State and prove the Demorgan’s theorems.
Answers
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.
In propositional logic and boolean algebra, De Morgan's laws[1][2][3] 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; and
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; and
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; and
not (A and B) = not A or not B
In set theory and Boolean algebra, these are written formally as
{\displaystyle {\begin{aligned}{\overline {A\cup B}}&={\overline {A}}\cap {\overline {B}},\\{\overline {A\cap B}}&={\overline {A}}\cup {\overline {B}},\end{aligned}}} {\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 and B are sets,
A is the complement of A,
∩ is the intersection, and
∪ is the union.
In formal language, the rules are written as
{\displaystyle \neg (P\lor Q)\iff (\neg P)\land (\neg Q),} \neg (P\lor Q)\iff (\neg P)\land (\neg Q),
and
{\displaystyle \neg (P\land Q)\iff (\neg P)\lor (\neg Q)} \neg (P\land Q)\iff (\neg P)\lor (\neg Q)
where
P and Q are propositions,
{\displaystyle \neg } \neg is the negation logic operator (NOT),
{\displaystyle \land } \land is the conjunction logic operator (AND),
{\displaystyle \lor } \lor is the disjunction logic operator (OR),
{\displaystyle \iff } \iff is a metalogical symbol meaning "can be replaced in a logical proof with".
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.
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