Verify any three de morgan's laws used in boolean algebra
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In propositional logic and boolean algebra, 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.
In set theory and Boolean algebra, these are written formally as
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
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.
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
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
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.
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