate>represented by the symbol</translate> <math>\cap</math>, <translate>indicates the elements belonging to both sets</translate>:

<math>(A\cap B)=\{\forall x\in U \mid x\in A \lor x\in B\}</math>

<math>(A\cap B)=\{\forall x\in U \mid x\in A \lor x\in B\}</math>

|-

|-

|[[File:Venn0010.svg|sinistra|80px]]

|[[File:Venn0010.svg|left|80px]]

|'''Difference:''' represented by the symbol <math>-</math>, for example <math>A-B</math> shows that all elements of <math>A</math> except those shared with <math>B</math>

|'''<translate>Difference</translate>:''' <translate>represented by the symbol</translate> <math>-</math>, <translate>for example</translate> <math>A-B</math> <translate>shows all elements of</translate> <math>A</math> <translate>except those shared with</translate> <math>B</math>

|-

|-

|[[File:Venn1000.svg|sinistra|80px]]

|[[File:Venn1000.svg|left|80px]]

|'''Complementary:''' represented by a bar above the name of the collection, it indicates by <math>\bar{A}</math> the complementary of <math>A</math>, that is, the set of elements that belong to the whole universe except those of <math>A</math>, in formulas: <math>\bar{A}=U-A</math><br />

|'''<translate>Complementary</translate>:''' <translate>represented by a bar above the name of the collection, it indicates by</translate> <math>\bar{A}</math> <translate>the complementary of</translate> <math>A</math>, <translate>that is</translate>, <translate>the set of elements that belong to the whole universe except those of</translate> <math>A</math>, <translate>in formulas</translate>: <math>\bar{A}=U-A</math><br />

|}The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid. Remember that in classical logic, given the set <math>A</math> and its complementary <math>\bar{A}</math>, the principle of non-contradiction states that if an element belongs to the whole <math>A</math> it cannot at the same time also belong to its complementary <math>\bar{A}</math>; according to the principle of the excluded third, however, the union of a whole <math>A</math> and its complementary <math>\bar{A}</math> constitutes the complete universe <math>U</math>.  

|}

 

<translate>The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid</translate>. <translate>Remember that in classical logic, given the set</translate> <math>A</math> <translate>and its complementary</translate> <math>\bar{A}</math>, <translate>the principle of non-contradiction states that if an element belongs to the whole</translate> <math>A</math> <translate>it cannot at the same time also belong to its complementary</translate> <math>\bar{A}</math>; <translate>according to the principle of the excluded third, however, the union of a whole</translate> <math>A</math> <translate>and its complementary</translate> <math>\bar{A}</math> <translate>constitutes the complete universe</translate> <math>U</math>.  

In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.

<translate>In other words, if any element does not belong to the whole, it must necessarily belong to its complementary</translate>.

==Fuzzy set <math>\tilde{A}</math> and membership function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>==

==<translate>Fuzzy set</translate> <math>\tilde{A}</math> <translate>and membership function</translate> <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.

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.