State and prove De-Morgan's laws.

De Morgan's Laws

De Morgan's laws are a pair of rules of Boolean algebra that deal with negation of conjunctions and disjunctions. They are named after Augustus De Morgan, a British mathematician who first formulated them in the 19th century.

Statement of De Morgan's Laws:

1. Negation of a Conjunction (AND):

>¬(p ∧ q) ≡ ¬p ∨ ¬q

>In words: The negation of a conjunction (AND) of two propositions is equivalent to the disjunction (OR) of their negations.

1. Negation of a Disjunction (OR):

>¬(p ∨ q) ≡ ¬p ∧ ¬q

>In words: The negation of a disjunction (OR) of two propositions is equivalent to the conjunction (AND) of their negations.

Proof of De Morgan's Laws:

• Proof of the First Law:

>Assume ¬(p ∧ q). This means that the conjunction (AND) of p and q is false.

>But a conjunction is false if and only if at least one of its conjuncts is false.

>Therefore, either p is false or q is false, or both.

>In other words, ¬p ∨ ¬q.

>Hence, ¬(p ∧ q) ≡ ¬p ∨ ¬q.

• Proof of the Second Law:

>Assume ¬(p ∨ q). This means that the disjunction (OR) of p and q is false.

>But a disjunction is false if and only if both of its disjuncts are false.

>Therefore, both p and q are false.

>In other words, ¬p ∧ ¬q.

>Hence, ¬(p ∨ q) ≡ ¬p ∧ ¬q.

Applications of De Morgan's Laws:

De Morgan's laws are used extensively in digital logic design, computer programming, and other areas of mathematics and computer science.

• Simplifying Boolean Expressions:

>De Morgan's laws can be used to simplify Boolean expressions by converting them into an equivalent form that is easier to understand and manipulate.

• Designing Logic Circuits:

>De Morgan's laws are used to design logic circuits that implement Boolean functions. By using De Morgan's laws, it is possible to minimize the number of gates required to implement a given Boolean function.

• Reasoning and Argumentation:

>De Morgan's laws can be used in reasoning and argumentation to derive new conclusions from given premises. For example, if we know that ¬(p ∨ q), we can use De Morgan's law to conclude that ¬p ∧ ¬q.

De Morgan's laws are fundamental laws of Boolean algebra that have wide applications in various fields. Their simplicity and elegance make them a powerful tool for solving a variety of problems.