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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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Set theory
As mentioned in the previous chapter, the basic concept of fuzzy logic is that of multivalence, i.e., in terms of set theory, of the possibility that an object can belong to a set even partially and, therefore, also to several sets with different degrees. Let us recall from the beginning the basic elements of the theory of ordinary sets. As will be seen, in them appear the formal expressions of the principles of Aristotelian logic, recalled in the previous chapter.
Quantifiers
- Membership: represented by the symbol
(belongs), - for example the number 13 belongs to the set of odd numbers 
- Non-membership: represented by the symbol
(It does not belong)
- Inclusion: Represented by the symbol
(is content), - for example the whole 
it is contained within the larger set 
, 
(in this case it is said that is a subset of )
- Universal quantifier, which is indicated by the symbol
(for each)
- Demonstration, which is indicated by the symbol
(such that)