De Morgan’s laws
De Morgan’s laws are a set of two fundamental rules in Boolean algebra, named after the British mathematician Augustus De Morgan
De Morgan’s laws are a set of two fundamental rules in Boolean algebra, named after the British mathematician Augustus De Morgan. These laws describe how logical operators (AND, OR, NOT) can be combined and manipulated.
The first law, also known as De Morgan’s law of negation, states that the negation of a logical conjunction (AND) is equivalent to the logical disjunction (OR) of the negations of the individual statements. Symbolically, it can be represented as:
¬(p ∧ q) ≡ ¬p ∨ ¬q
This means that if we have a statement that says both statement p and statement q are true, then the negation of that statement is the same as saying that either statement p is false or statement q is false.
For example, let’s say we have the statements:
p: It is raining.
q: I am carrying an umbrella.
The negation of “It is raining and I am carrying an umbrella” (~p ∧ ~q) is equivalent to saying “It is not raining or I am not carrying an umbrella” (~p ∨ ~q).
The second law, known as De Morgan’s law of negation continued, states that the negation of a logical disjunction (OR) is equivalent to the logical conjunction (AND) of the negations of the individual statements. Symbolically, it can be represented as:
¬(p ∨ q) ≡ ¬p ∧ ¬q
This means that if we have a statement that says either statement p or statement q is true, then the negation of that statement is the same as saying that both statement p is false and statement q is false.
Using the same example as before, the negation of “It is raining or I am carrying an umbrella” (~p ∨ ~q) is equivalent to saying “It is not raining and I am not carrying an umbrella” (~p ∧ ~q).
De Morgan’s laws are useful in simplifying and transforming logical statements, especially when dealing with complex expressions. They allow us to interchange logical operators and manipulate the negations of statements.
More Answers:
Understanding the Domination Laws | Manipulating Inequalities in MathematicsUnderstanding the Double Negation Law | A Fundamental Principle in Logic and Mathematics
Exploring the Complement Laws in Set Theory | Intersection and Double Complement Properties