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===Set operators===

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

<math>(A\cup B)=\{\forall x\in U \mid x\in A \land x\in B\}</math>
|-
|[[File:Venn0001.svg|sinistra|80px]]
|'''Intersection:''' represented by the symbol <math>\cap</math>, indicates the elements belonging to both sets:

<math>(A\cap B)=\{\forall x\in U \mid x\in A \lor x\in B\}</math>
|-
|[[File:Venn0010.svg|left|80px]]
|'''Difference:''' represented by the symbol <math>-</math>, for example <math>A-B</math> shows all elements of <math>A</math> except those shared with <math>B</math>
|-
|[[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 />
|}

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>.

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