Verify the De Morgan’s law i.e.
(i) (A ∪ B)
0 = A0 ∩ B0
(ii) (A ∩ B)
0 = A0 ∪ B0
in set theory by taking suitable sets.
Answers
Answer:
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.