Question

I want a note on de Morgans law

Answer 1

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; 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.
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 as
{\displaystyle \neg (P\lor Q)\iff (\neg P)\land (\neg Q),}
and
{\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".

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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