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| *<!--66-->Universal quantifier, <!--67-->which is indicated by the symbol <math>\forall</math> (for each) | | *<!--66-->Universal quantifier, <!--67-->which is indicated by the symbol <math>\forall</math> (for each) |
| *<!--68-->Demonstration, <!--69-->which is indicated by the symbol <math>\mid</math> (such that) | | *<!--68-->Demonstration, <!--69-->which is indicated by the symbol <math>\mid</math> (such that) |
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| ===<!--70-->Set operators===
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| <!--71-->Given the whole universe <math>U</math> <!--72-->we indicate with <math>x</math> <!--73-->its generic element so that <math>x \in U</math>; <!--74-->then, we consider two subsets <math>A</math> and <math>B</math> <!--75-->internal to <math>U</math> <!--76-->so that <math>A \subset U</math> <!--77-->and <math>B \subset U</math>
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| {|
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| |[[File:Venn0111.svg|left|80px]]
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| |'''<!--78-->Union:''' <!--79-->represented by the symbol <math>\cup</math>, <!--80-->indicates the union of the two sets <math>A</math> <!--81-->and <math>B</math> <math>(A\cup B)</math>. <!--82-->It is defined by all the elements that belong to <math>A</math> <!--83-->and <math>B</math> <!--84-->or both:
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| <math>(A\cup B)=\{\forall x\in U \mid x\in A \land x\in B\}</math>
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| |[[File:Venn0001.svg|sinistra|80px]]
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| |'''<!--85-->Intersection:''' <!--86-->represented by the symbol <math>\cap</math>, <!--87-->indicates the elements belonging to both sets:
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| <math>(A\cap B)=\{\forall x\in U \mid x\in A \lor x\in B\}</math>
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| |[[File:Venn0010.svg|left|80px]]
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| |'''<!--88-->Difference:''' <!--89-->represented by the symbol <math>-</math>, <!--90-->for example <math>A-B</math> <!--91-->shows all elements of <math>A</math> <!--92-->except those shared with <math>B</math>
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| |[[File:Venn1000.svg|left|80px]]
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| |'''<!--93-->Complementary:''' <!--94-->represented by a bar above the name of the collection, it indicates by <math>\bar{A}</math> <!--95-->the complementary of <math>A</math>, <!--96-->that is, <!--97-->the set of elements that belong to the whole universe except those of <math>A</math>, <!--98-->in formulas: <math>\bar{A}=U-A</math><br />
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| |}
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| <!--99-->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. <!--100-->Remember that in classical logic, given the set <math>A</math> <!--101-->and its complementary <math>\bar{A}</math>, <!--102-->the principle of non-contradiction states that if an element belongs to the whole <math>A</math> <!--103-->it cannot at the same time also belong to its complementary <math>\bar{A}</math>; <!--104-->according to the principle of the excluded third, however, the union of a whole <math>A</math> <!--105-->and its complementary <math>\bar{A}</math> <!--106-->constitutes the complete universe <math>U</math>.
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| <!--107-->In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.
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