Difference between revisions of "Store:FLen03"

Latest revision as of 15:05, 13 March 2023

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 $\in$ (belongs), - for example the number 13 belongs to the set of odd numbers $\in$ $13\in Odd$
Non-membership: represented by the symbol $\notin$ (It does not belong)
Inclusion: Represented by the symbol $\subset$ (is content), - for example the whole $A$ it is contained within the larger set $U$ , $A\subset U$ (in this case it is said that $A$ is a subset of $U$ )
Universal quantifier, which is indicated by the symbol $\forall$ (for each)
Demonstration, which is indicated by the symbol $\mid$ (such that)

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-==Mengenlehre==
+==Set theory==
-Wie im vorigen Kapitel erwähnt, ist der Grundbegriff der Fuzzy-Logik der der Multivalenz, d. h. im Sinne der Mengenlehre die Möglichkeit, dass ein Objekt auch nur teilweise zu einer Menge und damit auch zu mehreren Mengen mit unterschiedlichem Grad gehören kann . Erinnern wir uns von Anfang an an die Grundelemente der Theorie der gewöhnlichen Mengen. Wie man sehen wird, erscheinen in ihnen die formalen Ausdrücke der Prinzipien der aristotelischen Logik, an die im vorigen Kapitel erinnert wurde.
+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.
-===Quantifizierer===
+===Quantifiers===
-*Mitgliedschaft: dargestellt durch das Symbol <math>\in </math> (gehört dazu), - zum Beispiel gehört die Zahl 13 zur Menge der ungeraden Zahlen <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>
-*Nichtmitgliedschaft: dargestellt durch das Symbol <math>\notin </math> (Es gehört nicht dazu)
+*Non-membership: represented by the symbol <math>\notin </math> (It does not belong)
-*Inklusion: Dargestellt durch das Symbol <math>\subset</math> (ist Inhalt), - zum Beispiel ist die ganze <math>A</math> in der größeren Menge <math>U</math> enthalten <math>A \subset U</math> (in diesem Fall sagt man, dass <math>A</math> eine Teilmenge von <math>U</math> ist
+*Inclusion: Represented by the symbol <math>\subset</math> (is content), - for example the whole <math>A</math> it is contained within the larger set <math>U</math>, <math>A \subset U</math> (in this case it is said that <math>A</math> is a subset of <math>U</math>)
-*Universalquantor, der durch das Symbol <math>\forall</math> (für jeden) gekennzeichnet ist
+*Universal quantifier, which is indicated by the symbol <math>\forall</math> (for each)
-*Demonstration, die durch das Symbol <math>\mid</math> (so dass) angezeigt wird
+*Demonstration, which is indicated by the symbol <math>\mid</math> (such that)