State and verify De-Morgans law.
Answers
De Morgan's laws 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; andthe 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; andthe 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; andnot (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}}}
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
and
where
P and Q are propositions,
is the negation logic operator (NOT),
is the conjunction logic operator (AND),
is the disjunction logic operator (OR),
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.
