DeMorgan's Theorem - ορισμός. Τι είναι το DeMorgan's Theorem
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Τι (ποιος) είναι DeMorgan's Theorem - ορισμός

PAIR OF TRANSFORMATION RULES THAT ARE BOTH VALID RULES OF INFERENCE
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  • De Morgan's Laws represented as a circuit with logic gates ([[International Electrotechnical Commission]] diagrams).
  • De Morgan's laws represented with [[Venn diagrams]]. In each case, the resultant set is the set of all points in any shade of blue.
  • url=https://www.researchgate.net/publication/366867569}}
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DeMorgan's theorem         
<logic> A logical theorem which states that the complement of a conjunction is the disjunction of the complements or vice versa. In symbols: not (x and y) = (not x) or (not y) not (x or y) = (not x) and (not y) E.g. if it is not the case that I am tall and thin then I am either short or fat (or both). The theorem can be extended to combinations of more than two terms in the obvious way. The same laws also apply to sets, replacing logical complement with set complement, conjunction ("and") with set intersection, and disjunction ("or") with set union. A (C) programmer might use this to re-write if (!foo && !bar) ... as if (!(foo || bar)) ... thus saving one operator application (though an {optimising compiler} should do the same, leaving the programmer free to use whichever form seemed clearest). (1995-12-14)
De Morgan's laws         
In propositional logic and Boolean algebra, De Morgan's lawsCopi and CohenMoore and Parker are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician.
Divergence theorem         
  • n}}
  • A volume divided into two subvolumes. At right the two subvolumes are separated to show the flux out of the different surfaces.
  • The volume can be divided into any number of subvolumes and the flux out of ''V'' is equal to the sum of the flux out of each subvolume, because the flux through the <span style="color:green;">green</span> surfaces cancels out in the sum. In (b) the volumes are shown separated slightly, illustrating that each green partition is part of the boundary of two adjacent volumes
  • </math> approaches <math>\operatorname{div} \mathbf{F}</math>
  • The divergence theorem can be used to calculate a flux through a [[closed surface]] that fully encloses a volume, like any of the surfaces on the left. It can ''not'' directly be used to calculate the flux through surfaces with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)
  • The vector field corresponding to the example shown. Vectors may point into or out of the sphere.
GENERALIZATION OF THE FUNDAMENTAL THEOREM IN VECTOR CALCULUS
Gauss' theorem; Gauss's theorem; Gauss theorem; Ostrogradsky-Gauss theorem; Ostrogradsky's theorem; Gauss's Theorem; Divergence Theorem; Gauss' divergence theorem; Ostrogradsky theorem; Gauss-Ostrogradsky theorem; Gauss Ostrogradsky theorem; Gauss–Ostrogradsky theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

Βικιπαίδεια

De Morgan's laws

In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

The rules can be expressed in English as:

  • The negation of a disjunction is the conjunction of the negations
  • The negation of a conjunction is the disjunction of the negations

or

  • The complement of the union of two sets is the same as the intersection of their complements
  • The complement of the intersection of two sets is the same as the union of their complements

or

  • not (A or B) = (not A) and (not B)
  • not (A and B) = (not A) or (not B)

where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B.

In set theory and Boolean algebra, these are written formally as

A B ¯ = A ¯ B ¯ , A B ¯ = A ¯ B ¯ , {\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}}

where

  • A {\displaystyle A} and B {\displaystyle B} are sets,
  • A ¯ {\displaystyle {\overline {A}}} is the complement of A {\displaystyle A} ,
  • {\displaystyle \cap } is the intersection, and
  • {\displaystyle \cup } is the union.

In formal language, the rules are written as

¬ ( P Q ) ( ¬ P ) ( ¬ Q ) , {\displaystyle \neg (P\lor Q)\iff (\neg P)\land (\neg Q),}

and

¬ ( P Q ) ( ¬ P ) ( ¬ Q ) {\displaystyle \neg (P\land Q)\iff (\neg P)\lor (\neg Q)}

where

  • P and Q are propositions,
  • ¬ {\displaystyle \neg } is the negation logic operator (NOT),
  • {\displaystyle \land } is the conjunction logic operator (AND),
  • {\displaystyle \lor } is the disjunction logic operator (OR),
  • {\displaystyle \iff } is a metalogical symbol meaning "can be replaced in a logical proof with", often read as "if and only if". For any combination of true/false values for P and Q, the left and right sides of the arrow will hold the same truth value after evaluation.

Applications of the rules include simplification of logical expressions in computer programs and digital circuit designs. De Morgan's laws are an example of a more general concept of mathematical duality.