De Morgan's Law

De Morgan's Law is a fundamental principle in mathematics and logic that states that the negation of a conjunction (AND) or a disjunction (OR) is equivalent to the disjunction or conjunction, respectively, of the negations of the individual statements. In other words, it provides a way to simplify complex logical expressions by transforming them into equivalent expressions that are easier to work with.

Formally, De Morgan's Law can be stated as follows:

- The negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations of the individual statements. That is, ¬(P ∧ Q) ≡ (¬P) ∨ (¬Q).
- The negation of a disjunction (OR) is equivalent to the conjunction (AND) of the negations of the individual statements. That is, ¬(P ∨ Q) ≡ (¬P) ∧ (¬Q).

For example, suppose we have the statement "It is not raining and it is not cloudy." Using De Morgan's Law, we can transform this into the equivalent statement "It is either raining or cloudy." This is because ¬(not raining AND not cloudy) is the same as saying either it is raining (not not raining) OR it is cloudy (not not cloudy).

De Morgan's Law is a useful tool in many areas of mathematics and computer science, including Boolean algebra, logic circuits, and set theory.