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Set operators

Given the whole universe $U$ we indicate with $x$ its generic element so that $x\in U$ ; then, we consider two subsets $A$ and $B$ internal to $U$ so that $A\subset U$ and $B\subset U$

	Union: represented by the symbol $\cup$ , indicates the union of the two sets $A$ and $B$ $(A\cup B)$ . It is defined by all the elements that belong to $A$ and $B$ or both: $(A\cup B)=\{\forall x\in U\mid x\in A\land x\in B\}$
	Intersection: represented by the symbol $\cap$ , indicates the elements belonging to both sets: $(A\cap B)=\{\forall x\in U\mid x\in A\lor x\in B\}$
	Difference: represented by the symbol $-$ , for example $A-B$ shows all elements of $A$ except those shared with $B$
	Complementary: represented by a bar above the name of the collection, it indicates by ${\bar {A}}$ the complementary of $A$ , that is, the set of elements that belong to the whole universe except those of $A$ , in formulas: ${\bar {A}}=U-A$

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 $A$ and its complementary ${\bar {A}}$ , the principle of non-contradiction states that if an element belongs to the whole $A$ it cannot at the same time also belong to its complementary ${\bar {A}}$ ; according to the principle of the excluded third, however, the union of a whole $A$ and its complementary ${\bar {A}}$ constitutes the complete universe $U$ .

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