Understanding De Morgan’s Law | A Fundamental Principle in Logic and Set Theory

De Morgan’s Law

De Morgan’s Law is a fundamental principle in logic and set theory that describes the relationship between logical operators, such as negation, conjunction (AND), and disjunction (OR)

De Morgan’s Law is a fundamental principle in logic and set theory that describes the relationship between logical operators, such as negation, conjunction (AND), and disjunction (OR). It was named after the British mathematician and logician, Augustus De Morgan, who first formulated these laws.

De Morgan’s Law states that the negation of a conjunction (AND) or a disjunction (OR) is equivalent to the disjunction (OR) or conjunction (AND) of the negations of the individual terms, respectively.

In mathematical notation, De Morgan’s Law can be stated as follows:

1. The negation of a conjunction: ¬(P ∧ Q) is equivalent to (¬P) ∨ (¬Q).
This means that if both statements P and Q are false, then their conjunction (P ∧ Q) is false as well. On the other hand, if at least one of them is true, then the negation (¬(P ∧ Q)) will be true.

2. The negation of a disjunction: ¬(P ∨ Q) is equivalent to (¬P) ∧ (¬Q).
This means that if neither statement P nor statement Q is true, then their disjunction (P ∨ Q) is false. However, if at least one of them is true, then the negation (¬(P ∨ Q)) will be false.

De Morgan’s Law can also be extended to more than two terms. For example:

3. The negation of a conjunction of three terms: ¬(P ∧ Q ∧ R) is equivalent to (¬P) ∨ (¬Q) ∨ (¬R).

4. The negation of a disjunction of three terms: ¬(P ∨ Q ∨ R) is equivalent to (¬P) ∧ (¬Q) ∧ (¬R).

These laws are useful in simplifying logical expressions and manipulating Boolean algebra equivalences. They allow for the conversion between conjunctions and disjunctions, which can help in analyzing and solving logical problems.

More Answers: