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|[[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>

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|[[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>

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|[[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>.  

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<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.

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