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→Set theory
<translate>The importance and the charm of fuzzy logic arise from the fact that it is able to translate the uncertainty inherent in some data of human language into mathematical formalism, coding ‘elastic’ concepts (such as almost high, fairly good, etc.), in order to make them understandable and manageable by computers</translate>.
<translate>The importance and the charm of fuzzy logic arise from the fact that it is able to translate the uncertainty inherent in some data of human language into mathematical formalism, coding ‘elastic’ concepts (such as almost high, fairly good, etc.), in order to make them understandable and manageable by computers</translate>.
==Set theory==
==<translate>Set theory</translate>==
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.
<translate>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</translate>. <translate>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</translate>.
===Quantifiers===
===<translate>Quantifiers</translate>===
*Membership: represented by the symbol <math>\in </math> (belongs), - for example the number 13 belongs to the set of odd numbers <math>\in </math> <math>13\in Odd </math>
*Membership: represented by the symbol <math>\in </math> (belongs), - for example the number 13 belongs to the set of odd numbers <math>\in </math> <math>13\in Odd </math>
*Demonstration, which is indicated by the symbol <math>\mid</math> (such that)
*Demonstration, which is indicated by the symbol <math>\mid</math> (such that)
===Set operators===
===<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>
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>