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<translate>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</translate>. <translate>Remember that in classical logic, given the set</translate> <math>A</math> <translate>and its complementary</translate> <math>\bar{A}</math>, <translate>the principle of non-contradiction states that if an element belongs to the whole</translate> <math>A</math> <translate>it cannot at the same time also belong to its complementary</translate> <math>\bar{A}</math>; <translate>according to the principle of the excluded third, however, the union of a whole</translate> <math>A</math> <translate>and its complementary</translate> <math>\bar{A}</math> <translate>constitutes the complete universe</translate> <math>U</math>.
<translate>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</translate>. <translate>Remember that in classical logic, given the set</translate> <math>A</math> <translate>and its complementary</translate> <math>\bar{A}</math>, <translate>the principle of non-contradiction states that if an element belongs to the whole</translate> <math>A</math> <translate>it cannot at the same time also belong to its complementary</translate> <math>\bar{A}</math>; <translate>according to the principle of the excluded third, however, the union of a whole</translate> <math>A</math> <translate>and its complementary</translate> <math>\bar{A}</math> <translate>constitutes the complete universe</translate> <math>U</math>.
In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.
<translate>In other words, if any element does not belong to the whole, it must necessarily belong to its complementary</translate>.
==Fuzzy set <math>\tilde{A}</math> and membership function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>==
==<translate>Fuzzy set</translate> <math>\tilde{A}</math> <translate>and membership function</translate> <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>==
We choose - as a formalism - to represent a fuzzy set with the 'tilde':<math>\tilde{A}</math>. A fuzzy set is a set where the elements have a 'degree' of belonging (consistent with fuzzy logic): some can be included in the set at 100%, others in lower percentages.
We choose - as a formalism - to represent a fuzzy set with the 'tilde':<math>\tilde{A}</math>. A fuzzy set is a set where the elements have a 'degree' of belonging (consistent with fuzzy logic): some can be included in the set at 100%, others in lower percentages.